# Patent

## Title
Kernels and kernel methods for spectral data

## Abstract

Support vector machines are used to classify data contained within a structured dataset such as a plurality of signals generated by a spectral analyzer. The signals are pre-processed to ensure alignment of peaks across the spectra. Similarity measures are constructed to provide a basis for comparison of pairs of samples of the signal. A support vector machine is trained to discriminate between different classes of the samples. to identify the most predictive features within the spectra. In a preferred embodiment feature selection is performed to reduce the number of features that must be considered.

## Background
RELATED APPLICATIONS
The present application claims priority of U.S. provisional application Ser. No. 60/328,309, filed Oct. 9, 2001 and is a continuation-in-part of application Ser. No. 10/477,078, filed Nov. 7, 2003, now issued as U.S. Pat. No. 7,353,215, which was the U.S. national stage filing of International application No. PCT/US02/14311 filed in the U.S. Receiving Office on May 7, 2002 which claims priority to each of the following U.S. provisional patent applications: Ser. No. 60/289,163, filed May 7, 2001, and Ser. No. 60/329,874, filed Oct. 17, 2001. This application is also a continuation-in-part of U.S. patent application Ser. No. 10/057,849, filed Jan. 24, 2002, now U.S. Pat. No. 7,117,188, which is a continuation-in-part of application Ser. No. 09/633,410, filed Aug. 7, 2000, now U.S. Pat. No. 6,882,990.
This application is related to, but does not claim the priority of, application Ser. No. 09/578,011, filed May 24, 2000, now U.S. Pat. No. 6,658,395, which is a continuation-in-part of application Ser. No. 09/568,301, filed May 9, 2000, now issued as U.S. Pat. No. 6,427,141, which is a continuation of application Ser. No. 09/303,387, filed May 1, 1999, now issued as U.S. Pat. No. 6,128,608, which claims priority to U.S. provisional application Ser. No. 60/083,961, filed May 1, 1998. This application is also related to co-pending application Ser. No. 09/633,615, Ser. No. 09/633,616, now issued as U.S. Pat. No. 6,760,715, and Ser. No. 09/633,850, now issued as U.S. Pat. No. 6,789,069, all filed Aug. 7, 2000, which are also continuations-in-part of application Ser. No. 09/578,011, now issued as U.S. Pat. No. 6,658,395. This application is also related to application Ser. No. 09/303,386 and Ser. No. 09/305,345, now issued as U.S. Pat. No. 6,157,921, both filed May 1, 1999, and to U.S. application Ser. No. 09/715,832, filed Nov. 14, 2000, all of which also claim priority to provisional application Ser. No. 60/083,961. Each of the above-identified applications is incorporated herein by reference.


FIELD OF THE INVENTION
The present invention relates to the use of learning machines to identify relevant patterns in datasets containing large quantities of diverse data, and more particularly to a method and system for selection of kernels to be used in kernel machines which best enable identification of relevant patterns in spectral data.
BACKGROUND OF THE INVENTION
In recent years, machine-learning approaches for data analysis have been widely explored for recognizing patterns which, in turn, allow extraction of significant information contained within a large data set that may also include data consisting of nothing more than irrelevant detail. Learning machines comprise algorithms that may be trained to generalize using data with known outcomes. Trained learning machine algorithms may then be applied to predict the outcome in cases of unknown outcome, i.e., to classify the data according to learned patterns. Machine-learning approaches, which include neural networks, hidden Markov models, belief networks and kernel-based classifiers such as support vector machines, are ideally suited for domains characterized by the existence of large amounts of data, noisy patterns and the absence of general theories. Support vector machines are disclosed in U.S. Pat. Nos. 6,128,608 and 6,157,921, both of which are assigned to the assignee of the present application and are incorporated herein by reference.
Many successful approaches to pattern classification, regression, clustering, and novelty detection problems rely on kernels for determining the similarity of a pair of patterns. These kernels are usually defined for patterns that can be represented as a vector of real numbers. For example, the linear kernels, radial basis function kernels, and polynomial kernels all measure the similarity of a pair of real vectors. Such kernels are appropriate when the patterns are best represented in this way, as a sequence of real numbers. The choice of a kernel corresponds to the choice of representation of the data in a feature space. In many applications, the patterns have a greater degree of structure. This structure can be exploited to improve the performance of the learning system. Examples of the types of structured data that commonly occur in machine learning applications are strings, such as DNA sequences, and documents; trees, such as parse trees used in natural language processing; graphs, such as web sites or chemical molecules; signals, such as ECG signals and microarray expression profiles; spectra; images; spatio-temporal data; and relational data, among others.
For structural objects, kernels methods are often applied by first finding a mapping from the structured objects to a vector of real numbers. In one embodiment of the kernel selection method, the invention described herein provides an alternative approach which may be applied to the selection of kernels which may be used for structured objects.
Many problems in bioinformatics, chemistry and other industrial processes involve the measurement of some features of samples by means of an apparatus whose operation is subject to fluctuations, influenced for instance by the details of the preparation of the measurement, or by environmental conditions such as temperature. For example, analytical instruments that rely on scintillation crystals for detecting radiation are known to be temperature sensitive, with additional noise or signal drift occurring if the crystals are not maintained within a specified temperature range. Data recorded using such measurement devices is subject to problems in subsequent processing, which can include machine learning methods. Therefore, it is desirable to provide automated ways of dealing with such data.
In certain classification tasks, there is a priori knowledge about the invariances related to the task. For instance, in image classification, it is known that the label of a given image should not change after a small translation or rotation. In a second embodiment of the method for selecting kernels for kernel machines, to improve performance, prior knowledge such as known transformation invariances or noise distribution is incorporated in the kernels. A technique is disclosed in which noise is represented as vectors in a feature space. The noise vectors are taken into account in constructing invariant kernel classifiers to obtain a decision function that is invariant with respect to the noise.
BRIEF SUMMARY OF THE INVENTION
According to the present invention, methods are provided for selection and construction of kernels for use in kernel machines such that the kernels are suited to analysis of data which may posses characteristics such as structure, for example DNA sequences, documents; graphs, signals, such as ECG signals and microarray expression profiles; spectra; images; spatio-temporal data; and relational data, and which may possess invariances or noise components that can interfere with the ability to accurately extract the desired information.
In an exemplary embodiment, in a method for defining a similarity measure for structured objects, a location-dependent kernel is defined on one or more structures that comprise patterns in the structured data of a data set. This locational kernel defines a notion of similarity between structures that is relevant to particular locations in the structures. Using the locational kernel, a pair of structures can be viewed according to the similarities of their components. Multiple locational kernels can then be used to obtain yet another locational kernel. A kernel on the set of structures can be then obtained by combining the locational kernel(s). Different methods for combining the locational kernels include summing the locational kernels over the indices, fixing the indices, and taking the product over all index positions of locational kernels constructed from Gaussian radial basis function (RBF) kernels whose indices have been fixed. The resulting kernels are then used for processing the input data set.
The method of the present invention comprises the steps of: representing a structured object as a collection of component objects, together with their location within the structure; defining kernels, called locational kernels, that measure the similarity of (structure, location in structure) pairs; constructing locational kernels from kernels defined on the set of components; constructing locational kernels from other locational kernels; and constructing kernels on structured objects by combining locational kernels.
An iterative process comparing postprocessed training outputs or test outputs can be applied to make a determination as to which kernel configuration provides the optimal solution. If the test output is not the optimal solution, the selection of the kernel may be adjusted and the support vector machine may be retrained and retested. Once it is determined that the optimal solution has been identified, a live data set may be collected and pre-processed in the same manner as was the training data set to select the features that best represent the data. The pre-processed live data set is input into the learning machine for processing.
In an exemplary application of the kernel selection techniques of the first embodiment, a locational kernel construction scheme is used for comparing infrared spectra to classify disease states.
In a second embodiment, another component of kernel selection and design involves incorporating noise information into the kernel so that invariances are introduced into the learning system. The noise of the instrument that was used to generate the data can provide information that can be used in kernel design. For example, if certain peaks in a spectrum always have the same value while other peaks have values that vary with different experimental settings, the system takes this into account in its calculations. One method of compensation comprises normalizing every peak value. Analysis of control classes to infer the variability across different instruments can be incorporated into the kernel.
While the inventive methods disclosed herein are contemplated for use in different types of kernel machines, in an exemplary implementation, the kernel machine is a support vector machine. The exemplary system comprises a storage device for storing a training data set and a test data set, and a processor for executing a support vector machine. The processor is also operable for collecting the training data set from the database, pre-processing the training data set, training the support vector machine using the pre-processed training data set, collecting the test data set from the database, pre-processing the test data set in the same manner as was the training data set, testing the trained support vector machine using the pre-processed test data set, and in response to receiving the test output of the trained support vector machine, post-processing the test output to determine if the test output is an optimal solution. The exemplary system may also comprise a communications device for receiving the test data set and the training data set from a remote source. In such a case, the processor may be operable to store the training data set in the storage device prior pre-processing of the training data set and to store the test data set in the storage device prior pre-processing of the test data set. The exemplary system may also comprise a display device for displaying the post-processed test data. The processor of the exemplary system may further be operable for performing each additional function described above.
In an exemplary embodiment, a system and method are provided for enhancing knowledge discovery from data using multiple learning machines in general and multiple support vector machines in particular. Training data for a learning machine is pre-processed. Multiple support vector machines, each comprising distinct kernels, are trained with the pre-processed training data and are tested with test data that is pre-processed in the same manner. The test outputs from multiple support vector machines are compared in order to determine which of the test outputs if any represents an optimal solution. Selection of one or more kernels may be adjusted and one or more support vector machines may be retrained and retested. When it is determined that an optimal solution has been achieved, live data is pre-processed and input into the support vector machine comprising the kernel that produced the optimal solution. The live output from the learning machine may then be post-processed as needed to place the output in a format appropriate for interpretation by a human or another computer.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a flowchart illustrating an exemplary general method for increasing knowledge that may be discovered from data using a learning machine.
FIG. 2 is a flowchart illustrating an exemplary method for increasing knowledge that may be discovered from data using a support vector machine.
FIG. 3 is a flowchart illustrating an exemplary optimal categorization method that may be used in a stand-alone configuration or in conjunction with a learning machine for pre-processing or post-processing techniques in accordance with an exemplary embodiment of the present invention.
FIG. 4 is a functional block diagram illustrating an exemplary operating environment for an embodiment of the present invention.
FIG. 5 is a functional block diagram illustrating a hierarchical system of multiple support vector machines.
FIG. 6 is a plot showing an original spectra curve superimposed with a new representation of the spectra according to the present invention where the new representation comprises ±1 spikes.
FIG. 7 illustrates a toy problem where the true decision function is the circle. For each point a tangent vector corresponding to a small rotation is plotted.
FIGS. 8a and b are plot showing test errors where FIG. 8a represents test error for different learning algorithms against the width of a RBF kernel and γ fixed to 0.9, and FIG. 8b represents test error of IHKPCA across τ and for different values of σ. The test errors are averaged over the 100 splits and the error bars correspond to the standard deviation of the means.
FIG. 9 is a plot of the approximation ratio of Equation 100 for different values of σ. When the ratio is near 1, there is almost no loss of information using IHKPCA and thus in this case ISVM and IHKPCA are identical.
FIG. 10 is a plot of test error versus γ for LIH. Trying to incorporate invariance using this method fails on this toy example.
FIG. 11 is a plot of the value of the invariance enforcement part (from Equation 106 of the objective function across σ. A small value amounts to a better enforcement of the invariances.
FIG. 12 illustrates the numeral 9 with the original digit in the center and as translated from 1 pixel on the left, right, up and down.
FIGS. 13a and 13b are plots comparing ISVM, IHKPCA and VSV on the USPS dataset, where FIG. 13a (γ=0) corresponds to standard SVM and FIG. 13b (γ→1) indicates significant emphasis on the enforcement of the constraints. The test errors are averaged over the 23 splits and the error bars correspond to the standard deviation of the means.
FIG. 14 shows results for 4 invariances and different values of γ.
FIG. 15a illustrates an example of a mass spectrogram for a portion of a spectrum corresponding to small mass; FIG. 15b illustrates a portion of a spectrum corresponding to large mass.
FIG. 16: illustrates a mass spectrogram, where the spectrum concerning mass <1000 is not considered.
FIG. 17 is a plot of multi-class margin distribution on the test set for the multi-class SVM polynomial degree 2 with infinite C.
FIG. 18 is a plot showing the distribution of the output of the M-SVM.
FIG. 19 illustrates an example of the effect of the alignment algorithm comparing the two random samples before alignment (19a) and after alignment (19b).
FIG. 20 is a plot of test error versus the value of the smoothing parameter for smoothing of raw data.
FIG. 21 provides an example of the peak detection comparing the peaks detected with two random samples (FIG. 21a) where the vertical lines are the detected peaks over the two random samples. FIG. 21b shows a smoothed baseline signal.
FIG. 22 is a table showing error rates for peak detection on problem A versus B+C using a random signal and a radius r of features around each peak.
FIG. 23 is a table showing error rates for peak detection on problem A versus B+C using a random signal and a radius r of features around each peak.
FIG. 24 is a set of scatter plots showing the first four principal components for the Imacfx1 condition where the symbols correspond to the class variable.
FIG. 25 is a set of scatter plots showing the first four principal components for the Imacfx1 condition where the symbols correspond to the plate ID.
FIG. 26 is a set of scatter plots showing the first four principal components for the Imacfx1 condition with each spectrum normalized by its total ion current where the symbols correspond to plate ID.
FIG. 27 is a set of scatter plots showing the first four principal components for the Imacfx1 condition with each spectrum normalized by its total ion current where the symbols correspond to plate ID.
FIGS. 28a and 28b are sets of scatter plots showing the first four principal components for the Imacfx1 condition with each spectrum normalized standardizing separately in each plate where the symbols in FIG. 28a correspond to plate ID and in FIG. 28b correspond to class variable.
FIG. 29 is a histogram showing the distribution of Golub's statistic for the Imacfx1 condition.
FIG. 30 is a histogram showing the distribution of Golub's statistic for data with a Gaussian distribution and a random class variable where the dataset is the same size as the Imacfx1 data.
FIGS. 31a-31c are histograms showing the distribution of Golub's statistic for Plates 1, 2 and 3, respectively.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention provides methods, systems and devices for discovering knowledge from data using learning machines. Particularly, the present invention is directed to methods, systems and devices for knowledge discovery from data using learning machines that are provided information regarding changes in biological systems. More particularly, the present invention comprises methods of use of such knowledge for diagnosing and prognosing changes in biological systems such as diseases. Additionally, the present invention comprises methods, compositions and devices for applying such knowledge to the testing and treating of individuals with changes in their individual biological systems. Preferred embodiments comprise detection of genes involved with prostate cancer and use of such information for treatment of patients with prostate cancer.
As used herein, “biological data” means any data derived from measuring biological conditions of human, animals or other biological organisms including microorganisms, viruses, plants and other living organisms. The measurements may be made by any tests, assays or observations that are known to physicians, scientists, diagnosticians, or the like. Biological data may include, but is not limited to, clinical tests and observations, physical and chemical measurements, genomic determinations, proteomic determinations, drug levels, hormonal and immunological tests, neurochemical or neurophysical measurements, mineral and vitamin level determinations, genetic and familial histories, and other determinations that may give insight into the state of the individual or individuals that are undergoing testing. Herein, the use of the term “data” is used interchangeably with “biological data”.
While several examples of learning machines exist and advancements are expected in this field, the exemplary embodiments of the present invention focus on kernel-based learning machines and more particularly on the support vector machine.
The present invention can be used to analyze biological data generated at multiple stages of investigation into biological functions, and further, to integrate the different kinds of data for novel diagnostic and prognostic determinations. For example, biological data obtained from clinical case information, such as diagnostic test data, family or genetic histories, prior or current medical treatments, and the clinical outcomes of such activities, can be utilized in the methods, systems and devices of the present invention. Additionally, clinical samples such as diseased tissues or fluids, and normal tissues and fluids, and cell separations can provide biological data that can be utilized by the current invention. Proteomic determinations such as 2-D gel, mass spectrophotometry and antibody screening can be used to establish databases that can be utilized by the present invention. Genomic databases can also be used alone or in combination with the above-described data and databases by the present invention to provide comprehensive diagnosis, prognosis or predictive capabilities to the user of the present invention.
A first aspect of the present invention facilitates analysis of data by pre-processing the data prior to using the data to train a learning machine and/or optionally post-processing the output from a learning machine. Generally stated, pre-processing data comprises reformatting or augmenting the data in order to allow the learning machine to be applied most advantageously. In some cases, pre-processing involves selecting a method for reducing the dimensionality of the feature space, i.e., selecting the features which best represent the data. Methods which may be used for this purpose include recursive feature elimination (RFE) and elimination of data points based on expert knowledge, e.g., knowledge that measurement at ends of the measurement range tend to be predominantly noise. The features remaining after feature selection are then used to train a learning machine for purposes of pattern classification, regression, clustering and/or novelty detection.
The learning machine is a kernel machine in which the kernels are selected in consideration of certain characteristics of the data. In one embodiment, the data has a structure which permits locational kernels to be defined. These locational kernels are then used to identifying similarities between components of structures in the dataset. The locational kernels are then combined through one or more of a variety of possible operations to derive the kernel function. In another embodiment, the input dataset includes invariance or noise. The invariance or noise is taken into consideration when selecting the kernel function. Virtual support vectors may be added, or a set of tangent vectors may be generated. In preferred embodiment the kernel machine is a support vector machine.
In a manner similar to pre-processing, post-processing involves interpreting the output of a learning machine in order to discover meaningful characteristics thereof. The meaningful characteristics to be ascertained from the output may be problem- or data-specific. Post-processing involves interpreting the output into a form that, for example, may be understood by or is otherwise useful to a human observer, or converting the output into a form which may be readily received by another device for, e.g., archival or transmission.
FIG. 1 is a flowchart illustrating a general method 100 for analyzing data using learning machines. The method 100 begins at starting block 101 and progresses to step 102 where a specific problem is formalized for application of analysis through machine learning. Particularly important is a proper formulation of the desired output of the learning machine. For instance, in predicting future performance of an individual equity instrument, or a market index, a learning machine is likely to achieve better performance when predicting the expected future change rather than predicting the future price level. The future price expectation can later be derived in a post-processing step as will be discussed later in this specification.
After problem formalization, step 103 addresses training data collection. Training data comprises a set of data points having known characteristics. This data may come from customers, research facilities, academic institutions, national laboratories, commercial entities or other public or confidential sources. The source of the data and the types of data provided are not crucial to the methods. Training data may be collected from one or more local and/or remote sources. The data may be provided through any means such as via the internet, server linkages or discs, CD/ROMs, DVDs or other storage means. The collection of training data may be accomplished manually or by way of an automated process, such as known electronic data transfer methods. Accordingly, an exemplary embodiment of the learning machine for use in conjunction with the present invention may be implemented in a networked computer environment. Exemplary operating environments for implementing various embodiments of the learning machine will be described in detail with respect to FIGS. 4-5.
At step 104, the collected training data is optionally pre-processed in order to allow the learning machine to be applied most advantageously toward extraction of the knowledge inherent to the training data. During this preprocessing stage a variety of different transformations can be performed on the data to enhance its usefulness. Such transformations, examples of which include addition of expert information, labeling, binary conversion, Fourier transformations, etc., will be readily apparent to those of skill in the art. However, the preprocessing of interest in the present invention is the reduction of dimensionality by way of feature selection, different methods of which are described in detail below.
Returning to FIG. 1, an exemplary method 100 continues at step 106, where the learning machine is trained using the pre-processed data. As is known in the art, a learning machine is trained by adjusting its operating parameters until a desirable training output is achieved. The determination of whether a training output is desirable may be accomplished either manually or automatically by comparing the training output to the known characteristics of the training data. A learning machine is considered to be trained when its training output is within a predetermined error threshold from the known characteristics of the training data. In certain situations, it may be desirable, if not necessary, to post-process the training output of the learning machine at step 107. As mentioned, post-processing the output of a learning machine involves interpreting the output into a meaningful form. In the context of a regression problem, for example, it may be necessary to determine range categorizations for the output of a learning machine in order to determine if the input data points were correctly categorized. In the example of a pattern recognition problem, it is often not necessary to post-process the training output of a learning machine.
At step 108, test data is optionally collected in preparation for testing the trained learning machine. Test data may be collected from one or more local and/or remote sources. In practice, test data and training data may be collected from the same source(s) at the same time. Thus, test data and training data sets can be divided out of a common data set and stored in a local storage medium for use as different input data sets for a learning machine. Regardless of how the test data is collected, any test data used must be pre-processed at step 110 in the same manner as was the training data. As should be apparent to those skilled in the art, a proper test of the learning may only be accomplished by using testing data of the same format as the training data. Then, at step 112 the learning machine is tested using the pre-processed test data, if any. The test output of the learning machine is optionally post-processed at step 114 in order to determine if the results are desirable. Again, the post processing step involves interpreting the test output into a meaningful form. The meaningful form may be one that is readily understood by a human or one that is compatible with another processor. Regardless, the test output must be post-processed into a form which may be compared to the test data to determine whether the results were desirable. Examples of post-processing steps include but are not limited of the following: optimal categorization determinations, scaling techniques (linear and non-linear), transformations (linear and non-linear), and probability estimations. The method 100 ends at step 116.
FIG. 2 is a flow chart illustrating an exemplary method 200 for enhancing knowledge that may be discovered from data using a specific type of learning machine known as a support vector machine (SVM). A SVM implements a specialized algorithm for providing generalization when estimating a multi-dimensional function from a limited collection of data. A SVM may be particularly useful in solving dependency estimation problems. More specifically, a SVM may be used accurately in estimating indicator functions (e.g. pattern recognition problems) and real-valued functions (e.g. function approximation problems, regression estimation problems, density estimation problems, and solving inverse problems). The SVM was originally developed by Boser, Guyon and Vapnik (“A training algorithm for optimal margin classifiers”, Fifth Annual Workshop on Computational Learning Theory, Pittsburgh, ACM (1992) pp. 142-152). The concepts underlying the SVM are also explained in detail in Vapnik's book, entitled Statistical Learning Theory (John Wiley & Sons, Inc. 1998), which is herein incorporated by reference in its entirety. Accordingly, a familiarity with SVMs and the terminology used therewith are presumed throughout this specification.
The exemplary method 200 begins at starting block 201 and advances to step 202, where a problem is formulated and then to step 203, where a training data set is collected. As was described with reference to FIG. 1, training data may be collected from one or more local and/or remote sources, through a manual or automated process. At step 204 the training data is optionally pre-processed. Those skilled in the art should appreciate that SVMs are capable of processing input data having extremely large dimensionality, however, according to the present invention, pre-processing includes the use of feature selection methods to reduce the dimensionality of feature space.
At step 206 a kernel is selected for the SVM. As is known in the art, different kernels will cause a SVM to produce varying degrees of quality in the output for a given set of input data. Therefore, the selection of an appropriate kernel may be essential to the desired quality of the output of the SVM. In one embodiment of the learning machine, a kernel may be chosen based on prior performance knowledge. As is known in the art, exemplary kernels include polynomial kernels, radial basis classifier kernels, linear kernels, etc. In an alternate embodiment, a customized kernel may be created that is specific to a particular problem or type of data set. In yet another embodiment, the multiple SVMs may be trained and tested simultaneously, each using a different kernel. The quality of the outputs for each simultaneously trained and tested SVM may be compared using a variety of selectable or weighted metrics (see step 222) to determine the most desirable kernel. In one of the preferred embodiments of the invention which is particularly advantageous for use with structured data, locational kernels are defined to exploit patterns within the structure. The locational kernels are then used to construct kernels on the structured object. Further discussion of the kernels for structured data is provided below. Still another embodiment incorporates noise data into selection and construction of kernels in order to facilitate extraction of relevant knowledge from noise such as measurement artifacts and other variables that may impair the accuracy of the analysis. This method for construction kernels is also described below in more detail.
At step 208 the pre-processed training data is input into the SVM. At step 210, the SVM is trained using the pre-processed training data to generate an optimal hyperplane. Optionally, the training output of the SVM may then be post-processed at step 211. Again, post-processing of training output may be desirable, or even necessary, at this point in order to properly calculate ranges or categories for the output. At step 212 test data is collected similarly to previous descriptions of data collection. The test data is pre-processed at step 214 in the same manner as was the training data above. Then, at step 216 the pre-processed test data is input into the SVM for processing in order to determine whether the SVM was trained in a desirable manner. The test output is received from the SVM at step 218 and is optionally post-processed at step 220.
Based on the post-processed test output, it is determined at step 222 whether an optimal minimum was achieved by the SVM. Those skilled in the art should appreciate that a SVM is operable to ascertain an output having a global minimum error. However, as mentioned above, output results of a SVM for a given data set will typically vary with kernel selection. Therefore, there are in fact multiple global minimums that may be ascertained by a SVM for a given set of data. As used herein, the term “optimal minimum” or “optimal solution” refers to a selected global minimum that is considered to be optimal (e.g. the optimal solution for a given set of problem specific, pre-established criteria) when compared to other global minimums ascertained by a SVM. Accordingly, at step 222, determining whether the optimal minimum has been ascertained may involve comparing the output of a SVM with a historical or predetermined value. Such a predetermined value may be dependant on the test data set. For example, in the context of a pattern recognition problem where data points are classified by a SVM as either having a certain characteristic or not having the characteristic, a global minimum error of 50% would not be optimal. In this example, a global minimum of 50% is no better than the result that would be achieved by flipping a coin to determine whether the data point had that characteristic. As another example, in the case where multiple SVMs are trained and tested simultaneously with varying kernels, the outputs for each SVM may be compared with output of other SVM to determine the practical optimal solution for that particular set of kernels. The determination of whether an optimal solution has been ascertained may be performed manually or through an automated comparison process.
If it is determined that the optimal minimum has not been achieved by the trained SVM, the method advances to step 224, where the kernel selection is adjusted. Adjustment of the kernel selection may comprise selecting one or more new kernels or adjusting kernel parameters. Furthermore, in the case where multiple SVMs were trained and tested simultaneously, selected kernels may be replaced or modified while other kernels may be re-used for control purposes. After the kernel selection is adjusted, the method 200 is repeated from step 208, where the pre-processed training data is input into the SVM for training purposes. When it is determined at step 222 that the optimal minimum has been achieved, the method advances to step 226, where live data is collected similarly as described above. By definition, live data has not been previously evaluated, so that the desired output characteristics that were known with respect to the training data and the test data are not known.
At step 228 the live data is pre-processed in the same manner as was the training data and the test data. At step 230, the live pre-processed data is input into the SVM for processing. The live output of the SVM is received at step 232 and is post-processed at step 234. The method 200 ends at step 236.
FIG. 3 is a flow chart illustrating an exemplary optimal categorization method 300 that may be used for pre-processing data or post-processing output from a learning machine. Additionally, as will be described below, the exemplary optimal categorization method may be used as a stand-alone categorization technique, independent from learning machines. The exemplary optimal categorization method 300 begins at starting block 301 and progresses to step 302, where an input data set is received. The input data set comprises a sequence of data samples from a continuous variable. The data samples fall within two or more classification categories. Next, at step 304 the bin and class-tracking variables are initialized. As is known in the art, bin variables relate to resolution, while class-tracking variables relate to the number of classifications within the data set. Determining the values for initialization of the bin and class-tracking variables may be performed manually or through an automated process, such as a computer program for analyzing the input data set. At step 306, the data entropy for each bin is calculated. Entropy is a mathematical quantity that measures the uncertainty of a random distribution. In the exemplary method 300, entropy is used to gauge the gradations of the input variable so that maximum classification capability is achieved.
The method 300 produces a series of “cuts” on the continuous variable, such that the continuous variable may be divided into discrete categories. The cuts selected by the exemplary method 300 are optimal in the sense that the average entropy of each resulting discrete category is minimized. At step 308, a determination is made as to whether all cuts have been placed within input data set comprising the continuous variable. If all cuts have not been placed, sequential bin combinations are tested for cutoff determination at step 310. From step 310, the exemplary method 300 loops back through step 306 and returns to step 308 where it is again determined whether all cuts have been placed within input data set comprising the continuous variable. When all cuts have been placed, the entropy for the entire system is evaluated at step 309 and compared to previous results from testing more or fewer cuts. If it cannot be concluded that a minimum entropy state has been determined, then other possible cut selections must be evaluated and the method proceeds to step 311. From step 311 a heretofore untested selection for number of cuts is chosen and the above process is repeated from step 304. When either the limits of the resolution determined by the bin width has been tested or the convergence to a minimum solution has been identified, the optimal classification criteria is output at step 312 and the exemplary optimal categorization method 300 ends at step 314.
The optimal categorization method 300 takes advantage of dynamic programming techniques. As is known in the art, dynamic programming techniques may be used to significantly improve the efficiency of solving certain complex problems through carefully structuring an algorithm to reduce redundant calculations. In the optimal categorization problem, the straightforward approach of exhaustively searching through all possible cuts in the continuous variable data would result in an algorithm of exponential complexity and would render the problem intractable for even moderate sized inputs. By taking advantage of the additive property of the target function, in this problem the average entropy, the problem may be divided into a series of sub-problems. By properly formulating algorithmic sub-structures for solving each sub-problem and storing the solutions of the sub-problems, a significant amount of redundant computation may be identified and avoided. As a result of using the dynamic programming approach, the exemplary optimal categorization method 300 may be implemented as an algorithm having a polynomial complexity, which may be used to solve large sized problems.
As mentioned above, the exemplary optimal categorization method 300 may be used in pre-processing data and/or post-processing the output of a learning machine. For example, as a pre-processing transformation step, the exemplary optimal categorization method 300 may be used to extract classification information from raw data. As a post-processing technique, the exemplary optimal range categorization method may be used to determine the optimal cut-off values for markers objectively based on data, rather than relying on ad hoc approaches. As should be apparent, the exemplary optimal categorization method 300 has applications in pattern recognition, classification, regression problems, etc. The exemplary optimal categorization method 300 may also be used as a stand-alone categorization technique, independent from SVMs and other learning machines.
FIG. 4 and the following discussion are intended to provide a brief and general description of a suitable computing environment for implementing biological data analysis according to the present invention. Although the system shown in FIG. 4 is a conventional personal computer 1000, those skilled in the art will recognize that the invention also may be implemented using other types of computer system configurations. The computer 1000 includes a central processing unit 1022, a system memory 1020, and an Input/Output (“I/O”) bus 1026. A system bus 1021 couples the central processing unit 1022 to the system memory 1020. A bus controller 1023 controls the flow of data on the I/O bus 1026 and between the central processing unit 1022 and a variety of internal and external I/O devices. The I/O devices connected to the I/O bus 1026 may have direct access to the system memory 1020 using a Direct Memory Access (“DMA”) controller 1024.
The I/O devices are connected to the I/O bus 1026 via a set of device interfaces. The device interfaces may include both hardware components and software components. For instance, a hard disk drive 1030 and a floppy disk drive 1032 for reading or writing removable media 1050 may be connected to the I/O bus 1026 through disk drive controllers 1040. An optical disk drive 1034 for reading or writing optical media 1052 may be connected to the I/O bus 1026 using a Small Computer System Interface (“SCSI”) 1041. Alternatively, an IDE (Integrated Drive Electronics, i.e., a hard disk drive interface for PCs), ATAPI (ATtAchment Packet Interface, i.e., CD-ROM and tape drive interface), or EIDE (Enhanced IDE) interface may be associated with an optical drive such as may be the case with a CD-ROM drive. The drives and their associated computer-readable media provide nonvolatile storage for the computer 1000. In addition to the computer-readable media described above, other types of computer-readable media may also be used, such as ZIP drives, or the like.
A display device 1053, such as a monitor, is connected to the I/O bus 1026 via another interface, such as a video adapter 1042. A parallel interface 1043 connects synchronous peripheral devices, such as a laser printer 1056, to the I/O bus 1026. A serial interface 1044 connects communication devices to the I/O bus 1026. A user may enter commands and information into the computer 1000 via the serial interface 1044 or by using an input device, such as a keyboard 1038, a mouse 1036 or a modem 1057. Other peripheral devices (not shown) may also be connected to the computer 1000, such as audio input/output devices or image capture devices.
A number of program modules may be stored on the drives and in the system memory 1020. The system memory 1020 can include both Random Access Memory (“RAM”) and Read Only Memory (“ROM”). The program modules control how the computer 1000 functions and interacts with the user, with I/O devices or with other computers. Program modules include routines, operating systems 1065, application programs, data structures, and other software or firmware components. In an illustrative embodiment, the learning machine may comprise one or more pre-processing program modules 1075A, one or more post-processing program modules 1075B, and/or one or more optimal categorization program modules 1077 and one or more SVM program modules 1070 stored on the drives or in the system memory 1020 of the computer 1000. Specifically, pre-processing program modules 1075A, post-processing program modules 1075B, together with the SVM program modules 1070 may comprise computer-executable instructions for pre-processing data and post-processing output from a learning machine and implementing the learning algorithm according to the exemplary methods described with reference to FIGS. 1 and 2. Furthermore, optimal categorization program modules 1077 may comprise computer-executable instructions for optimally categorizing a data set according to the exemplary methods described with reference to FIG. 3.
The computer 1000 may operate in a networked environment using logical connections to one or more remote computers, such as remote computer 1060. The remote computer 1060 may be a server, a router, a peer device or other common network node, and typically includes many or all of the elements described in connection with the computer 1000. In a networked environment, program modules and data may be stored on the remote computer 1060. The logical connections depicted in FIG. 5 include a local area network (“LAN”) 1054 and a wide area network (“WAN”) 1055. In a LAN environment, a network interface 1045, such as an Ethernet adapter card, can be used to connect the computer 1000 to the remote computer 1060. In a WAN environment, the computer 1000 may use a telecommunications device, such as a modem 1057, to establish a connection. It will be appreciated that the network connections shown are illustrative and other devices of establishing a communications link between the computers may be used.
In another embodiment, a plurality of SVMs can be configured to hierarchically process multiple data sets in parallel or sequentially. In particular, one or more first-level SVMs may be trained and tested to process a first type of data and one or more first-level SVMs can be trained and tested to process a second type of data. Additional types of data may be processed by other first-level SVMs. The output from some or all of the first-level SVMs may be combined in a logical manner to produce an input data set for one or more second-level SVMs. In a similar fashion, output from a plurality of second-level SVMs may be combined in a logical manner to produce input data for one or more third-level SVM. The hierarchy of SVMs may be expanded to any number of levels as may be appropriate. In this manner, lower hierarchical level SVMs may be used to pre-process data that is to be input into higher level SVMs. Also, higher hierarchical level SVMs may be used to post-process data that is output from lower hierarchical level SVMs.
Each SVM in the hierarchy or each hierarchical level of SVMs may be configured with a distinct kernel. For example, SVMs used to process a first type of data may be configured with a first type of kernel while SVMs used to process a second type of data may utilize a second, different type of kernel. In addition, multiple SVMs in the same or different hierarchical level may be configured to process the same type of data using distinct kernels.
FIG. 5 illustrates an exemplary hierarchical system of SVMs. As shown, one or more first-level SVMs 1302a and 1302b may be trained and tested to process a first type of input data 1304a, such as mammography data, pertaining to a sample of medical patients. One or more of these SVMs may comprise a distinct kernel, indicated as “KERNEL 1” and “KERNEL 2”. Also, one or more additional first-level SVMs 1302c and 1302d may be trained and tested to process a second type of data 1304b, which may be, for example, genomic data for the same or a different sample of medical patients. Again, one or more of the additional SVMs may comprise a distinct kernel, indicated as “KERNEL 1” and “KERNEL 3”. The output from each of the like first-level SVMs may be compared with each other, e.g., 1306a compared with 1306b; 1306c compared with 1306d, in order to determine optimal outputs 1308a and 1308b. Then, the optimal outputs from the two groups or first-level SVMs, i.e., outputs 1308a and 1308b, may be combined to form a new multi-dimensional input data set 1310, for example, relating to mammography and genomic data. The new data set may then be processed by one or more appropriately trained and tested second-level SVMs 1312a and 1312b. The resulting outputs 1314a and 1314b from second-level SVMs 1312a and 1312b may be compared to determine an optimal output 1316. Optimal output 1316 may identify causal relationships between the mammography and genomic data points. As should be apparent to those of skill in the art, other combinations of hierarchical SVMs may be used to process either in parallel or serially, data of different types in any field or industry in which analysis of data is desired.
Selection of Kernels
In a first embodiment, a method is provided for construction of kernels for structured data. Specifically, a similarity measure is defined for structured objects in which a location-dependent kernel is defined on one or more structures that comprise patterns in the structured data of a data set. This locational kernel defines a notion of similarity between structures that is relevant to particular locations in the structures. Using the locational kernel, a pair of structures can be viewed as a rich collection of similarities of their components. Multiple locational kernels can then be used to obtain yet another locational kernel. A kernel on the set of structures can be then obtained by combining the locational kernel(s). Different methods for combining the locational kernels include summing the locational kernels over the indices, fixing the indices, and taking the product over all index positions of locational kernels constructed from Gaussian radial basis function (RBF) kernels whose indices have been fixed. The resulting kernels are then used for processing the input data set.
A structured object is represented as a collection of components objects together with their locations within the structure. Given a component set A and a path set P, a structure on A and P is a triple s=(I, c, p), where I is the index set, c:I→A is the component function, and p: I×I→P is the path function. When I is finite, s can be said to be a finite structure.
The index set and component and path functions associated with s are denoted by Is, cs, and ps.
Let S be a set of structures on A and P. A vicinity function for S is a function h:

{(s,i,i′):sεS,i,i′εIs}→R of the form h(s,i,i′)=φ(ps(i,i′))  (1)

for some φ: P→R.

This vicinity function h is stable if, for all: sεS and iεIs,





  
    
      
        
          
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Notice that every vicinity function on a set of finite structures is stable. The following are a few examples of structures and vicinity functions:
1. If A=R and P=Z, xεRn is defined as a finite structure s, where Is={1, . . . , n}, cs(i)=xi, and ps(i−i′)=i−i′. For example, x might be noisy estimates of the energy in n adjacent frequency bands. In this case, a vicinity function might be defined as

h(s,i,i′)=e−c(i−i′)2  (3)

for some c>0. This function models closeness in the frequency domain, since small values of (i−i′) correspond to nearby frequency bands.

2. If A is finite alphabet and P=Z, a finite string t on A can be defined as the finite structure s, where Is={1, . . . , |t|}, with |t| the length of the string t, cs(i)=t[i], and ps(i,i′)=i−i′. For example, t could be a DNA sequence, a protein sequence, or a document, represented as a sequence of words. In such cases, a vicinity function might be defined as

h(s,i,i′)=1(i>i′)λi−i′  (4)

for some λε[0,1). For a DNA sequence, one might choose a vicinity function such as

h(s,i,i′)=1(i>i′)λ1|(i−i′+6 mod 12)−6|λ2(i−i′)  (5)

for some λ1, λ2ε[0,1). In this case, the vicinity function reflects closeness in the molecule, since elements of the sequence separated by about 12 base pairs are in adjacent coils of the helix.

3. If A=R, P=Z, and tεl2 is a discrete-time real signal, then one could define the structure s where Is=N, cs(i)=t[i], and ps(i,i′)=i−i′. This is not a finite structure. An example of a vicinity function might be the impulse response h(s,i,i′)=φ(i−i′). This leads to some kind of matched filter.
4. If A=[0,1], P=Z2, and gεAxsxys is a grayscale image of width xs pixels and height ys pixels, then one could define the structure s with Is={1, . . . , xs}×{1, . . . , ys}, cs(x,y)=g[x,y], and

ps((x,y),(x′,y′))=(x−x′,y−y′)  (6)

An example of a vicinity function is

h(s,(x,y),(x′,y′))=e−((x−x′)2+(y−y′)2)/(2σ2)  (7)

5. For a directed acyclic graph (“DAG”) (V,E) with vertices labeled with elements of A, one could define the structure s with Is=V, component function cs(i) defined to be the label associated with vertex i, path set P defined as the set of all connected DAGs, and ps(i,i′) is the DAG consisting of all paths from i to i′. For example, A might be a set of part-of-speech tags, and s might correspond to a parse tree, with each vertex labeled with a tag. An example of a vicinity function is

h(s,i,i′)=λ−d(i,i′),  (8)

where d(i,i′) is the number of edges in the shortest directed path from i to i′ (or ∞ if there is no such path.

Next, locational kernels are defined to measure the similarity of pairs, e.g., structure or location in the structure. A kernel on a set T means a nonnegative-definite, symmetric, real function on T2.
For concreteness, consider the comparison of two documents. where the documents are represented as sequences of words. The key idea is to construct kernels comparing the two documents centered at particular words, that is, to define a measure of the similarity of two words, surrounded by the rest of the documents.
For a set S of structures, define

={(s,i):sεS, iεIs}  (9)

A locational kernel on S corresponds to a kernel on .

A locational kernel on S is said to be bounded if, for all sεIs.




  
    
      
        
          
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Notice that every locational kernel on a set of finite structures is bounded.
If k is a locational kernel on a set S of finite structures, and h is a vicinity function on S, then h*k is a locational kernel on S. This can be shown as follows:
Let Φ:→l2 be the feature map for k, so that

k((s,i),(t,j))=Φ(s,i),(t,j)).  (11)

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and both factors are finite, since k is bounded and h is stable. The boundedness of k and the stability of h are implied by the finiteness of Is.

Note that the function ((s, i, i′), (t, j, j′))→h(s, i, i′)h(t, j, j′) is a kernel, and so for the case of finite structures one could express h*k as a special case of an R-convolution.
An interesting class of vicinity functions is obtained by considering smoothness properties of the component function. For example, suppose that xεRn is a vector of the energy contained in n band-pass filtered versions of a signal, so that it is a smoothed version of the power of the signal in n adjacent frequency bands. (We can consider x as a structure by defining A=R′P=Z, Is, {1, . . . , n}, cs(i)=xi and ps(i,i′)=i−i′.) Since components of the vector x already contain information about the neighboring components, it might be useful to consider a representation that focuses on how each component deviates from the value implied by a smooth interpolation of the neighboring components. One could, for example, represent each component by the coefficients of a low-degree polynomial that accurately approximates the nearby components, and by the error of such an approximation. In fact, these representations correspond to convolutions of certain vicinity functions with the component kernel.
To see this more generally, consider a kernel k0 on the component space A, and let Φ be the corresponding feature map from A to l2. Choose some functions φ1, . . . , φn that map from P to R. For each sεS and iεIs define the parameterized map f:Is→l2 by




  
    
      
        
          
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in which case one can loosely think of fs,i as an approximation of Φ in the vicinity of i.

Also, for these optimal parameters, define as,i,0=Φ(cs(i))−fs,i(i). Then, for each 0≦n≦N. define the kernel {tilde over (k)}n, ((s,i),(t,j))=as,i,n,at,j,n.
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                          j
                          ′
                        
                      
                      )
                    
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        16
        )
      
    
  




the proof of which follows:

Fix sε and I. For the objective function




  
    
      
        
          J
          =
          
            
              1
              2
            
            ⁢
            
              
                ∑
                
                  
                    i
                    ′
                  
                  ∈
                  
                    I
                    s
                  
                
              
              ⁢
              
                
                  w
                  ⁡
                  
                    (
                    
                      s
                      ,
                      i
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                    
                      
                        
                          f
                          
                            s
                            ,
                            i
                          
                        
                        ⁡
                        
                          (
                          
                            i
                            ′
                          
                          )
                        
                      
                      -
                      
                        Φ
                        ⁡
                        
                          (
                          
                            
                              c
                              s
                            
                            ⁡
                            
                              (
                              
                                i
                                ′
                              
                              )
                            
                          
                          )
                        
                      
                    
                    
                  
                  2
                
              
            
          
        
        ,
      
    
    
      
        (
        17
        )
      
    
  




and for nε{1, . . . , N}, the partial derivative of J with respect to the parameter as,i,n is





  
    
      
        
          
            ∂
            
              a
              
                s
                ,
                i
                ,
                n
              
            
          
          ⁢
          J
        
        =
        
          
            ∑
            
              
                i
                ′
              
              ∈
              
                I
                s
              
            
          
          ⁢
          
            
              w
              ⁡
              
                (
                
                  s
                  ,
                  i
                  ,
                  
                    i
                    ′
                  
                
                )
              
            
            ⁢
            
              
                φ
                n
              
              ⁡
              
                (
                
                  
                    p
                    s
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                
                )
              
            
            ⁢
            
              
                (
                
                  
                    
                      f
                      
                        s
                        ,
                        i
                      
                    
                    ⁡
                    
                      (
                      
                        i
                        ′
                      
                      )
                    
                  
                  -
                  
                    Φ
                    ⁡
                    
                      (
                      
                        
                          c
                          s
                        
                        ⁡
                        
                          (
                          
                            i
                            ′
                          
                          )
                        
                      
                      )
                    
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




These N partial derivatives are zero when





  
    
      
        
          
            
              ∑
              
                
                  i
                  ′
                
                ∈
                
                  I
                  s
                
              
            
            ⁢
            
              
                w
                ⁡
                
                  (
                  
                    s
                    ,
                    i
                    ,
                    
                      i
                      ′
                    
                  
                  )
                
              
              ⁢
              
                
                  φ
                  n
                
                ⁡
                
                  (
                  
                    
                      p
                      s
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        
                          i
                          ′
                        
                      
                      )
                    
                  
                  )
                
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    a
                    
                      s
                      ,
                      i
                      ,
                      n
                    
                  
                  ⁢
                  
                    
                      φ
                      n
                    
                    ⁡
                    
                      (
                      
                        
                          p
                          s
                        
                        ⁡
                        
                          (
                          
                            i
                            ,
                            
                              i
                              ′
                            
                          
                          )
                        
                      
                      )
                    
                  
                
              
            
          
          =
          
            
              
                
                  ∑
                  
                    
                      i
                      ′
                    
                    ∈
                    
                      I
                      s
                    
                  
                
                ⁢
                
                  
                    w
                    ⁡
                    
                      (
                      
                        s
                        ,
                        i
                        ,
                        
                          i
                          ′
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      φ
                      n
                    
                    ⁡
                    
                      (
                      
                        
                          p
                          s
                        
                        ⁡
                        
                          (
                          
                            i
                            ,
                            
                              i
                              ′
                            
                          
                          )
                        
                      
                      )
                    
                  
                  ⁢
                  
                    Φ
                    ⁡
                    
                      (
                      
                        
                          c
                          s
                        
                        ⁡
                        
                          (
                          
                            i
                            ′
                          
                          )
                        
                      
                      )
                    
                  
                
              
              ⇔
              
                W
                ⁡
                
                  (
                  
                    
                      
                        
                          a
                          
                            s
                            ,
                            in
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        
                          ·
                          
                            a
                            
                              s
                              ,
                              i
                              ,
                              N
                            
                          
                        
                      
                    
                  
                  )
                
              
            
            =
            
              
                ∑
                
                  
                    i
                    ′
                  
                  ∈
                  
                    I
                    s
                  
                
              
              ⁢
              
                
                  w
                  ⁡
                  
                    (
                    
                      s
                      ,
                      i
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                
                ⁢
                
                  Φ
                  ⁡
                  
                    (
                    
                      
                        c
                        s
                      
                      ⁡
                      
                        (
                        
                          i
                          ′
                        
                        )
                      
                    
                    )
                  
                
                ⁢
                
                  (
                  
                    
                      
                        
                          
                            φ
                            1
                          
                          ⁡
                          
                            (
                            
                              
                                p
                                s
                              
                              ⁡
                              
                                (
                                
                                  i
                                  ,
                                  
                                    i
                                    ′
                                  
                                
                                )
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        
                          
                            φ
                            N
                          
                          ⁡
                          
                            (
                            
                              
                                p
                                s
                              
                              ⁡
                              
                                (
                                
                                  i
                                  ,
                                  
                                    i
                                    ′
                                  
                                
                                )
                              
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        19
        )
      
    
  




where the N×N real matrix W has entries





  
    
      
        
          W
          
            n
            ,
            
              n
              ′
            
          
        
        =
        
          
            ∑
            
              
                i
                ′
              
              ∈
              
                I
                s
              
            
          
          ⁢
          
            
              w
              ⁡
              
                (
                
                  s
                  ,
                  i
                  ,
                  
                    i
                    ′
                  
                
                )
              
            
            ⁢
            
              
                φ
                
                  n
                  ′
                
              
              ⁡
              
                (
                
                  
                    p
                    s
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                
                )
              
            
            ⁢
            
              
                
                  φ
                  n
                
                ⁡
                
                  (
                  
                    
                      p
                      s
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        
                          i
                          ′
                        
                      
                      )
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




If the matrix is invertible, one can write




   
  
    
      
        
          
            
              
                
                  a
                  
                    s
                    ,
                    i
                    ,
                    n
                  
                
                =
                
                ⁢
                
                  
                    ∑
                    
                      
                        i
                        ′
                      
                      ∈
                      
                        I
                        s
                      
                    
                  
                  ⁢
                  
                    
                      Φ
                      ⁡
                      
                        (
                        
                          
                            c
                            s
                          
                          ⁡
                          
                            (
                            
                              i
                              ′
                            
                            )
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        
                          w
                          ⁡
                          
                            (
                            
                              s
                              ,
                              i
                              ,
                              
                                i
                                ′
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              W
                              
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      
                                        φ
                                        1
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            p
                                            s
                                          
                                          ⁡
                                          
                                            (
                                            
                                              i
                                              ,
                                              
                                                i
                                                ′
                                              
                                            
                                            )
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                                
                                  
                                    ⋮
                                  
                                
                                
                                  
                                    
                                      
                                        φ
                                        N
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            p
                                            s
                                          
                                          ⁡
                                          
                                            (
                                            
                                              i
                                              ,
                                              
                                                i
                                                ′
                                              
                                            
                                            )
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              )
                            
                          
                          ]
                        
                      
                      n
                    
                  
                
              
            
          
          
            
              
                
                  =
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          ′
                        
                        ∈
                        
                          I
                          s
                        
                      
                    
                    ⁢
                    
                      
                        Φ
                        ⁡
                        
                          (
                          
                            
                              c
                              s
                            
                            ⁡
                            
                              (
                              
                                i
                                ′
                              
                              )
                            
                          
                          )
                        
                      
                      ⁢
                      
                        
                          h
                          n
                        
                        ⁡
                        
                          (
                          
                            s
                            ,
                            i
                            ,
                            
                              i
                              ′
                            
                          
                          )
                        
                      
                    
                  
                
                ,
              
            
          
        
      
      
        
          (
          21
          )
        
      
    
  




where hn is defined appropriately. This relationship follows for n=1 . . . , N and the case n=0 follows from the definition of aa,s,0 

Locational kernels define a notion of similarity between structures s and t that is relevant to particular locations i and j in the structures. For a locational kernel k, k((s,i), (t,j)) should be thought of as the similarity of the components of s in the locality of i with the components of t in the locality of j.
If KA is a kernel on a component set A, and S is a class of structures on A, then one can define a locational kernel on S via

k((s,i), (t,j))=KA(cs(i),ct(j))  (22)

where s, tεS, iεIs, and jεIt.

Using a locational kernel, two structures s, tεS can be viewed as a rich collection of similarities of their components. Then, a variety of operations can be performed on locational kernels k0, k1, . . . on S to obtain another locational kernel. For example:
1. Addition:

(k0+ki)((s,i),(t,j))=k0((s,i),(t,j))+k1((s,i),(t,j))  (23)

2. Scalar multiplication: for λ≧0,

(λk0)((s,i),(t,j))=λk0((s,i),(t,j))  (24)

3. Multiplication:

(k0k1)((s,i),(t,j))=k0((s,i),(t,j))k1((s,i),(t,j))  (25)

4. Pointwise limits:




  
    
      
        
          
            (
            
              
                lim
                ⁢
                
                  
                
                ⁢
                
                  k
                  n
                
              
              
                n
                ->
                ∞
              
            
            )
          
          ⁢
          
            (
            
              
                (
                
                  s
                  ,
                  i
                
                )
              
              ,
              
                (
                
                  t
                  ,
                  j
                
                )
              
            
            )
          
        
        =
        
          
            lim
            
              n
              ->
              ∞
            
          
          ⁢
          
            
              k
              n
            
            (
            
              
                (
                
                  s
                  ,
                  i
                
                )
              
              ,
              
                (
                
                  t
                  ,
                  j
                
                )
              
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




provided the limit exists for all s, tεS and Is,jεIt 

5. Argument transformations: if T is a set of structures and f:→, then

kf((s,i),(t,j))=k0(f(s,i),f(t,j))  (27)

is a locational kernel on T. As a special case, if T=S and Is=It for all s, tεS, one might have f(s,i)=(fs(s),fI(i)) for some functions fs:S→S and fI:Is→Is.

6. Convolution with a vicinity function h for S:




  
    
      
        
          
            (
            
              h
              *
              
                k
                0
              
            
            )
          
          ⁢
          
            (
            
              
                (
                
                  s
                  ,
                  i
                
                )
              
              ,
              
                (
                
                  t
                  ,
                  j
                
                )
              
            
            )
          
        
        =
        
          
            ∑
            
              
                
                  
                    
                      i
                      ′
                    
                    ∈
                    
                      I
                      ⁢
                      s
                    
                  
                
              
              
                
                  
                    
                      j
                      ′
                    
                    ∈
                    
                      I
                      t
                    
                  
                
              
            
          
          ⁢
          
            
              h
              ⁡
              
                (
                
                  s
                  ,
                  i
                  ,
                  
                    i
                    ′
                  
                
                )
              
            
            ⁢
            
              h
              ⁡
              
                (
                
                  t
                  ,
                  j
                  ,
                  
                    j
                    ′
                  
                
                )
              
            
            ⁢
            
              
                k
                0
              
              ⁡
              
                (
                
                  
                    (
                    
                      s
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                  ,
                  
                    (
                    
                      t
                      ,
                      
                        j
                        ′
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




7. Convolution with a real-valued function v:{(s,t,i,i′,j, j′) s, tεS, i,i′εIs,j,j′εIt}→R:




  
    
      
        
          
            (
            
              v
              *
              
                k
                0
              
            
            )
          
          ⁢
          
            (
            
              
                (
                
                  s
                  ,
                  i
                
                )
              
              ,
              
                (
                
                  t
                  ,
                  j
                
                )
              
            
            )
          
        
        =
        
          
            ∑
            
              
                
                  
                    
                      i
                      ′
                    
                    ∈
                    
                      I
                      s
                    
                  
                
              
              
                
                  
                    
                      j
                      ′
                    
                    ∈
                    
                      I
                      t
                    
                  
                
              
            
          
          ⁢
          
            
              v
              ⁡
              
                (
                
                  s
                  ,
                  t
                  ,
                  i
                  ,
                  
                    i
                    ′
                  
                  ,
                  j
                  ,
                  
                    j
                    ′
                  
                
                )
              
            
            ⁢
            
              
                k
                0
              
              ⁡
              
                (
                
                  
                    (
                    
                      s
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                  ,
                  
                    (
                    
                      t
                      ,
                      
                        j
                        ′
                      
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




These operations can be combined to obtain, e.g., power series (with nonnegative coefficients) of locational kernels. Moreover, the convolution can be iterated. For example, starting with a kernel kA on the component set A, k0 can be defined as the corresponding locational kernel on S. The multiplication and convolution operations are iterated as follows. For n=1, 2, . . . , define

kn=k0(h*kn−1),  (30)

and, with k′0=k′′0,

k′n=k′n−1(h*k0).  (31)

Note that k1=k′1, and k′n=k0(h*k0)n.

Loosely speaking, k′n defines the similarity of s at i and t at j as a weighted product of the k0-similarities of n nearby components of s and t, where the weights are given by the product h(s,i1,i) . . . h(s,in,i). The kernel kn works in a similar way, but in this case the weights are given by the product h(s,i1,i2) . . . h(s,in−1,in).h(s,in,i).
Let kd(a,b)=Φ(a), Φ(b) for some feature map Φ: A→l2,
Where dεN∪{∞}. One can write

kn((s,i),(t,i))=Ψ(s,i),Ψ(t,j))  (32)

where Ψ(s,i) is an element of l2 with components





  
    
      
        
          
            
              Ψ
              ⁡
              
                (
                
                  s
                  ,
                  i
                
                )
              
            
            
              
                i
                1
              
              ,
              ...
              ,
              
                i
                n
              
            
          
          =
          
            
              ∑
              
                
                  j
                  1
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  j
                  n
                
              
            
            ⁢
            
              
                ∏
                
                  l
                  =
                  1
                
                n
              
              ⁢
              
                
              
              ⁢
              
                
                  h
                  ⁡
                  
                    (
                    
                      s
                      ,
                      
                        j
                        l
                      
                      ,
                      
                        j
                        
                          l
                          -
                          1
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    Φ
                    ⁡
                    
                      (
                      
                        
                          c
                          s
                        
                        ⁡
                        
                          (
                          
                            j
                            l
                          
                          )
                        
                      
                      )
                    
                  
                  
                    i
                    l
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        33
        )
      
    
  




where i1ε{1, . . . , d}. for l=1, . . . , n, and the sum is over (j1, . . . , jn)εIsn, and j0=i.

Further,

k′n((s,i),(t,j))=Ψ(s,i),Ψ′(t,j),  (34)

where Ψ(s,i) is an element of l2 with components





  
    
      
        
          
            
              Ψ
              ′
            
            ⁡
            
              (
              
                s
                ,
                i
              
              )
            
          
          
            
              i
              1
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              i
              n
            
          
        
        =
        
          
            ∑
            
              
                j
                1
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                j
                n
              
            
          
          ⁢
          
            
              ∏
              
                l
                =
                1
              
              n
            
            ⁢
            
              
                h
                ⁡
                
                  (
                  
                    s
                    ,
                    
                      j
                      l
                    
                    ,
                    i
                  
                  )
                
              
              ⁢
              
                
                  Φ
                  ⁡
                  
                    (
                    
                      
                        c
                        s
                      
                      ⁡
                      
                        (
                        
                          j
                          l
                        
                        )
                      
                    
                    )
                  
                
                
                  1
                  l
                
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




where the sum is over (j1, . . . , jn)εIsn.

The next step in the process for constructing kernels for structured data is to obtain kernels from the locational kernels. Given a locational kernel, one can obtain a kernel on the set of structures in using a number of different methods.
The first method comprises summing over the indices:
If S is a set of finite structures and k is a locational kernel on S, the kernel K on S can be defined as




  
    
      
        
          K
          ⁡
          
            (
            
              s
              ,
              t
            
            )
          
        
        =
        
          
            ∑
            
              
                
                  
                    i
                    ∈
                    
                      I
                      s
                    
                  
                
              
              
                
                  
                    j
                    ∈
                    
                      I
                      t
                    
                  
                
              
            
          
          ⁢
          
            k
            ⁡
            
              (
              
                
                  (
                  
                    s
                    ,
                    i
                  
                  )
                
                ,
                
                  (
                  
                    t
                    ,
                    j
                  
                  )
                
              
              )
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




for s, tεS. More generally, if for every sεS there is a weighting function ws:Is→R, then one can define the kernels K on S as





  
    
      
        
          K
          ⁡
          
            (
            
              s
              ,
              t
            
            )
          
        
        =
        
          
            ∑
            
              
                
                  
                    i
                    ∈
                    
                      I
                      s
                    
                  
                
              
              
                
                  
                    j
                    ∈
                    
                      I
                      t
                    
                  
                
              
            
          
          ⁢
          
            
              
                w
                s
              
              ⁡
              
                (
                i
                )
              
            
            ⁢
            
              
                w
                t
              
              ⁡
              
                (
                j
                )
              
            
            ⁢
            
              
                k
                ⁡
                
                  (
                  
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                    ,
                    
                      (
                      
                        t
                        ,
                        j
                      
                      )
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




If the structures in S are not all finite, but k is bounded and each ws is absolutely summable, then this definition still gives a kernel. Viewing the kernel as some kind of average of locational kernels looking at the structure's components might yield principled ways of dealing with missing data, i.e., missing components.

Proof of the definition in Equation 37 is as follows: Let Φ:→l2 be the feature map for k, so that

k((s,i),(t,j))=Φ(s,i),Φ(t,j).  (38)

Then,





   
  
    
      
        
          
            
              K
              ⁡
              
                (
                
                  s
                  ,
                  t
                
                )
              
            
            
              i
              ∈
              
                
                  I
                  s
                
                ⁢
                
                  
                
                ⁢
                j
              
              ∈
              
                I
                t
              
            
          
          =
          
            
              ∑
              
                i
                ∈
                
                  I
                  s
                
              
            
            ⁢
            
              
                ∑
                
                  j
                  ∈
                  
                    I
                    t
                  
                
              
              ⁢
              
                
                  
                    w
                    s
                  
                  ⁡
                  
                    (
                    i
                    )
                  
                
                ⁢
                
                  
                    w
                    t
                  
                  ⁡
                  
                    (
                    j
                    )
                  
                
                ⁢
                
                  〈
                  
                    
                      Φ
                      ⁡
                      
                        (
                        
                          s
                          ,
                          i
                        
                        )
                      
                    
                    ,
                    
                      Φ
                      ⁡
                      
                        (
                        
                          t
                          ,
                          j
                        
                        )
                      
                    
                  
                  〉
                
              
            
          
        
      
      
        
          (
          39
          )
        
      
    
    
      
        
          
            
          
          ⁢
          
            =
            
              〈
              
                
                  (
                  
                    
                      ∑
                      
                        i
                        ∈
                        
                          I
                          s
                        
                      
                    
                    ⁢
                    
                      
                        
                          w
                          s
                        
                        ⁡
                        
                          (
                          i
                          )
                        
                      
                      ⁢
                      
                        Φ
                        ⁡
                        
                          (
                          
                            s
                            ,
                            i
                          
                          )
                        
                      
                    
                  
                  )
                
                ,
                
                  (
                  
                    
                      ∑
                      
                        j
                        ∈
                        
                          I
                          t
                        
                      
                    
                    ⁢
                    
                      
                        
                          w
                          t
                        
                        ⁡
                        
                          (
                          j
                          )
                        
                      
                      ⁢
                      
                        Φ
                        ⁡
                        
                          (
                          
                            t
                            ,
                            j
                          
                          )
                        
                      
                    
                  
                  )
                
              
              〉
            
          
        
      
      
        
          (
          40
          )
        
      
    
    
      
        
          
            
          
          ⁢
          
            
              =
              
                〈
                
                  
                    ψ
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  ,
                  
                    ψ
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                〉
              
            
            ,
          
        
      
      
        
          (
          41
          )
        
      
    
  




where





  
    ψ
    ⁡
    
      (
      s
      )
    
  
  =
  
    
      ∑
      
        i
        ∈
        
          I
          s
        
      
    
    ⁢
    
      
        
          
            w
            s
          
          
            i
            ∈
            
              I
              s
            
          
        
        ⁡
        
          (
          i
          )
        
      
      ⁢
      
        
          Φ
          ⁡
          
            (
            
              s
              ,
              i
            
            )
          
        
        .
      
    
  




Thus, K provided that S maps ψ to l2.

But





  
    
      
        
          
            
              
              
                ψ
                ⁡
                
                  (
                  s
                  )
                
              
              
            
            ≤
            
              
                ∑
                
                  i
                  ∈
                  
                    I
                    s
                  
                
              
              ⁢
              
                
                
                  
                    
                      w
                      s
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                  ⁢
                  
                    Φ
                    ⁡
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                  
                
                
              
            
          
          =
          
            
              max
              
                i
                ∈
                
                  I
                  s
                
              
            
            ⁢
            
              
                
                
                  Φ
                  ⁡
                  
                    (
                    
                      s
                      ,
                      i
                    
                    )
                  
                
                
              
              ⁢
              
                
                  ∑
                  
                    i
                    ∈
                    
                      I
                      s
                    
                  
                
                ⁢
                
                  
                  
                    
                      w
                      s
                    
                    ⁡
                    
                      (
                      i
                      )
                    
                  
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        42
        )
      
    
  




and both factors are finite. The boundedness of k and the absolute summability of the constant weighting function ws(i)≡1 are implied by the finiteness of Is.

A second method for obtaining the kernel on the set of structures given a location kernel is fixing the indices.
If i is a function on S satisfying i(s)εIs, then

K(s,t)=k((s,i(s)),(t,i(t)))  (43)

is a kernel. Note that one must substitute the same function i in both of k's arguments, to ensure that the result is symmetric in s,t. As a special case, if Is=It for all s,tεS, one can choose a constant function i.

As a third method, one could use

K(s,t)=max k((s,i),(t,j)),  (44)

to obtain a notion of similarity that is invariant with respect to certain transformations of the index sets (e.g., translations of images), cf. the notion of ‘kernel jittering’ as discussed by D. DeCoste and M. C. Burl in “Distortion-invariant recognition via jittered queries”, Computer Vision and Pattern Recognition (CVPR-2000), June 2000.

In addition to the foregoing operations, one can apply all known manipulations allowed for kernels. Thus, a wealth of different kernels can be obtained. Further, some ‘classical’ kernels can be used as special cases:

    
    
        The Gaussian RBF (radial basis function) kernel is a product (over all index positions) of locational kernels constructed from Gaussian RBF kernels kA whose indices have been fixed
        Polynomial kernels of degree n can be obtained via convolutions as follows: consider the case where A=R and Is=It=I, use the shorthands cs(i)=si,ct(i)=ti, and define

k0((s,i),(t,j))=δij  (45)

as well as, for some real-valued function u(i,i′), and nεN,

    
    






  
    
      
        
          
            k
            n
          
          ⁡
          
            (
            
              
                (
                
                  s
                  ,
                  i
                
                )
              
              ,
              
                (
                
                  t
                  ,
                  j
                
                )
              
            
            )
          
        
        =
        
          
            δ
            
              i
              ⁢
              
                
              
              ⁢
              j
            
          
          ⁢
          
            
              ∑
              
                
                  i
                  ′
                
                ,
                
                  j
                  ∈
                  I
                
              
            
            ⁢
            
              
                s
                i
              
              ⁢
              
                u
                ⁡
                
                  (
                  
                    i
                    ,
                    
                      i
                      ′
                    
                  
                  )
                
              
              ⁢
              
                t
                j
              
              ⁢
              
                u
                ⁡
                
                  (
                  
                    j
                    ,
                    
                      j
                      ′
                    
                  
                  )
                
              
              ⁢
              
                
                  
                    k
                    
                      n
                      -
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          s
                          ,
                          
                            i
                            ′
                          
                        
                        )
                      
                      ,
                      
                        (
                        
                          t
                          ,
                          
                            j
                            ′
                          
                        
                        )
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




Note that this has the form kn=k0(h*kn−1), as introduced above, where the vicinity function h(s,i,i′)=siu(i,i′) is used. For constant u, the result is the homogeneous polynomial kernel as shown in the following:
For nεN0, the vicinity function h(s,i,i′)=si and the component kernel k0(s,i),(t,j))=δij lead to a locational kernel kn with the property that averaged over all indices, the homogeneous polynomial kernel is recovered.




  
    
      
        
          
            1
            
              
              I
              
            
          
          ⁢
          
            
              ∑
              
                i
                ,
                
                  j
                  ∈
                  I
                
              
            
            ⁢
            
              
                k
                n
              
              ⁡
              
                (
                
                  
                    (
                    
                      s
                      ,
                      i
                    
                    )
                  
                  ,
                  
                    (
                    
                      t
                      ,
                      j
                    
                    )
                  
                
                )
              
            
          
        
        =
        
          
            
              1
              
                
                I
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  ∈
                  I
                
              
              ⁢
              
                
                  k
                  n
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                    ,
                    
                      (
                      
                        t
                        ,
                        j
                      
                      )
                    
                  
                  )
                
              
            
          
          =
          
            
              
                (
                
                  
                    ∑
                    
                      i
                      ∈
                      I
                    
                  
                  ⁢
                  
                    
                      s
                      i
                    
                    ⁢
                    
                      t
                      i
                    
                  
                
                )
              
              n
            
            .
          
        
      
    
    
      
        (
        47
        )
      
    
  




The locational kernel kn((s,i),(t,j)) can thus be thought of as the kernel computing all n-th order product features ‘belonging’ to index i.
As proof of the preceding, by induction on n, for n=0, Equation 47 holds true. Supposing, then, that it also holds true for n−1, where n≧1, yielding:




   
  
    
      
        
          
            
              
                
                  
                    1
                    
                      
                      I
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        ,
                        
                          j
                          ∈
                          I
                        
                      
                    
                    ⁢
                    
                      
                        k
                        n
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              s
                              ,
                              i
                            
                            )
                          
                          ,
                          
                            (
                            
                              t
                              ,
                              j
                            
                            )
                          
                        
                        )
                      
                    
                  
                
                =
                
                ⁢
                
                  
                    I
                    
                      
                      I
                      
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        ,
                        j
                      
                    
                    ⁢
                    
                      
                        δ
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          j
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              ′
                            
                            ,
                            
                              j
                              ′
                            
                          
                        
                        ⁢
                        
                          
                            s
                            i
                          
                          ⁢
                          
                            t
                            j
                          
                          ⁢
                          
                            
                              k
                              
                                n
                                -
                                1
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  (
                                  
                                    s
                                    ,
                                    
                                      i
                                      ′
                                    
                                  
                                  )
                                
                                ,
                                
                                  (
                                  
                                    t
                                    ,
                                    
                                      j
                                      ′
                                    
                                  
                                  )
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    ∑
                    i
                  
                  ⁢
                  
                    
                      s
                      i
                    
                    ⁢
                    
                      t
                      i
                    
                    ⁢
                    
                      1
                      
                        
                        I
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            ′
                          
                          ,
                          
                            j
                            ′
                          
                        
                      
                      ⁢
                      
                        
                          k
                          
                            n
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            
                              (
                              
                                s
                                ,
                                
                                  i
                                  ′
                                
                              
                              )
                            
                            ,
                            
                              (
                              
                                t
                                ,
                                
                                  j
                                  ′
                                
                              
                              )
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    ∑
                    i
                  
                  ⁢
                  
                    
                      s
                      i
                    
                    ⁢
                    
                      
                        
                          t
                          i
                        
                        ⁡
                        
                          (
                          
                            
                              ∑
                              i
                            
                            ⁢
                            
                              
                                s
                                i
                              
                              ⁢
                              
                                t
                                i
                              
                            
                          
                          )
                        
                      
                      
                        n
                        -
                        1
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      (
                      
                        
                          ∑
                          i
                        
                        ⁢
                        
                          
                            s
                            i
                          
                          ⁢
                          
                            t
                            i
                          
                        
                      
                      )
                    
                    n
                  
                  .
                
              
            
          
        
      
      
        
          (
          48
          )
        
      
    
  




For non-constant functions u, polynomial kernels are obtained where different product features are weighted differently. For example, in the processing of genetic strings, one could use a function u which decays with the distance (on the string) of i and j, e.g., for some c>0,

u(i,j)=max {0, 1−c·|i−j|}.  (49)

Thus, one can obtain “locality-improved” kernels such as the ones that have successfully been used in start codon recognition and image processing.
Another interesting generalization of polynomial kernels suggested by the above view is the use of another function instead of δij, such as 1|i−j|<c, for some cεn. This way, one should obtain polynomial kernels which compare, e.g., sequences that are not precisely registered, thus obtaining some degree of transitional invariance.




  
    
      
        
          〈
          
            
              a
              
                s
                ,
                i
                ,
                n
              
            
            ,
            
              a
              
                t
                ,
                j
                ,
                n
              
            
          
          〉
        
        =
        
          
            ∑
            
              
                i
                ′
              
              ,
              
                j
                ′
              
            
          
          ⁢
          
            
              
                h
                n
              
              ⁡
              
                (
                
                  s
                  ,
                  i
                  ,
                  
                    i
                    ′
                  
                
                )
              
            
            ⁢
            
              
                h
                n
              
              ⁡
              
                (
                
                  t
                  ,
                  j
                  ,
                  
                    j
                    ′
                  
                
                )
              
            
            ⁢
            
              
                k
                0
              
              ⁡
              
                (
                
                  
                    (
                    
                      s
                      ,
                      
                        i
                        ′
                      
                    
                    )
                  
                  ,
                  
                    (
                    
                      t
                      ,
                      j
                    
                    )
                  
                
                )
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




To provide an example, in the homogeneous polynomial case, where h(s,i,i′)=si, h(s,jl, jl−1)=h(s,jl,i)=sjl. Therefore, ψ=ψ′, and thus Equation 47 implies the corresponding result for k′n: 
For nεE N0, h(s,i,i′)=si, and k0((s,i),(t,j))=δij,




  
    
      
        
          
            1
            
              
              I
              
            
          
          ⁢
          
            
              ∑
              
                i
                ,
                
                  j
                  ∈
                  I
                
              
            
            ⁢
            
              
                k
                n
                ′
              
              ⁡
              
                (
                
                  
                    (
                    
                      s
                      ,
                      i
                    
                    )
                  
                  ,
                  
                    (
                    
                      t
                      ,
                      j
                    
                    )
                  
                
                )
              
            
          
        
        =
        
          
            
              1
              
                
                I
                
              
            
            ⁢
            
              
                ∑
                
                  i
                  ∈
                  I
                
              
              ⁢
              
                
                  k
                  n
                  ′
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                    ,
                    
                      (
                      
                        t
                        ,
                        i
                      
                      )
                    
                  
                  )
                
              
            
          
          =
          
            
              
                (
                
                  
                    ∑
                    
                      i
                      ∈
                      I
                    
                  
                  ⁢
                  
                    
                      s
                      i
                    
                    ⁢
                    
                      t
                      i
                    
                  
                
                )
              
              n
            
            .
          
        
      
    
    
      
        (
        51
        )
      
    
  




Moreover, the component kernel k0((s,i),(t,j))=kA(si,tj)=δij can trivially be represented as a dot product in l2d, where d=|I|, via δij=ei, ej, with {e1, . . . , ed} an orthonormal basis of l2d. Therefore,




   
  
    
      
        
          
            
              
                
                  
                    Ψ
                    ⁡
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                  
                  
                    
                      i
                      0
                    
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      i
                      n
                    
                  
                
                =
                
                  
                    ∑
                    
                      
                        j
                        1
                      
                      ,
                      ...
                      ,
                      
                        j
                        n
                      
                    
                  
                  ⁢
                  
                    
                      ∏
                      
                        l
                        =
                        1
                      
                      n
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          s
                          
                            j
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            e
                            
                              j
                              1
                            
                          
                          )
                        
                      
                      
                        t
                        1
                      
                    
                  
                
              
            
          
          
            
              
                
                  =
                  
                    
                      ∑
                      
                        
                          j
                          1
                        
                        ,
                        ...
                        ,
                        
                          j
                          n
                        
                      
                    
                    ⁢
                    
                      
                        ∏
                        
                          l
                          =
                          1
                        
                        n
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          s
                          
                            j
                            1
                          
                        
                        ⁢
                        
                          δ
                          
                            
                              j
                              1
                            
                            ⁢
                            
                              i
                              1
                            
                          
                        
                      
                    
                  
                
                ,
              
            
          
          
            
              
                =
                
                  
                    ∏
                    
                      l
                      =
                      1
                    
                    n
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    s
                    
                      i
                      1
                    
                  
                
              
            
          
        
      
      
        
          (
          52
          )
        
      
    
  




the well-known feature map of the homogeneous polynomial kernel, computing all possible products of n components. Note that this is independent of i. Therefore, when summing over all i, the result is a kernel which has the same feature map, up to the overall factor |I|.

Suppose S is the set of finite strings over the finite alphabet A, h is defined as

h(s,i,i′)=1(i>i′)λi−i′  (53)

for some λε[0,1), and k(a,b)=1(a=b). Then, each component of Φ(a) has the form

Φ(a)i=1(a=ai),  (54)

where A={a1, . . . , a|A|}. In this case,

kn((s,i)(t,j)=ψ(s,i),ψ(t,j),  (55)

where each component of ψ(s,i) is of the form





   
  
    
      
        
          
            
              
                
                  
                    
                      Ψ
                      ′
                    
                    ⁡
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                  
                  
                    
                      i
                      0
                    
                    ,
                    
                      i
                      1
                    
                    ,
                    ...
                    ,
                    
                      i
                      n
                    
                  
                
                =
                
                  
                    ∑
                    
                      
                        i
                        =
                        
                          j
                          1
                        
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        
                          j
                          n
                        
                        >
                        i
                      
                    
                    n
                  
                  ⁢
                  
                    
                      λ
                      
                        
                          
                            ∑
                            
                              l
                              =
                              1
                            
                            n
                          
                          ⁢
                          
                            j
                            l
                          
                        
                        -
                        i
                      
                    
                    ⁢
                    
                      
                        ∏
                        
                          l
                          =
                          1
                        
                        n
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          Φ
                          ⁡
                          
                            (
                            
                              
                                c
                                s
                              
                              ⁡
                              
                                (
                                
                                  j
                                  l
                                
                                )
                              
                            
                            )
                          
                        
                        
                          i
                          l
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                  
                    ∑
                    
                      
                        i
                        =
                        
                          
                            j
                            1
                          
                          <
                        
                      
                      ,
                      
                        ...
                        ⁢
                        
                          
                        
                        ⁢
                        
                          j
                          n
                        
                      
                    
                  
                  ⁢
                  
                    
                      λ
                      
                        
                          
                            ∑
                            
                              i
                              =
                              1
                            
                            n
                          
                          ⁢
                          
                            j
                            1
                          
                        
                        -
                        i
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      (
                      
                        
                          ∀
                          l
                        
                        ,
                        
                          s
                          ⁡
                          
                            [
                            
                              j
                              l
                            
                            ]
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                  a
                  
                    i
                    l
                  
                
              
            
          
        
      
      
        
          (
          56
          )
        
      
    
  




That is, the feature space has a component corresponding to each sequence (ai0, . . . , ain) and n+1 characters from A, and the value of that component is the sum over all matching subsequences

(s[i], s[ji], . . . , s[jn])=(ai0, . . . , ain)  (57)

of λ raised to the power of the length of the gap between the first and last of these characters. This is the string kernel described by H. Lodhi, J. Shawe-Taylor, N. Cristianini, and C. Watkins in “Text classification using string kernels”, Technical Report 2000-79, NeuroCOLT, 2000. Published in: T. K. Leen, T. G. Dietterich and V. Tresp (eds.), Advances in Neural Information Processing Systems 13, MIT Press, 2001, which is incorporated herein by reference.

Similarly,

k′n((s,i),(t,j))=ψ(s,i),ψ(t,j),  (58)

Where each component of ψ(s,i) is of the form





   
  
    
      
        
          
            
              
                
                  
                    
                      Ψ
                      ′
                    
                    ⁡
                    
                      (
                      
                        s
                        ,
                        i
                      
                      )
                    
                  
                  
                    
                      i
                      0
                    
                    ,
                    
                      i
                      1
                    
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      i
                      n
                    
                  
                
                =
                
                  
                    ∑
                    
                      
                        j
                        1
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        
                          j
                          n
                        
                        >
                        i
                      
                    
                  
                  ⁢
                  
                    
                      λ
                      
                        
                          
                            ∑
                            
                              l
                              =
                              1
                            
                            n
                          
                          ⁢
                          
                            j
                            l
                          
                        
                        -
                        i
                      
                    
                    ⁢
                    
                      
                        ∏
                        
                          l
                          =
                          1
                        
                        n
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          Φ
                          ⁡
                          
                            (
                            
                              
                                c
                                s
                              
                              ⁡
                              
                                (
                                
                                  j
                                  1
                                
                                )
                              
                            
                            )
                          
                        
                        
                          i
                          l
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                  
                    ∑
                    
                      
                        i
                        =
                        
                          j
                          1
                        
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        
                          j
                          n
                        
                        >
                        i
                      
                    
                  
                  ⁢
                  
                    
                      λ
                      
                        
                          
                            ∑
                            
                              i
                              =
                              1
                            
                            n
                          
                          ⁢
                          
                            j
                            1
                          
                        
                        -
                        i
                      
                    
                    ⁢
                    1
                    ⁢
                    
                      
                        (
                        
                          
                            ∀
                            l
                          
                          ,
                          
                            
                              s
                              ⁢
                              
                                
                                
                                  j
                                  l
                                
                                
                              
                            
                            =
                            
                              a
                              
                                i
                                l
                              
                            
                          
                        
                        )
                      
                      .
                    
                  
                
              
            
          
        
      
      
        
          (
          59
          )
        
      
    
  




In this case, the value of the component for each sequence (ai0, . . . , a1n) of n+1 characters from A is a sum over all matching, but not necessarily ordered, subsequences (s[i],s[j1], . . . s[n]=(ai0, . . . , ain). For each of these sequences, the component is incremented by λ raised to a power that depends on the distances from i to each matching character. This is a bit like a bag of words approach, in the sense that the characters can appear in any order, but each character's distance from i incurs a penalty. Using another viewpoint, each i,j pair defines the centers of clusters in the two structures, and the value of the locational kernel for those centers is a sum over all sequences of n characters of λ raised to the power of the sum over the sequence of the distances from the character to the center.

The nth order string kernel described by H. Lodhi, et al can be defined for fixed strings s, t using:




  
    
      
        
          
            
              K
              0
              ′
            
            ⁡
            
              (
              
                i
                ,
                j
              
              )
            
          
          =
          
            1
            ⁢
            
              
            
            ⁢
            for
            ⁢
            
              
            
            ⁢
            all
            ⁢
            
              
            
            ⁢
            i
          
        
        ,
        
          
            j
            ⁢
            
              
            
            ⁢
            and
            ⁢
            
              
            
            ⁢
            
              
                K
                k
                ′
              
              ⁡
              
                (
                
                  i
                  ,
                  j
                
                )
              
            
          
          =
          
            
              0
              ⁢
              
                
              
              ⁢
              for
              ⁢
              
                
              
              ⁢
              
                min
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
            <
            k
          
        
        ,
        
          

        
        ⁢
        
          
            
              
                
                  
                    K
                    k
                    ′
                  
                  ⁡
                  
                    (
                    
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                    )
                  
                
                =
                
                  
                    λ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        K
                        k
                        ′
                      
                      ⁡
                      
                        (
                        
                          i
                          ,
                          j
                        
                        )
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        
                          j
                          ′
                        
                        =
                        1
                      
                      j
                    
                    ⁢
                    
                      1
                      ⁢
                      
                        (
                        
                          s
                          ⁡
                          
                            [
                            
                              i
                              +
                              1
                            
                            ]
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      =
                      
                        t
                        ⁡
                        
                          [
                          
                            j
                            ′
                          
                          ]
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      K
                      
                        k
                        -
                        1
                      
                      ′
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        
                          
                            j
                            ′
                          
                          -
                          1
                        
                      
                      )
                    
                  
                  ⁢
                  
                    λ
                    
                      j
                      -
                      
                        j
                        ′
                      
                      +
                      2
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        60
        )
      
    
  
  
    
      
        
          
            
              K
              n
            
            ⁡
            
              (
              
                i
                ,
                j
              
              )
            
          
          =
          
            
              0
              ⁢
              
                
              
              ⁢
              for
              ⁢
              
                
              
              ⁢
              
                min
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
            <
            n
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    K
                    n
                  
                  ⁡
                  
                    (
                    
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                    )
                  
                
                =
                
                  
                    
                      K
                      n
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                  +
                  
                    
                      ∑
                      
                        
                          j
                          ′
                        
                        =
                        1
                      
                      j
                    
                    ⁢
                    
                      1
                      ⁢
                      
                        (
                        
                          s
                          ⁡
                          
                            [
                            
                              i
                              +
                              1
                            
                            ]
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    =
                    
                      t
                      ⁡
                      
                        [
                        
                          j
                          ′
                        
                        ]
                      
                    
                  
                  )
                
                ⁢
                
                  
                    K
                    
                      n
                      -
                      1
                    
                    ′
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      
                        
                          j
                          ′
                        
                        -
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  
                    λ
                    2
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        61
        )
      
    
  




Then, the nth order kernel for two strings s, t is defined as Kn(|s|,|t|).

The following result shows that the string kernel is of the form described in Equation 30, with a simple exponential weighting function.
The nth order string kernel of Lodhi et al is given by




  
    
      
        
          
            
              K
              n
            
            ⁡
            
              (
              
                
                  
                  s
                  
                
                ,
                
                  
                  t
                  
                
              
              )
            
          
          =
          
            
              λ
              
                2
                ⁢
                n
              
            
            ⁢
            
              
                ∑
                
                  i
                  ,
                  j
                
              
              ⁢
              
                
                  L
                  
                    n
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        62
        )
      
    
  




where, for all integer i, j,





  
    
      
        
          
            L
            0
          
          ⁡
          
            (
            
              i
              ,
              j
            
            )
          
        
        =
        
          {
          
            
              
                
                  
                    
                      1
                      ⁢
                      
                        (
                        
                          
                            s
                            ⁡
                            
                              [
                              i
                              ]
                            
                          
                          =
                          
                            t
                            ⁡
                            
                              [
                              j
                              ]
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        
                          if
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        ≤
                        i
                        ≤
                        
                          
                            
                            s
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          and
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        ≤
                        j
                        ≤
                        
                          
                          t
                          
                        
                      
                      ,
                    
                  
                
                
                  
                    0
                  
                  
                    otherwise
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
                  L
                  k
                
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
            
            =
            
              
                
                  L
                  0
                
                ⁡
                
                  (
                  
                    i
                    ,
                    j
                  
                  )
                
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      ′
                    
                    ≤
                    
                      i
                      -
                      1
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      
                        j
                        ′
                      
                      ≤
                      
                        j
                        -
                        1
                      
                    
                  
                  ⁢
                  
                    
                      λ
                      
                        
                          (
                          
                            i
                            -
                            1
                          
                          )
                        
                        -
                        
                          i
                          ′
                        
                      
                    
                    ⁢
                    
                      λ
                      
                        
                          (
                          
                            j
                            -
                            1
                          
                          )
                        
                        -
                        
                          j
                          ′
                        
                      
                    
                    ⁢
                    
                      
                        
                          L
                          
                            k
                            -
                            1
                          
                        
                        ⁡
                        
                          (
                          
                            
                              i
                              ′
                            
                            ,
                            
                              j
                              ′
                            
                          
                          )
                        
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        63
        )
      
    
  




This can be proven by first showing by induction that

Lk(i,j)=λ−2kL0(i,j)Kk′(i−1,j−1)  (64)

for all i, j, k. (Arbitrary values can be assigned to K′k when its arguments are outside its range of definition, since L0(i,j) is zero in that case.) Clearly, Equation 64 is true for k=0 and for all i, j. Suppose it is true for some k≧0 and consider Lk+1. By Equation 63,





   
  
    
      
        
          
            
              
                
                  
                    L
                    
                      k
                      +
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      L
                      0
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          ′
                        
                        ≤
                        
                          i
                          -
                          1
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            j
                            ′
                          
                          ≤
                          
                            j
                            -
                            1
                          
                        
                      
                      ⁢
                      
                        
                          λ
                          
                            
                              (
                              
                                i
                                -
                                1
                              
                              )
                            
                            -
                            
                              i
                              ′
                            
                          
                        
                        ⁢
                        
                          λ
                          
                            
                              (
                              
                                j
                                -
                                1
                              
                              )
                            
                            -
                            
                              j
                              ′
                            
                          
                        
                        ⁢
                        
                          
                            L
                            
                              k
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              
                                i
                                ′
                              
                              ,
                              
                                j
                                ′
                              
                            
                            )
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      L
                      0
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          ′
                        
                        ≤
                        
                          i
                          -
                          1
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            j
                            ′
                          
                          ≤
                          
                            j
                            -
                            1
                          
                        
                      
                      ⁢
                      
                        λ
                        
                          i
                          -
                          1
                          -
                          
                            i
                            ′
                          
                          +
                          j
                          -
                          1
                          -
                          
                            j
                            ′
                          
                          -
                          
                            2
                            ⁢
                            k
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      L
                      0
                    
                    ⁡
                    
                      (
                      
                        
                          i
                          ′
                        
                        ,
                        
                          j
                          ′
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      K
                      k
                      ′
                    
                    ⁡
                    
                      (
                      
                        
                          
                            i
                            ′
                          
                          -
                          1
                        
                        ,
                        
                          
                            j
                            ′
                          
                          -
                          1
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    λ
                    
                      
                        
                          -
                          2
                        
                        ⁢
                        k
                      
                      -
                      2
                    
                  
                  ⁢
                  
                    
                      L
                      0
                    
                    ⁡
                    
                      (
                      
                        i
                        ,
                        j
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        K
                        
                          k
                          +
                          1
                        
                        ′
                      
                      ⁡
                      
                        (
                        
                          
                            i
                            -
                            1
                          
                          ,
                          
                            j
                            -
                            1
                          
                        
                        )
                      
                    
                    .
                  
                
              
            
          
        
      
      
        
          (
          65
          )
        
      
    
  




Finally, proving Equation 62, from the definition of Kn, it can be shown by induction that




   
  
    
      
        
          
            
              
                
                  
                    K
                    n
                  
                  ⁡
                  
                    (
                    
                      i
                      ,
                      j
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    λ
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          ′
                        
                        =
                        1
                      
                      i
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            j
                            ′
                          
                          =
                          1
                        
                        j
                      
                      ⁢
                      
                        1
                        ⁢
                        
                          (
                          
                            
                              s
                              ⁢
                              
                                
                                
                                  i
                                  ′
                                
                                
                              
                            
                            =
                            
                              t
                              ⁢
                              
                                
                                
                                  j
                                  ′
                                
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          
                            K
                            
                              n
                              -
                              1
                            
                            ′
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  i
                                  ′
                                
                                -
                                1
                              
                              ,
                              
                                
                                  j
                                  ′
                                
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    λ
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          ′
                        
                        =
                        1
                      
                      i
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            j
                            ′
                          
                          =
                          1
                        
                        j
                      
                      ⁢
                      
                        
                          
                            L
                            0
                          
                          ⁡
                          
                            (
                            
                              
                                i
                                ′
                              
                              ,
                              
                                j
                                ′
                              
                            
                            )
                          
                        
                        ⁢
                        
                          
                            K
                            
                              n
                              -
                              1
                            
                            ′
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  i
                                  ′
                                
                                -
                                1
                              
                              ,
                              
                                
                                  j
                                  ′
                                
                                -
                                1
                              
                            
                            )
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    λ
                    
                      2
                      ⁢
                      n
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        
                          i
                          ′
                        
                        =
                        1
                      
                      i
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            j
                            ′
                          
                          =
                          1
                        
                        j
                      
                      ⁢
                      
                        
                          
                            L
                            
                              n
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              
                                i
                                ′
                              
                              ,
                              
                                j
                                ′
                              
                            
                            )
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        by
                        ⁢
                        
                          
                        
                        ⁢
                        Equation
                        ⁢
                        
                          
                        
                        ⁢
                        64.
                      
                    
                  
                
              
            
          
        
      
      
        
          (
          66
          )
        
      
    
  




The above-described locational kernels and method for building on the locational kernels provide a method for defining a kernel similarity measure for general structure objects by comparing the objects centered on individual components thereof. Such kernels and methods may be applied to text documents, DNA sequences, images, time-series data, and spectra, and may be used for analyses including pattern recognition, regression estimation, novelty detection, density estimation and clustering. In applications to biological data, the kernel similarity measure can be used for microarray gene expression measurements, protein measurements and medical images for classification of diseases and disease treatment.
EXAMPLE 1
Analysis of Spectral Data
The dataset consists of a set of curves coded as vectors of length 1688 and that are described as spectra. The task is to discriminate these spectra from a learning set built of 151 spectra of class 0 and 145 spectra of class 1. Thus, the learning set has a size of 296. The generalization set is not known: it is formed only by the spectra but not by the corresponding labels.
A linear SVM has been trained on the whole learning set. The dataset is linearly separable with a margin of 1.3e-3.
In the experiments, the dataset is used as it is (referred as the “original dataset”) or it is normalized (referred as the “normalized dataset”). The normalization consists in subtracting for each spectrum its mean over its coordinates and then to divide all spectra by their own standard deviation.
To deal with this discrimination problem, a linear SVM is applied as follows:




  
    
      
        min
        
          w
          ,
          b
        
      
      ⁢
      
        
          
          w
          
        
        2
        2
      
    
    +
    
      C
      ⁢
      
        
          ∑
          
            i
            =
            1
          
          m
        
        ⁢
        
          
            ξ
            i
          
          ⁢
          
            
          
          ⁢
          subject
          ⁢
          
            
          
          ⁢
          to
          ⁢
          
            :
          
          ⁢
          
            
          
          ⁢
          
            
              y
              i
            
            ⁡
            
              (
              
                
                  〈
                  
                    w
                    ,
                    
                      x
                      i
                    
                  
                  〉
                
                +
                b
              
              )
            
          
        
      
    
  
  ≥
  
    1
    -
    
      
        ξ
        i
      
      ⁢
      
        
      
      ⁢
      and
      ⁢
      
        
      
      ⁢
      
        ξ
        i
      
    
  
  ≥
  0




where m=296 is the number of learning points, ., .is the dot product, xi are the spectra of the learning set and yi are their classes.

The protocol that was used is as follows:













1.
Do 100 times:


2.
 Draw 80% of the learning set and put it in X.


3.
 Train a linear SVM on the set X.


4.
 Compute the error on the 20% remaining points of the learning set.


5.
 Store the error in err.


6.
End of Do (1).


7.
Output the mean of err over the 100 runs.









Table 1 provides the generalized error rates for normalized and non-normalized data. The number in parenthesis represents the standard deviation.









TABLE 1





Dataset
C = +∞
C = 300







Normalized dataset
23.3% (±4.9%)
22.7% (±5.2%)


Non-Normalized dataset
23.3% (±4.9%)
25.3% (±4.8%)









The same experiments were performed with a reduced number of features (the first 543 coordinates of each spectra and the coordinates 1177 to 1500, selected by taking the regions of the graphs that were most different between the two classes 1 and 0), resulting in a generalization error of 22.7% (±5%) with C=∞ on the non-normalized learning set. (The same experiment was not performed with the normalized dataset.)
Another experiment involved splitting the original learning set with all the coordinates into three pieces. The first one (148 spectra) was the set used to define the linear SVM. The second one (59) was used to find the hyperparameter C and the third one (89) was used to test the SVM. The error obtained with a linear SVM on the last set (non-normalized) is 22.5% with C=200. The same experiments with the normalized dataset generated a generalization error of 23.6% with C=20.
Two additional methods were tested:
1) The 1-Nearest Neighbor (“1NN”) classifier: the output of this system on a new spectrum is computed by taking the class of the nearest spectrum in the learning set.
2) The Least Min Square (“LMS”) classifier: linear whose parameters are obtained with the following optimization problem:




  
    min
    
      w
      ,
      b
    
  
  ⁢
  
    
      ∑
      
        i
        =
        1
      
      m
    
    ⁢
    
      
        
          (
          
            
              〈
              
                w
                ,
                x
              
              〉
            
            +
            b
            -
            
              y
              i
            
          
          )
        
        2
      
      .
    
  




The experiments are the same as for the linear SVM. Table 2 shows that both methods have a greater generalization error than the linear SVM. The number in parenthesis represents the standard deviation.









TABLE 2





Dataset
I-NN
LMS







Normalized dataset
23.5% (±5.2%)
27.2% (±5.7%)


Non-Normalized dataset
35.8% (±6.3%)
28.1% (±9.l%)









The preliminary results support the fact that this problem may not be solved easily despite the fact that the problem is linearly separable. Non-linear SVM with accurate kernels should be used and a preprocessing step could also be useful. These two steps in the process of setting up a good system require some knowledge on the dataset. It is apparent that the linear system still learns something from this dataset since its error rate is less than 50% (the best solution is 22.5% error rate).
Table 3 sums up the results of the preceding experiments with the results in increasing order. “Norm” indicates that the normalized dataset was used, while “Non-norm”indicates the non-normalized data. “One-run means one run of a SVM whose coefficient C has been defined on a validation set. The latter are presented in decreasing order.










TABLE 3







Description
Generalization error (std)









SVM (C = 200, Non-Norm, one run)
22.5%



SVM (C = 20, Norm, one run)
23.6%



SVM (C = 300, Norm)
22.7% (±5.2%)



SVM (C = ∞, Norm)
23.3% (±4.9%)



SVM (C = ∞, Non-Norm)
23.3% (±4.9%)



1-NN (Norm)
23.5% (±5.2%)



SVM (C = 300, Non-Norm)
25.3% (±4.8%)



LMS (Norm)
27.2% (±5.7%)



LMS (Non-Norm)
28.1% (±9.1%)



1-NN (Non-Norm)
35.8% (±6.3%)










Next, experiments were performed with non-linear SVMs on the same dataset as in the previous experiments, where the data consists of mid IR spectra of biological samples at 4 wave number resolution. During data collection, care was taken to avoid instrument drifting between the measurements of the two classes. The data are properly labeled and various linear discriminants, including LDA and “partial least square” were used directly on the spectrum used as a feature vector. A feature selection technique was also used.
The kernels used for this series of experiments are the following:




  
    k
    ⁡
    
      (
      
        f
        ,
        g
      
      )
    
  
  =
  
    ∫
    
      
        f
        ⁡
        
          (
          x
          )
        
      
      ⁢
      
        g
        ⁡
        
          (
          
            x
            -
            t
          
          )
        
      
      ⁢
      
        exp
        ⁡
        
          (
          
            -
            
              
                t
                2
              
              σ
            
          
          )
        
      
      ⁢
      
        d
        x
      
      ⁢
      
        d
        t
      
    
  




where f and g are spectra. The value of σ controls the smoothness of the representation. Experiments were performed for both normalized and non-normalized data.










TABLE 4





Value of σ
Generalization error (std)
Learning error (std)

















2.5
32.1% (±6.3%)
23.2% (±5.0%)


5
36.3% (±5.5%)
35.8% (±3.7%)


7.5
38.7% (±11.5%)
35.5% (±10.6%)


10
38.8% (±7.6%)
34.7% (±6.2%)









The results displayed in Table 4 show that such a kernel does not provide good systems for normalized and non-normalized data. The generalization error is always greater than for linear systems. Smoothness may then appear as a bad property for this particular problem. The fact the learning error is not zero is strong evidence that the smooth representation induced by such kernel is not adapted for this discrimination task. The same experiments were run with a very small σ1, producing the same result as for a linear SVM.
Next, the same protocol as for linear SVM previously described was used, but with polynomial kernels:




  
    K
    ⁡
    
      (
      
        f
        ,
        g
      
      )
    
  
  =
  
    
      (
      
        
          
            〈
            
              f
              ,
              g
            
            〉
          
          1688
        
        +
        1
      
      )
    
    d
  




where d is the degree of the polynomial. All experiments were performed with C=∞. The normalization factor 1/1688 has been used in other experimental settings and has been found to be useful.

The results shown in Tables 5 and 6 are that the learning error for polynomial of degree 1 with C=∞ is not zero. This unusual result may be related to numerical anomalies in the implementation of the SVM software. The best result obtained here is a generalization error of 21.2% with the polynomial of degree 2 such that the improvement from the linear model is not significant.









TABLE 5





Value of d
Generalization error (std)
Learning error (std)







1
22.9% (±5.4%)
4.6% (±4.7%)


2
21.2% (±4.7%)
0.3% (±0.3%)


3
22.0% (±5.2%)
  0% (±0%)


4
23.4% (±5.1%)
  0% (±0%)


5
23.4% (±5.0%)
  0% (±0%)


















TABLE 6





Value of d
Generalization error (std)
Learning error (std)







1
23.5% (±5.2%)
4.2% (±0.9%)


2
20.8% (±4.7%)
0.3% (±0.3%)


3
22.6% (±4.7%)
  0% (±0%)


4
22.9% (±5.6%)
  0% (±0%)


5
22.3% (±4.7%)
  0% (±0%)









Next, Gaussian kernels were used with C=∞. The kernel used is as follows:




  
    k
    ⁡
    
      (
      
        f
        ,
        g
      
      )
    
  
  =
  
    exp
    ⁡
    
      (
      
        
          
            
            
              f
              -
              g
            
            
          
          2
          2
        
        
          1688
          ⁢
          
            σ
            2
          
        
      
      )
    
  




where σ is the standard deviation of the kernel. Results of this experiment are presented in Tables 7 and 8. In order to ensure that an accurate value for σ was used, a SVM was run repeatedly for sigma between 0.1 to 10 at increments of 0.1. The best value for σ was obtained for σ=1. Nonetheless, the improvement from the linear model is not significant.










TABLE 7





Value of σ
Generalization error (std)
Learning error (std)

















0.0001
52.9% (±4.4%)
  0% (±0%)


0.001
52.1% (±5.1%)
  0% (±0%)


0.01
46.7% (±5.5%)
  0% (±0%)


0.1
23.3% (±5.5%)
  0% (±0%)


1
21.4% (±5.4%)
  0% (±0%)


10
22.6% (±5.6%)
6.5% (±0.9%)


















TABLE 8





Value of σ
Generalization error (std)
Learning error (std)

















0.0001
53.6% (±4.2%)
  0% (±0%)


0.001
51.2% (±5.0%)
  0% (±0%)


0.01
45.0% (±5.7%)
  0% (±0%)


0.1
28.2% (±5.9%)
  0% (±0%)


1
21.1% (±5.5%)
  0% (±0%)


10
28.7% (±5.2%)
17.0% (±1.5%)









According to the nature of the learning problem, it can be useful to use kernels for distribution since a spectrum can be interpreted as the distribution of frequencies. To exploit this approach, a χ2 kernel was built as follows:




  
    
      k
      χ
    
    ⁡
    
      (
      
        f
        ,
        g
      
      )
    
  
  =
  
    
      
        exp
        ⁡
        
          (
          
            -
            
              
                χ
                2
              
              ⁡
              
                (
                
                  f
                  ,
                  g
                
                )
              
            
          
          )
        
      
      ⁢
      
        
      
      ⁢
      where
      ⁢
      
        
      
      ⁢
      
        
          χ
          2
        
        ⁡
        
          (
          
            f
            ,
            g
          
          )
        
      
    
    =
    
      
        ∑
        
          i
          =
          1
        
        1688
      
      ⁢
      
        
          
            (
            
              
                f
                i
              
              -
              
                g
                i
              
            
            )
          
          2
        
        
          
            f
            i
          
          +
          
            g
            i
          
        
      
    
  




Choosing C=∞ and using the same protocol as before, the result is presented in Table 9.











TABLE 9







Generalization error (std)
Learning error (std)


















x2 kernel (normalized
   % (±0 %)
  0% (± 0%)


data)


x2 kernel (non-
40.7% (±6.5%)
2.1% (±9.5%)


normalized data)









These results are poor, showing that this kernel, which is not proved to be a Mercer kernel, is not adapted for this task. Many times, the optimization procedure has not converged which indicates that the kernel is not only unacceptable from a learning point of view but also from a computational one.
In these experiments, the representation of the data is changed in which each point of the original curves is replaced by the parameters of a quadratic polynomial that fits the data in a window width 11 centered at the point. More formally, let f be a spectrum, and let fi be one of its features. Consider the vector v=(vi=fi−5, . . . , v6=fi, . . . , v11=fi+5) and compute the coefficients of the polynomial aix2+bix+ci such that:




  
    ∑
    
      j
      =
      1
    
    11
  
  ⁢
  
    
      (
      
        
          
            
              a
              i
            
            ⁡
            
              (
              
                j
                11
              
              )
            
          
          2
        
        +
        
          
            b
            i
          
          ⁡
          
            (
            
              j
              11
            
            )
          
        
        +
        
          c
          i
        
        -
        
          v
          j
        
      
      )
    
    2
  




is minimum. Then, the spectrum is coded as the sequence (ai, bi, ci, ei) where





  
    e
    i
  
  =
  
    
      
        
          a
          i
        
        ⁡
        
          (
          
            6
            11
          
          )
        
      
      2
    
    +
    
      
        b
        i
      
      ⁡
      
        (
        
          6
          11
        
        )
      
    
    +
    
      c
      i
    
    -
    
      f
      i
    
  




is the error of the polynomial approximation for x=i which corresponds to v6=fi. The result of the experiments are described in Table 10. The best result so far is obtained with this representation (19.9% of generalization error). However, this is not significantly different from the result obtained with the same kernel on the normalized dataset (20.8%).











TABLE 10







Kernel
Generalization error (std)









Linear
26.7% (±5.1%)



Polynomial (d =2)
22.5% (±5.3%)



Polynomial (d =3)
20.9% (±5.4%)



Polynomial (d =4)
20.3% (±5.7%)



Polynomial (d =5)
19.9% (±5.1%)



Polynomial (d =6)
20.3% (±5.3%)



Polynomial (d =7)
20.2% (±4.5%)



Polynomial (d =8)
20.1% (±4.4)



Polynomial (d =9)
19.9% (±5.0%)










Because the above-described method are still far away from the 2% desired error, improved models are explored.
First, each spectrum is coded by the locations of the different maximum. This approach is motivated by the way physicists use a spectrum to analyze the physical properties of a molecule.
Second, data cleaning is used to remove many outliers that make it difficult to generalize.
In the first method, the spectra is preprocessed in order to compute the location of their maximum/minimum. Each point in a spectrum that is identified as a maximum (resp. minimum) is affected with a value of +1 (resp. −1). All the other points are set to zero. The hope is that this representation codes the number of maximum/minimum as well as their locations. The latter are usually used by physicists to study such spectra. FIG. 6 illustrates an example of one spectrum, showing the original curve and its new representation as a {0,±1}-vector, which are represented as ±1 spikes. Note that there are more extrema coded than can be seen on the curve This bias is added to avoid loss of information by imposing overly strict conditions for maximum/minimum. Also the last 166 features were removed since they were mainly noise.
The polynomial kernels are normalized by 10, so that:




  
    k
    ⁡
    
      (
      
        f
        ,
        g
      
      )
    
  
  =
  
    
      (
      
        
          
            〈
            
              f
              ,
              g
            
            〉
          
          10
        
        +
        1
      
      )
    
    d
  




Table 11 provides the results for leave-one-out error for different polynomial kernels with C=∞.










TABLE 11







Value of the degree d
Leave-one-out error









1
31.4%



2
29.4%



3
27.4%



4
28.0%



5
30.1%










The normalization number has been chosen in terms of f,g that is of the order of 10. To compute the maximum and minimum of the spectra, the derivatives of the spectra were first computed, then the position of the zero of these derivatives was identified. The computation of the derivative was done by taking a window of width 5 and by fitting a linear model with bias on this window. The linear part was identified as being the derivative of the spectrum for the point corresponding to the center of the window. The zero elements of the derivative of the spectrum were identified simply by looking at the sign changes.
The results do not provide improvement over the results of earlier tests. However, this does not mean that this kind of approach is not adapted to this problem. Rather, there are many parameters that must be set up in these experiments, including the manners in which the derivative is computed and the extrema are located. Both parameters depend on hand-tuned parameters and can influence the final performance of the model.
Since all of the foregoing models gave roughly the same results for different experimental settings, it appeared that some of the data in the learning set may have been improperly labeled. In order to remove these data, the following data cleaning strategy was used:













1.
Do 100 times


2.
 Define a random partition of the learning set into two sets: X(80%



 of the data) and Xt (20% of the data).


3.
 Do 100 times,


4.
   Define a random partition of X into Xl (60% of the data in



   X), and Xv (40% of the data in X).


5.
   Learn on Xl.


6.
   Store the error on Xv as a matrix E:



     E(i, j) = { Validation error of the ith run if j is in Xl


7.
  End For   0 Otherwise


8.
  Compute the mean error = mean (column-wise) of the matrix E.


9.
  Order the elements in X with increasing mean error.


10.
  Learn the model on the first 80% elements of the ordered set X.


11.
  Store the error of the model on Xt in vector err.


12.
End Do


13.
Output the mean and the standard deviation of err.









The data cleaning step removed 20% of the elements in the sets X (which corresponds to 35 data) before learning the final model. The number 20% was chosen arbitrarily. The experiments were performed with the quadratic representation since it produced the best results thus far. A polynomial kernel of degree 2 was selected, and C=∞ as before. Table 12 summarizes the test error results.










TABLE 12







Value of the degree d
Generalization error (std)









1
25.0% (±5.6%)



2
21.3% (±4.4%)



3
21.1% (±4.7%)










Once again, the results do not show improvement over earlier methods. However, removing 20% of data from the set X may have been too much and better results could be obtained by removing a smaller percentage of the data. To test this assertion, the same experiment was performed by removing from one to seven data in X. using a linear SVM only. The improvement is not significant.
Rather than learning with all the features, an experiment was performed where seven SVMs trained on different subsets of the features were combined. Actually, seven SVMs were trained on seven segments of the input space. The first SVM concerns the first 251 features, the second concerns the features at position 252-502. The results included a generalization error of 27.5% (±5%) on the original dataset.
A bagging method was applied on the quadratic representation. Fifty splits of the data were computed into a training set of size 251 and a test sets 45. Then, a bagging of 30 SVMs was performed by drawing with replacements 251 points from the training sets according to the uniform distribution over the learning set. For the test set, there was an estimated generalization error of 18% (±5.9%) which represents a slight improvement compared to what has been presented so far.
A different way bagging SVM was tried in which, rather than taking the sign of each single SVM before bagging them, the real output was considered and combined by taking the sign of the sum of their outputs. Also, each single SVM was trained on a set of size 213. The results are shown in Table 13.










TABLE 13







Representation
Generalization error (std)









Quadratic (C =∞)
  20% (±5.2%)



Fourier (C =∞)
22.4% (±6.7%)










Finally, the data was split into a learning set (85% of the data) and a test set. Thirty runs of a polynomial SVM (degree 2) were performed on a subset of the training set (80% elements were drawn without replacement in the learning set). The best node was identified by computing its error on the whole learning set, resulting in a generalization error of 19% (±4.2%).
Noise and Invariance. The selection of a kernel corresponds to the choice of representation of the data in a feature space and, to improve performance, it should therefore incorporate prior knowledge such as known transformation invariances or noise distribution. In a second embodiment of kernel selection according to the present invention, an invariance or noise is represented as vectors in a feature space. The noise vectors are taken into account in constructing invariant support vector classifiers to obtain a decision function that is invariant with respect to the noise.
In certain classification tasks, there is a priori knowledge about the invariances related to the task. For example, in image classification, it is known that the label of a given image should not change after a small translation or rotation.
Generally, a local transformation Lt depending on a parameter t (for instance, a vertical translation of t pixels) is assumed such that any point x should be considered equivalent to Ltx, the transformed point. Ideally, the output of the learned function should be constant when its inputs are transformed by the desired invariance.
It has been shown that one cannot find a non-trivial kernel which is globally invariant (see C. J. C. Burges, “Geometry and invariance in kernel based methods”, B. Schölkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods—Support Vector Learning. MIT Press, 1999.0) For this reason, local invariances are considered for which each training point xi is associated with a tangent vector dxi,




   
  
    
      
        
          
            
              
                
                  d
                  
                    x
                    i
                  
                
                =
                
                ⁢
                
                  
                    lim
                    
                      t
                      ->
                      0
                    
                  
                  ⁢
                  
                    
                      1
                      t
                    
                    ⁢
                    
                      (
                      
                        
                          
                            L
                            t
                          
                          ⁢
                          
                            x
                            i
                          
                        
                        -
                        
                          x
                          i
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    ∂
                    
                      ∂
                      t
                    
                  
                  ⁢
                  
                    ❘
                    
                      t
                      =
                      0
                    
                  
                  ⁢
                  
                    
                      L
                      t
                    
                    ⁢
                    
                      x
                      i
                    
                  
                
              
            
          
        
      
      
        
          (
          67
          )
        
      
    
  




In practice, dxi can be either computed by finite difference or by differentiation. Note that generally one can consider more than one invariance transformation. The multi invariance case is usually a straightforward extension of the single one.
A common method for introducing invariances in a learning system is to add the perturbed examples Ltxi in the training set (see P. Niyogi, T. Poggio, and F. Girosi, “Incorporating prior information in machine learning by creating virtual examples”, IEEE Proceedings on Intelligent Signal Processing, 86(11):2196-2209, November 1998). These points are often labeled “virtual examples”. In the SVM (support vector machine) framework, when applied only to the SVs (support vectors), it leads to the “Virtual Support Vector” (“VSV”) method. See, e.g., B. Schölkopf, C. Burges, and V. Vapnik. “Incorporating invariances in support vector learning machines”, Artificial Neural Networks—ICANN'96, volume 1112, pages 47-52, Berlin, 1996. Springer Lecture Notes in Computer Science.
An alternative to the foregoing is to directly modify the cost function in order to take into account the tangent vectors. This has been successfully applied to neural networks (P. Simard, Y. LeCun, J. Denker, and B. Victorri. “Transformation invariance in pattern recognition, tangent distance and tangent propagation”, G. Orr and K. Müller, editors, Neural Networks: Tricks of the Trade. Springer, 1998.) and linear Support Vector Machines (B. Schölkopf, P. Y. Simard, A. J. Smola, and V. N. Vapnik. Prior knowledge in support vector kernels. In MIT Press, editor, NIPS, volume 10, 1998), both of which are incorporated herein by reference. According to the present embodiment of the invention, such methods are extended to the case of nonlinear SVMs, primarily by using kernel PCA (principal components analysis) methods. See, e.g., B. Schölkopf, A. Smola, and K. R. Müller. Nonlinear component analysis as a kernel eigenvatue problem. Neural Computation, 10:1299-1310, 1998, which is incorporated herein by reference.
The following description provides a few of the standard notations used for SVMs. Let {(xiyi)}1≦i≦n be a set of training examples, xi{dot over (ε)}X, belonging to classes labeled by yi{dot over (ε)}{−1,1}. The domain X is often, but not necessarily, taken to be  d{dot over (ε)} In kernel methods, these vectors are mapped into a feature space using a kernel function K(xi,xj) that defines an inner product in this feature space. The decision function given by an SVM is the maximal margin hyperplane in this space,




  
    
      
        
          
            g
            ⁡
            
              (
              x
              )
            
          
          =
          
            sign
            ⁡
            
              (
              
                f
                ⁡
                
                  (
                  x
                  )
                
              
              )
            
          
        
        ,
        
          
            where
            ⁢
            
              
            
            ⁢
            
              f
              ⁡
              
                (
                x
                )
              
            
          
          =
          
            
              (
              
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    n
                  
                  ⁢
                  
                    
                      α
                      i
                      0
                    
                    ⁢
                    
                      y
                      i
                    
                    ⁢
                    
                      K
                      ⁡
                      
                        (
                        
                          
                            x
                            i
                          
                          ,
                          x
                        
                        )
                      
                    
                  
                
                +
                b
              
              )
            
            .
          
        
      
    
    
      
        (
        68
        )
      
    
  




The coefficients αi0 are obtained by maximizing the functional





  
    
      
        
          W
          ⁡
          
            (
            α
            )
          
        
        -
        
          
            ∑
            
              i
              =
              l
            
            n
          
          ⁢
          
            α
            i
          
        
        -
        
          
            1
            2
          
          ⁢
          
            
              ∑
              
                i
                ,
                
                  j
                  =
                  1
                
              
              n
            
            ⁢
            
              
                α
                i
              
              ⁢
              
                α
                j
              
              ⁢
              
                y
                i
              
              ⁢
              
                y
                j
              
              ⁢
              
                K
                ⁡
                
                  (
                  
                    
                      x
                      i
                    
                    ,
                    
                      x
                      j
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        69
        )
      
    
  




under the constraints





  
    
      ∑
      
        i
        =
        1
      
      n
    
    ⁢
    
      
        α
        i
      
      ⁢
      
        y
        i
      
    
  
  =
  
    
      0
      ⁢
      
        
      
      ⁢
      and
      ⁢
      
        
      
      ⁢
      
        α
        i
      
    
    ≥
    0.
  




This formulation of the SVM optimization problem is referred to as the “hard margin” formulation since no training errors are allowed. Every training point satisfies the inequality yif(xi)≧1; points xi with αi>0 satisfy it as an equality. These points are called “support vectors.”
For the non-separable case, training errors must be permitted, which results in the so-called “soft margin” SVM algorithm (see C. Cortes and V. Vapnik, “Support vector networks”, Machine Learning, 20:273-297, 1995). It can be shown that soft margin SVMs with quadratic penalization of errors can be considered as a special case of the hard margin version with the modified kernel (N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines. Cambridge University Press, 2000).




  
    
      
        
          K
          ←
          
            K
            +
            
              
                1
                C
              
              ⁢
              I
            
          
        
        ,
      
    
    
      
        (
        70
        )
      
    
  




where I is the identity matrix and C a parameter penalizing the training errors. However, the primary focus of the present invention is on the hard margin SVM.

For linear SVMs, one wishes to find a hyperplane whose normal vector w is as orthogonal as possible to the tangent vectors. This can be readily understood from the equality

f(xi+dxi)−f(xi)=w·dxi.

For this purpose, it has been suggested to minimize the functional




  
    
      
        
          
            (
            
              1
              -
              γ
            
            )
          
          ⁢
          
            w
            2
          
        
        +
        
          γ
          ⁢
          
            
              ∑
              
                i
                =
                1
              
              n
            
            ⁢
            
              
                (
                
                  w
                  ·
                  
                    d
                    
                      x
                      i
                    
                  
                
                )
              
              2
            
          
        
      
    
    
      
        (
        71
        )
      
    
  




subject to the constraints

yi(w·xi+b)≧1.

The parameter γ trades off between normal SVM training (γ=0) and full enforcement of the orthogonality between the hyperplane and the invariance directions (γ→1).

The square root of the regularized covariance matrix of the tangent vectors is:




  
    
      
        
          
            C
            γ
          
          =
          
            
              (
              
                
                  
                    (
                    
                      1
                      -
                      γ
                    
                    )
                  
                  ⁢
                  I
                
                +
                
                  γ
                  ⁢
                  
                    
                      ∑
                      i
                    
                    ⁢
                    
                      
                        d
                        
                          x
                          i
                        
                      
                      ⁢
                      
                        d
                        
                          x
                          i
                          T
                        
                      
                    
                  
                
              
              )
            
            
              1
              /
              2
            
          
        
        ,
      
    
    
      
        (
        72
        )
      
    
  




where T indicates the transpose.

Equation 71 then reads

wTCγ2w.

Introducing {tilde over (w)}=Cγw and {tilde over (x)}i=Cγ−1xi, the equivalent optimization problem is:




  
    
      
        
          min
          
            w
            ~
          
        
        ⁢
        
          
            w
            ~
          
          2
        
      
    
    
      
        (
        73
        )
      
    
  




under constraints

yi({tilde over (w)}·{tilde over (x)}i+b)≧1.

The preceding makes use of the equality Cγ−1w·xi=w·Cγ−1xi, which is valid because Cγ is symmetric. Note also that Cγ is strictly positive definite (and thus invertible) if γ<1. For this reason, for the remainder of this description, it will be assumed that γ<1.
The optimization problem of Equation (73) is the standard SVM situation where the training points xi have been linearly preprocessed using the matrix Cγ−1.
The output value of the decision function on a test point is:




   
  
    
      
        
          
            
              
                
                  f
                  ⁡
                  
                    (
                    x
                    )
                  
                
                =
                
                ⁢
                
                  
                    w
                    ·
                    x
                  
                  +
                  b
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        w
                        ~
                      
                      ·
                      
                        C
                        γ
                        
                          -
                          1
                        
                      
                    
                    ⁢
                    x
                  
                  +
                  b
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      ∑
                      
                        i
                        =
                        1
                      
                      n
                    
                    ⁢
                    
                      
                        α
                        i
                      
                      ⁢
                      
                        y
                        i
                      
                      ⁢
                      
                        C
                        γ
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        
                          x
                          i
                        
                        ·
                        
                          C
                          γ
                          
                            -
                            1
                          
                        
                      
                      ⁢
                      x
                    
                  
                  +
                  b
                
              
            
          
        
      
      
        
          (
          74
          )
        
      
    
  




The standard formulation of an SVM output is also obtained, however, the test point is first multiplied by Cγ−1.
To conclude, it can be said that a linear invariant SVM is equivalent to a standard SVM where the input space has been transformed through the linear mapping

x→Cγ−1x  (75)

It has been shown that this method leads to significant improvements in linear SVMs, and to small improvements when used as a linear preprocessing step in nonlinear SVMs. The latter, however, was a hybrid system with unclear theoretical foundations. The following description provides a method for dealing with the nonlinear case in a principled way.
In the nonlinear case, the data are first mapped into a high-dimensional feature space where a linear decision boundary is computed. To extend the previous analysis to the nonlinear case, one would need to compute the matrix Cγ in feature space,




  
    
      
        
          C
          γ
        
        =
        
          (
          
            
              
                (
                
                  1
                  -
                  γ
                
                )
              
              ⁢
              I
            
            +
            
              γ
              ⁢
              
                
                  ∑
                  i
                
                ⁢
                
                  
                    d
                    
                      
                    
                    ⁢
                    
                      Φ
                      ⁡
                      
                        (
                        
                          x
                          i
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    
                      
                    
                    ⁢
                    
                      
                        Φ
                        ⁡
                        
                          (
                          
                            x
                            i
                          
                          )
                        
                      
                      T
                    
                  
                
              
            
          
          )
        
      
    
    
      
        (
        76
        )
      
    
  




and the new kernel function

{tilde over (K)}(x,y)=Cγ−1Φ(x)·Cγ−1(y)=Φ(x)TCγ−2Φ(y)  (77)

However, due to the high dimension of the feature space, it is not practical to do it directly. There are two different techniques for overcoming this difficulty, however, the “kernel PCA map” must first be introduced.
Even in the case of a high dimensional feature space H, a training set (x1, . . . , xn) of size n when mapped to this feature space spans a subspace E⊂H . whose dimension is at most n. The “kernel PCA map”, described by Schölkopf, Smola and Müller (supra) and extended in K. Tsuda, “Support vector classifier with asymmetric kernel function”, In M. Verleysen, editor, Proceedings of ESANN'99, pages 183-188, 1999, incorporated herein by reference, makes use of this idea.
Let (v1, . . . , vn)εEn be an orthonormal basis of E with each vi being a principal axis of {Φ(x1), . . . , Φ(xn)). The kernel PCA map ψ: X→Rn is defined coordinatewise as

ψp(x)=Φ(xi)·vp, 1≦p≦n  (78)

Note that by definition, for all i and j, Φ(xi) and Φ(xj) lie in E and thus

K(xi,xj)=Φ(xi)·Φ(xj)=ψ(xi)·ψ(xj).  (79)

Equation 79 reflects the fact that if all principal components are retained, kernel PCA is merely a basis transform in E, leaving the dot product of training points invariant.

In the decomposition of the principal directions (v1, . . . , vn), for 1≦p≦n, vp can be written as




  
    
      
        
          v
          p
        
        =
        
          
            ∑
            
              i
              =
              1
            
            n
          
          ⁢
          
            
              a
              
                i
                ⁢
                
                  
                
                ⁢
                p
              
            
            ⁢
            
              
                Φ
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        80
        )
      
    
  




The fact that (v1, . . . , vn) are the principal directions in feature space of the training set means that they are eigenvectors of the covariance matrix,

Cvp=λpvp, 1≦p≦n  (81)

where





  
    
      
        C
        =
        
          
            ∑
            
              i
              =
              1
            
            n
          
          ⁢
          
            
              Φ
              ⁡
              
                (
                
                  x
                  i
                
                )
              
            
            ⁢
            
              
                
                  Φ
                  ⁡
                  
                    (
                    
                      x
                      i
                    
                    )
                  
                
                T
              
              .
            
          
        
      
    
    
      
        (
        82
        )
      
    
  




Note that it is assumed that the data are centered in feature space. It is actually always possible to center the data. See, e.g., Schölkopf, Smola and Müller (supra) for details on how this may be accomplished.

Combining Equations (80), (81) and (82), and multiplying the left side by φ(xk)T, yields




  
    
      
        
          
            
              
                Φ
                ⁡
                
                  (
                  
                    x
                    k
                  
                  )
                
              
              T
            
            ⁢
            
              (
              
                
                  ∑
                  
                    i
                    =
                    1
                  
                  n
                
                ⁢
                
                  
                    Φ
                    ⁡
                    
                      (
                      
                        x
                        i
                      
                      )
                    
                  
                  ⁢
                  
                    
                      Φ
                      ⁡
                      
                        (
                        
                          x
                          i
                        
                        )
                      
                    
                    T
                  
                
              
              )
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  1
                
                n
              
              ⁢
              
                
                  a
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    p
                  
                
                ⁢
                
                  Φ
                  ⁡
                  
                    (
                    
                      x
                      j
                    
                    )
                  
                
              
            
          
          =
          
            
              λ
              p
            
            ⁢
            
              
                Φ
                ⁡
                
                  (
                  
                    x
                    k
                  
                  )
                
              
              T
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  1
                
                n
              
              ⁢
              
                
                  a
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    p
                  
                
                ⁢
                
                  Φ
                  ⁡
                  
                    (
                    
                      x
                      j
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        83
        )
      
    
  




which can be written as





  
    
      
        
          
            
              ∑
              
                i
                ,
                
                  j
                  =
                  1
                
              
              n
            
            ⁢
            
              
                K
                ⁡
                
                  (
                  
                    
                      x
                      i
                    
                    ,
                    
                      x
                      k
                    
                  
                  )
                
              
              ⁢
              
                K
                ⁡
                
                  (
                  
                    
                      x
                      i
                    
                    ,
                    
                      x
                      j
                    
                  
                  )
                
              
              ⁢
              
                a
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  p
                
              
            
          
          =
          
            
              λ
              p
            
            ⁢
            
              
                ∑
                
                  j
                  -
                  1
                
                n
              
              ⁢
              
                
                  a
                  
                    jp
                    ⁢
                    
                      
                    
                  
                
                ⁢
                
                  K
                  ⁡
                  
                    (
                    
                      
                        x
                        k
                      
                      ,
                      
                        x
                        j
                      
                    
                    )
                  
                
              
            
          
        
        ,
        
          1
          ≤
          p
        
        ,
        
          k
          ≤
          
            n
            .
          
        
      
    
    
      
        (
        84
        )
      
    
  




In matrix notation, this last equality reads

K2ap=λpap 

which is satisfied whenever

Kap=λpKap.  (85)

Thus, if ap is an eigenvector of the kernel matrix K, its components can be used in the Equation 80. Note that the eigenvalues of K are the same as those in the covariance matrix of Equation 82.
Enforcing the constraint that vp has unit length, Equations 80 and 85 yield

vp2=apTKap=λpap2.  (86)

Writing the eigendecomposition of K as

K=UΛUT,  (87)

with U being an orthonormal matrix and Λ a diagonal matrix, aip=Uip/√{square root over (λp,)} so the kernel PCA map reads

ψ(x)=Λ−1/2UTk(x)  (88)

where

k(x)=(K(x,x1), . . . , K(x,xn))T.  (89)

In order to be able to compute the new kernel according to Equation 77, the matrix Cγ (Equation 76) is diagonalized using the same approach as that described in the preceding section. In that section, it was shown to diagonalize the feature space covariance matrix in the PCA subspace (spanned by a training sample) by computing the eigendecomposition of the Gram matrix of those points. Now, rather than having a set of training points {Φ(xi)}, there is given a set of tangent vectors {dΦ(xi)} and a tangent covariance matrix




  
    
      
        
          C
          t
        
        =
        
          
            ∑
            
              i
              =
              1
            
            n
          
          ⁢
          
            
              d
              
                Φ
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                d
                
                  
                    Φ
                    ⁡
                    
                      (
                      
                        x
                        i
                      
                      )
                    
                  
                  T
                
              
              .
            
          
        
      
    
    
      
        (
        90
        )
      
    
  




Considering the Gram matrix Kt of the tangent vectors:




   
  
    
      
        
          
            
              
                
                  K
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    j
                  
                  t
                
                =
                
                ⁢
                
                  
                    d
                    
                      Φ
                      ⁡
                      
                        (
                        
                          x
                          i
                        
                        )
                      
                    
                  
                  ·
                  
                    d
                    
                      Φ
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    K
                    ⁡
                    
                      (
                      
                        
                          
                            x
                            i
                          
                          +
                          
                            d
                            
                              x
                              i
                            
                          
                        
                        ,
                        
                          
                            x
                            j
                          
                          +
                          
                            d
                            
                              x
                              j
                            
                          
                        
                      
                      )
                    
                  
                  -
                  
                    K
                    ⁡
                    
                      (
                      
                        
                          
                            x
                            i
                          
                          +
                          
                            d
                            
                              x
                              i
                            
                          
                        
                        ,
                        
                          x
                          j
                        
                      
                      )
                    
                  
                  -
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    K
                    ⁡
                    
                      (
                      
                        
                          x
                          i
                        
                        ,
                        
                          
                            x
                            j
                          
                          +
                          
                            d
                            
                              x
                              j
                            
                          
                        
                      
                      )
                    
                  
                  +
                  
                    K
                    ⁡
                    
                      (
                      
                        
                          x
                          i
                        
                        ,
                        
                          x
                          j
                        
                      
                      )
                    
                  
                
              
            
          
        
      
      
        
          (
          91
          )
        
      
    
    
      
        
          
            
          
          ⁢
          
            =
            
              
                d
                
                  x
                  i
                  T
                
              
              ⁢
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    K
                    ⁡
                    
                      (
                      
                        
                          x
                          i
                        
                        ,
                        
                          x
                          j
                        
                      
                      )
                    
                  
                
                
                  
                    ∂
                    
                      x
                      i
                    
                  
                  ⁢
                  
                    ∂
                    
                      x
                      j
                    
                  
                
              
              ⁢
              
                d
                
                  x
                  j
                
              
            
          
        
      
      
        
          (
          92
          )
        
      
    
  




This matrix Kt can be computed either by finite differences (Equation 91) or with the analytical derivative expression given by Equation 92. Note that for a linear kernel, K(x, y)=xTy, and (17) reads Kijt=dxiTdxj, which is a standard dot product between the tangent vectors.
As in the foregoing description of the kernel PCA map, the eigendecomposition of Kl is written as Kt=UΛUT, which allows the expansion of the eigenvectors (v1, . . . , vn) of Ct corresponding to non-zero eigenvalues to be found.




  
    
      
        
          v
          p
        
        =
        
          
            1
            
              
                λ
                p
              
            
          
          ⁢
          
            
              ∑
              
                i
                =
                1
              
              n
            
            ⁢
            
              
                U
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  p
                
              
              ⁢
              
                
                  d
                  
                    Φ
                    ⁡
                    
                      (
                      
                        x
                        i
                      
                      )
                    
                  
                
                .
              
            
          
        
      
    
    
      
        (
        93
        )
      
    
  




The orthonormal family (v1, . . . , vn) to an orthonormal basis of H, (V1, . . . , vn, vn+1, . . . , vdH) is completed, with dH=dim H≦∞.
In this basis, Ct is diagonal

Ct=VD(λ1, . . . , λn, 0, . . . , 0)VT,

where V is an orthonormal matrix whose columns are the vi. Referring back to Equation 77 the matrix of interest is Cγ−2




  
    
      
        
          
            C
            γ
            
              -
              2
            
          
          =
          
            V
            ⁢
            
              
            
            ⁢
            
              D
              ⁡
              
                (
                
                  
                    1
                    
                      
                        γλ
                        1
                      
                      +
                      1
                      -
                      γ
                    
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    1
                    
                      
                        γλ
                        n
                      
                      +
                      1
                      -
                      γ
                    
                  
                  ,
                  
                    1
                    
                      1
                      -
                      γ
                    
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    1
                    
                      1
                      -
                      γ
                    
                  
                
                )
              
            
            ⁢
            
              V
              T
            
          
        
        ,
      
    
    
      
        (
        94
        )
      
    
  




To be able to compute the new kernel function according to Equation 77, the projection of a point in feature space Φ(xi) onto one of the vectors vp must be determined. From Equation 93, we have, for 1≦p≦n,




  
    
      
        
          
            
              
                
                  Φ
                  ⁡
                  
                    (
                    x
                    )
                  
                
                ·
                
                  v
                  p
                
              
              =
              
              ⁢
              
                
                  1
                  
                    
                      λ
                      p
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    n
                  
                  ⁢
                  
                    
                      U
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        p
                      
                    
                    ⁡
                    
                      (
                      
                        
                          K
                          ⁡
                          
                            (
                            
                              
                                
                                  x
                                  i
                                
                                +
                                
                                  d
                                  
                                    x
                                    i
                                  
                                
                              
                              ,
                              x
                            
                            )
                          
                        
                        -
                        
                          K
                          ⁡
                          
                            (
                            
                              
                                x
                                i
                              
                              ,
                              x
                            
                            )
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  1
                  
                    
                      λ
                      p
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      1
                    
                    n
                  
                  ⁢
                  
                    
                      U
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        p
                      
                    
                    ⁢
                    
                      d
                      
                        x
                        i
                        T
                      
                    
                    ⁢
                    
                      
                        
                          ∂
                          
                            K
                            ⁡
                            
                              (
                              
                                
                                  x
                                  i
                                
                                ,
                                x
                              
                              )
                            
                          
                        
                        
                          ∂
                          
                            x
                            i
                          
                        
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        95
        )
      
    
  




For p>n, however, the dot product Φ(x)·vp is unknown. Therefore, Equation 77 must be rewritten as




  
    
      
        
          
            K
            ~
          
          ⁡
          
            (
            
              x
              ,
              y
            
            )
          
        
        =
        
          
            
              
                Φ
                ⁡
                
                  (
                  x
                  )
                
              
              T
            
            ⁢
            
              (
              
                
                  C
                  γ
                  
                    -
                    2
                  
                
                -
                
                  
                    1
                    
                      1
                      -
                      γ
                    
                  
                  ⁢
                  I
                
              
              )
            
            ⁢
            
              Φ
              ⁡
              
                (
                y
                )
              
            
          
          +
          
            
              1
              
                1
                -
                γ
              
            
            ⁢
            
              K
              ⁡
              
                (
                
                  x
                  ,
                  y
                
                )
              
            
          
        
      
    
    
      
        (
        96
        )
      
    
  




From Equation 94, it can be seen that the eigenvalues of




  
    C
    γ
    
      -
      2
    
  
  -
  
    
      1
      
        1
        -
        γ
      
    
    ⁢
    I
  




associated with the eigenvectors vp are zero for p>n.

Combining Equations 94, 95 and 96 produces




  
    
      
        
          
            
              
                
                  K
                  ~
                
                ⁡
                
                  (
                  
                    x
                    ,
                    y
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    1
                    
                      1
                      -
                      γ
                    
                  
                  ⁢
                  
                    K
                    ⁡
                    
                      (
                      
                        x
                        ,
                        y
                      
                      )
                    
                  
                
                +
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    n
                  
                  ⁢
                  
                    
                      Φ
                      ⁡
                      
                        (
                        x
                        )
                      
                    
                    ·
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    v
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        1
                        
                          
                            γλ
                            p
                          
                          +
                          1
                          -
                          γ
                        
                      
                      -
                      
                        1
                        
                          1
                          -
                          γ
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    Φ
                    ⁡
                    
                      (
                      y
                      )
                    
                  
                  ·
                  
                    v
                    p
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    1
                    
                      1
                      -
                      γ
                    
                  
                  ⁢
                  
                    K
                    ⁡
                    
                      (
                      
                        x
                        ,
                        y
                      
                      )
                    
                  
                
                +
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    n
                  
                  ⁢
                  
                    
                      1
                      
                        γ
                        p
                      
                    
                    ⁢
                    
                      (
                      
                        
                          1
                          
                            
                              γλ
                              p
                            
                            +
                            1
                            -
                            γ
                          
                        
                        -
                        
                          1
                          
                            1
                            -
                            γ
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  (
                  
                    
                      ∑
                      
                        i
                        =
                        1
                      
                      n
                    
                    ⁢
                    
                      
                        U
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          p
                        
                      
                      ⁢
                      
                        d
                        
                          x
                          i
                          T
                        
                      
                      ⁢
                      
                        
                          ∂
                          
                            K
                            ⁡
                            
                              (
                              
                                
                                  x
                                  i
                                
                                ,
                                x
                              
                              )
                            
                          
                        
                        
                          ∂
                          
                            x
                            i
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      ∑
                      
                        i
                        =
                        1
                      
                      n
                    
                    ⁢
                    
                      
                        U
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          p
                        
                      
                      ⁢
                      
                        d
                        
                          x
                          i
                          T
                        
                      
                      ⁢
                      
                        
                          ∂
                          
                            K
                            ⁡
                            
                              (
                              
                                
                                  x
                                  i
                                
                                ,
                                y
                              
                              )
                            
                          
                        
                        
                          ∂
                          
                            x
                            i
                          
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        97
        )
      
    
  




One drawback of the previous approach appears when dealing with multiple invariances (i.e., more than one tangent vector per training point). Under such conditions, one must diagonalize the matrix Kt (see Equation 91) whose size is equal to the number of invariances.
To introduce multiple invariances, an alternative method can be used which consists of decomposing the Gram matrix of the input vectors. This technique uses the PCA kernel map described above. In that method, a training sample of size n spans a subspace E whose dimension is at most n. The empirical kernel map ψ (see Equation 88) gives in an orthonormal basis the components of a point Φ(x) projected on E.
From Equation 79, it is can be seen that training a nonlinear SVM on {x1, . . . , xn} is equivalent to training a linear SVM on {ψ(x1), . . . , ψ(xn)} and thus, due to the nonlinear mapping ψ, one can work directly in the linear space E and use the same technique described above for invariant linear SVMs (see Equations 71-73). However, the invariance direction dΦ(xi) does not necessarily belong to E. By projecting them onto E, some information might be lost. However, this approximation gives a similar decision function to that obtained using decomposition of the Gram matrix of the tangent vectors.
Finally, the proposed algorithm consists of training an invariant linear SVM as previously described for invariant linear SVMs with training set {Ψ(x1), . . . Ψ(xn)} and with invariance directions {dΨ(x1), . . . , Ψ(xn)}, where

dψ(xi)=ψ(xi+dxi)−ψ(xi),  (98)

which can be expressed from Equation 88 as





  
    
      
        
          d
          
            
              ψ
              ⁡
              
                (
                
                  x
                  i
                
                )
              
            
            p
          
        
        =
        
          
            
              d
              
                x
                i
                T
              
            
            
              
                λ
                p
              
            
          
          ⁢
          
            
              ∑
              
                j
                =
                1
              
              n
            
            ⁢
            
              
                U
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  p
                
              
              ⁢
              
                
                  
                    ∂
                    
                      K
                      ⁡
                      
                        (
                        
                          
                            x
                            i
                          
                          ,
                          
                            x
                            j
                          
                        
                        )
                      
                    
                  
                  
                    ∂
                    
                      x
                      i
                    
                  
                
                .
              
            
          
        
      
    
    
      
        
          (
          99
          )
        
        ⁢
        
          
        
      
    
  




It is possible to estimate how much information has been lost by projecting dΦ(xi) on E with the ratio




  
    
      
        
          
            
            
              d
              
                ψ
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
            
          
          
            
            
              d
              
                Φ
                ⁡
                
                  (
                  
                    x
                    i
                  
                  )
                
              
            
            
          
        
        ≤
        1
      
    
    
      
        (
        100
        )
      
    
  




The techniques of enforcing an invariance and adding the virtual examples Ltxi in the training set are related and some equivalence can be shown (see Todd K. Leen, “From data distributions to regularization in invariant learning”, Nips, volume 7. The MIT Press, 1995), however, there are some differences.

When using the Virtual Support Vector (VSV) method, if a training point xi is far from the margin, adding the virtual example Ltxi will not change the decision boundary since neither of the points can become a support vector. Hence, adding virtual examples in the SVM framework enforces invariance only around the decision boundary, which is the main reason that the virtual SV method only adds virtual examples generated from points that were support vectors in the earlier iteration.
It might be argued that the points that are far from the decision boundary do not provide any significant information. On the other hand, there is some merit in not only keeping the output label invariant under the transformation Lt but also the real-valued output. This can be justified by interpreting the distance of a given point to the margin as an indication of its class-conditional probability (see John Platt, “Probabilities for support vector machines”, A. Smola, P. Bartlett, B. Schölkopf, and D. Scbuurmans, editors, Advances in Large Margin Classifiers. MIT Press, Cambridge, Mass., 2000). It appears reasonable that an invariance transformation should not affect this probability too much.
This advantage can be illustrated using the Example toy picture of FIG. 7. Referring to the bottom right corner of the figure, the point might not be a support vector, however, it is still worth enforcing that the level curves around that point follow the direction given by its tangent vector.
The complexity of computing the kernel PCA map Ψ(x) (Equation 88) is of the order n2: E is an n-dimensional space and each basis vector is expressed as a. combination of n training points (see also Equation 93). There are, thus, two ways of reducing the complexity:

    
    
        1. Reduce the dimension of E by keeping only the leading principal components; and
        2. Express each basis vector vp as a sparse linear combination of the training points.
    
    


Additional details can be obtained from C. Williams and M. Seeger., “Using the Nystrom method to speed up kernel machines”, Advances in Neural Information Processing Systems 13, pages 682-688. MIT Press, 2001.
A straightforward way to reduce the complexity is based on the method consisting of decomposition of the Gram matrix of the input vector described above. Here, a random subset of the training points of size m<<n is selected. Work is performed in the subspace E′ spanned by those m points. In practice, this means that the kernel PCA map (Equation 88) can be computed only from the m training examples. However, there is a loss of information by projecting on E′ instead of E, but if the eigenvalues of the kernel function decay quickly enough and m is large enough, ∥PE(X)−PE′(x)∥ should be small, where P is the projection operator. (See A. J. Smola and B. Schölkopf, “Sparse greedy matrix approximation for machine learning”, P. Langley, editor, Proceedings of the 17th International Conference on Machine Learning, pages 911-918, San Francisco, 2000. Morgan Kaufman.)
The method using the Gram matrix of the tangent vectors (Equations 90-95) described above amounts to finding the linear invariant hyperplane in the subspace spanned by the tangent vectors (in feature space). The method using the Gram matrix of the input vectors (training points) amounts to doing the same thing, but in the subspace spanned by the training points. More generally, one could consider an hybrid method where the linear invariant hyperplane is constructed in the linear space spanned by a subset of the tangent vectors and a subset of the training points.
EXAMPLES
In the following examples, a standard SVM is compared with several methods taking into account invariances:

    
    
        Standard SVM with virtual example (VSV).
        Invariant SVM as described in above (ISVM).
        Invariant hyperplane in kernel PCA coordinates as described above (IHKPCA).
        SVM trained with points preprocessed through the linear invariant hyperplane method (LIH)
    
    


The last method refers to disclosure provided by Schölkopf, Simard Smola and Vapnik (supra) where the authors describe the invariant linear SVM. In the following examples, the training and test points were linearly preprocessed using Cγ−1 (Equation 73). Afterwards, instead of training a linear SVM, a nonlinear SVM was trained. Even though there is no guarantee that such an approach will work, experimental results on the NIST (National Institute of Standards and Technology) set showed a slight improvement over the standard nonlinear SVM. The examples include tests run on a toy problem and a digit recognition dataset.
In all methods described for training an invariant SVM, there is a parameter γ that must be chosen, which is a trade-off between maximization of the margin and enforcement of the invariances (see, e.g., Equation 71). One can set this parameter using a validation set or by gradient descent. (O. Chapelle, V. Vapnik, V. Bousquet, and S. Mukherjee, “Choosing multiple parameters for support vector machines”, Machine Learning, 2001.)
In order to have a functional which is scale invariant with respect to the size of dxi (in other words, the parameterization of Lt, instead of minimizing the functional of Equation 71, the following is minimized:




  
    
      
        
          
            
              (
              
                1
                -
                γ
              
              )
            
            ⁢
            
              w
              2
            
          
          +
          
            γ
            ⁢
            
              
                ∑
                
                  i
                  =
                  1
                
                n
              
              ⁢
              
                
                  
                    (
                    
                      w
                      ·
                      
                        d
                        
                          x
                          i
                        
                      
                    
                    )
                  
                  2
                
                S
              
            
          
        
        ,
      
    
    
      
        (
        101
        )
      
    
  




with





  S
  =
  
    
      ∑
      
        i
        -
        1
      
      n
    
    ⁢
    
      
        d
        
          x
          i
          2
        
      
      .
    
  




This is equivalent to minimizing the functional of Equation 71 where γ is replaced by




  
    
      
        γ
        ←
        
          
            γ
            
              S
              +
              
                γ
                ⁡
                
                  (
                  
                    1
                    -
                    S
                  
                  )
                
              
            
          
          .
        
      
    
    
      
        (
        102
        )
      
    
  




Example 2
Toy Problem
Referring to FIG. 7, the toy problem is as follows: the training data has been generated uniformly from [−1,1]2. The[.] true decision boundary is a circle centered at the origin:

f(x)=sign(x2−0.7).  (103)

The a priori knowledge to be encoded in this toy problem is local invariance under rotations. Therefore, the output of the decision function on a given training point xi and on its image R(xi,ε) obtained by a small rotation should be as similar as possible. To each training point, a tangent vector dxi is associated which is actually orthogonal to xi 
A training set of 30 points was generated and the experiments were repeated 100 times. A Gaussian kernel




  
    
      
        
          K
          ⁡
          
            (
            
              x
              ,
              y
            
            )
          
        
        =
        
          exp
          ⁡
          
            (
            
              
                
                  
                  
                    x
                    -
                    y
                  
                  
                
                2
              
              
                2
                ⁢
                
                  σ
                  2
                
              
            
            )
          
        
      
    
    
      
        (
        104
        )
      
    
  




was chosen.

The results are summarized in FIGS. 8a and 8b, with the following observations:

    
    
        For each σ, adding virtual training examples reduces the test error.
        The ISVM and IHKPCA methods yield almost identical performances, especially for large values of σ, which can be explained by FIG. 9. The ratio of Equation 100 has been computed for different values of σ. For large values, this ratio is almost 1, which means that there is almost no loss of information by projecting dΦ(xi) on E. Thus, ISVM and IHKPCA produce the same decision boundary.
        The LIH method slightly impaired the performance. Nearly the same results were expected as would be obtained from a standard SVM. In this example, the covariance matrix of tangent vectors in input space should be roughly a constant times the identity matrix:
    
    






  
    
      
        
          
            ∑
            
              i
              =
              1
            
            n
          
          ⁢
          
            
              d
              
                x
                i
              
            
            ⁢
            
              d
              
                x
                i
                T
              
            
          
        
        ≈
        
          C
          ⁢
          
            
          
          ⁢
          
            
              I
              2
            
            .
          
        
      
    
    
      
        (
        105
        )
      
    
  




    
    
        Different values of γ were tried for this method, but none of them provided improvement, as shown in FIG. 10.
    
    


Certain results plotted in FIG. 8a were somewhat surprising, but may be specific to this problem:

    
    
        The test error of a normal SVM does not increase when σ has large values. This is not the traditional “bowl” shape.
        While the test error for SVM and VSV do not increase for large σ, the test errors of ISVM and IHKPCA do. Possibly a larger value of γ (more invariance) should be applied in this case. This can be seen from FIG. 11, where the following quantity has been plotted
    
    






  
    
      
        
          
            1
            n
          
          ⁢
          
            
              ∑
              
                i
                =
                1
              
              n
            
            ⁢
            
              
                (
                
                  w
                  ·
                  
                    d
                    
                      Φ
                      ⁡
                      
                        (
                        
                          x
                          i
                        
                        )
                      
                    
                  
                
                )
              
              2
            
          
        
      
    
    
      
        (
        106
        )
      
    
  




    
    
        However, the reason for a larger value of γ remains unclear. Note that Equation 106 gives a rule of thumb on how to select a good value for γ. Indeed, after training, a reasonable value for Equation 106 should be in the range [0.2-0.4]. This represents the average difference between the output of a training point and its transformed Ltxi.

Referring to FIG. 8b: 

        The more invariances, the better (and it converges to the same value for different σ). However, it might be due to the nature of this toy problem.
        When comparing log σ=1.4 and log σ=0, one notices that the decrease in the test error does not have the same speed. This is actually the dual of the phenomenon observed in FIG. 8a: for a same value of gamma, the test error tends to increase, when σ is larger.
    
    


Table 14 summarizes the test error of the different learning algorithms using optimal parameter settings.











TABLE 14





SVM
VSV
LIH
ISVM
IHKPCA







6.25
3.81
6.87
1.11
1.11









Example 3
Handwritten Digit Recognition
Invariances were incorporated for a hand-written digit recognition task. The USPS dataset have been used extensively in the past for this purpose by workers in the SVM field. This dataset consists of 7291 training and 2007 test examples.
According to B. Schölkopf, C. Burges, and V. Vapnik, “Extracting support data for a given task”, U. M. Fayyad and R. Uthurusamy, editors, First International Conference on Knowledge Discovery & Data Mining. AAAI Press, 1995, the best performance has been obtained for a polynomial kernel of degree 3,




  
    
      
        
          K
          ⁡
          
            (
            
              x
              ,
              y
            
            )
          
        
        =
        
          
            
              (
              
                
                  x
                  ·
                  y
                
                256
              
              )
            
            3
          
          .
        
      
    
    
      
        (
        107
        )
      
    
  




The factor 256, equaling the dimension of the data, has been included to avoid numerical instabilities. Since the choice of the base kernel is not the primary concern, the results described in this section were performed using this kernel. The local transformations are horizontal and vertical translations. All of the tangent vectors have been computed by a finite difference between the original digit and its 1-pixel translated, as shown in FIG. 12.
Following the experimental protocol described in B. Schölkopf, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson, “Generalization bounds via eigenvalues of the Gram matrix”, Technical Report 99-035, NeuroColt, 1999, the training set was split into 23 subsets of 317 training examples after a random permutation of the training and test set. The focus of the experiment was on a binary classification problem, namely separating digits 0 to 4 against 5 to 9. The gain in performance should also be valid for the multiclass case.
FIGS. 13a and 13b compare ISVM, IHKPCA and VSV for different values of γ. From these figures, it can be seen that the difference between ISVM (the original method) and IHKPCA (the approximation) is much larger than in the toy example. The quality of the approximation can be estimated through the approximation ratio of Equation 100. For the vertical translation, this value is 0.2. For horizontal translation, the value is 0.33. The fact that these values are much less than 1 explains the difference in performance between the 2 methods (especially when γ→1). A possible reason for the apparent performance disparity relative to the toy example (Example 2) may be due to the differences in input dimensionality. In 2 dimensions with a radial basis function (RBF) kernel, the 30 examples of the toy problem “almost span” the whole feature space, whereas with 256 dimensions, this is no longer the case.
What is noteworthy in these experiments is that the proposed method is much better than the standard VSV. The reason for this improvement may be that invariance is enforced around all training points, and not only around support vectors It should be noted that “VSV” refers to a standard SVM with a double size training set containing the original dates points and their translates.
The horizontal invariance yields larger improvements than the vertical one. This may be due to the fact that the digits in the USPS database are already centered vertically. (See FIG. 12.)
After having studied single invariances, gain was evaluated in the presence of multiple invariances. For the digit recognition task, 4 invariances (one pixel in each direction) were considered. FIG. 14 provides a comparison of IHKPCA (which scales better than ISVM for multiple invariances) with LIH. As indicated, IHKPCA provides significant improvement in performance while LIH fails.
Example 4
Application to Noisy Measurements
The different components of a training point in input space usually correspond to some noisy measurements associated with a physical device. Ideally, the learning machine should not to be affected by this noise, i.e., the noise should not cause the output of the decision function to vary.
If the noise model were known, it would be straightforward to train such a learning machine; one could simply consider the effect of the noise as an invariant transformation and use the non-linear invariant SVM techniques described above.
In practice, one does not know the noise but can perform several measurements of the same sample. In this way, it is possible to construct “noise vectors”.
Suppose that for each training sample xi, p additional measurements of the one sample yielding p replicates of xi, {xi(1), . . . , xi(p)} are performed. One can construct p noise vectors,

dxi(k)=xi(k)−xi, 1≦k≦p.  (108)

The replicates should be seen as identical to the original training point since they are measurements of the same object. This goal is achieved by enforcing in feature space the orthonormality of the noise vector to the hyperplane.
Note that the digit recognition experiments can be considered as an instance of this problem: the camera is never positioned in such a way that the digit falls exactly in the middle of the bitmap. In other words, there is a “position noise”.
The favorable results obtained in the digit recognition task prompted additional experiments on another dataset consisting of mass spectrograms. For each training sample there were two measurements, allowing construction of one noise vector. The techniques described herein yielded significant improvements over standard SVMs and the VSV method.
The method of the second embodiment of the present invention extends a method for constructing invariant hyperplanes to the nonlinear case. Results obtained with this new method are superior to the virtual SV method, the latter of which has recently broken the record on the NIST database which is the “gold standard” of handwritten digit benchmarks (see D. DeCoste and B. Schölkopf, “Training invariant support vector machines”, Machine Learning, 2001.)
Example 5
Prostate Disease Classification
SVMs were applied to analysis of laser desorption mass spectrometry for classification of patients with prostate disease to provide a significant improvement over prior methods, with 85% correct classification. The training set was formed by input/target pairs where the inputs are spectrograms of proteins obtained using ProteinChip® arrays and the SELDI (surface emitting laser desorption ionization) system, both developed by Ciphergen Biosystems, Inc. (Fremont, Calif.). The proteins were extracted from serum samples from patients being evaluated for prostate cancer. The targets are four prostate cancer classes denoted as A, B, C and D, where class A is Benign Prostate Hyperplasia (BPH), class D is normal and classes B and C are two levels of prostate cancer.
The ProteinChip® array comprises eight small spots that contain active surfaces retaining proteins with specific chemical properties. By pouring a solution with different proteins in these spots, one is able to do a protein selection based on their chemical properties. For a detailed description of the SELDI procedure, see, e.g., U.S. Pat. No. 5,894,063 of Hutchens, et al., which is incorporated herein by reference. Briefly, to analyze spot composition, laser light is directed onto each spot; the laser desorbs (releases) the proteins from the chip; and the ionized molecules are then accelerated by an electric field. The acceleration is inversely proportional to the mass, so that heavy proteins are slower. The time to reach the detector, referred to as “time of flight”, is then used to infer the mass. The SELDI mass spectrometer instrument measures the incoming ion current (ionized molecules) in a time resolution of 4 nano-seconds. An example of the resulting mass spectrogram is provided in FIGS. 15a and 15b, where FIG. 15a shows the small mass component of the spectrum and FIG. 15b shows the large mass portion. The range of the detected mass is generally between 1 to 200,000 daltons (1 dalton=1 atomic mass unit). Note that the peaks for large mass values are wider and smaller than for small mass values.
Large proteins have a wider peak width because they contain more isotopes. (This effect should be linear in the weight). High mass proteins have wider peaks also due to instrument dependent factors: large proteins tend to have more collisions with other molecules during the flight from the spot to the sensor. Another effect that widens protein peak width, and creates an uncertainty about the position of the peak is protein glycosylation: a protein can have a varying number of sugar groups attached to it. The sugar groups control the cellular activity of the protein, and at any given moment there will be a distribution of glycosylation states. In addition to these widening effects, there is the instrument error which is 0.1-0.3% times the mass, i.e., it gives the right distribution with a possible local shift of ±0.1-0.3% of the mass value. Thus, if a peak detection is applied, its sensitivity should be dependent on the mass value: peaks on large mass values are wide while peaks on small mass values are narrow. Many peaks have “shoulders” (see, e.g., FIG. 15b), which can represent system effects, different protein modifications, or different proteins.
The first part of the spectrograms (see FIG. 16), referred to as the “matrix”, is generally cut. The matrix corresponds to small molecules that are typically more noise than true information and are present in all solutions. Moreover, proteins usually weigh more than 10 kilo dalton. Another step that is performed is baseline subtraction, which is performed by locally approximating the convex hull of the points of the curve, then subtracting for each point a value such that the bottom frontier of this convex hull is the abscises axis.
The data were derived from patients with different “levels” of disease. The inputs represent the spectra of samples from different patients and are acquired according to the method described above. For each sample, two replicates were processed.
A manual peak detection scheme was performed by first averaging for each sample over the two replicates to yield one spectrum for one sample, then taking a random sampling of spectra from all the patient groups and manually picking peaks, ensuring consistent sets, e.g., that #56 in one spectrum roughly corresponds in mass with peak #56 in another spectrum. The average mass of each peak across the selected spectra is calculated, producing a list of 299 peak masses as a template of peaks that can be used to select all the peak values from all the samples. This procedure generates a set of vectors with 299 components representing the peak values of all the patient spectra. The dataset is summarized in Table 15, where each sample has two replicates. Thus, class A has 93 samples, class B 98 samples, etc., which yields the 385 pre-processed spectra (averaged over each two replicates).










TABLE 15









Class















A
B
C
D







# Instances
186
196
196
192










A pre-determined split provided 325 points for learning and 60 for testing. The test set contains 15 points of each class. The first experiments concerned the pre-processed dataset. The inputs xi are vectors of length 299 representing peak values. The targets yi are numbers 1, . . . , 4 representing the four classes A, B, C and D. The learning task is a multi-class classification where a machine must be designed to predict one of the four classes from the peak values contained in the inputs. While a number of different approaches could be used, in this example, the following two important families of approaches were applied.
The first family of approaches consists of the decomposition methods. These approaches rely on the decomposition of a multi-class problem into many binary problems. In the present example, this corresponds to designing one machine per class that learns how to discriminate class A against all the other classes, then class B against all the others and so on. At the end, four machines are built that are related to each class individually. The whole system predicts which class corresponds to one input by taking the class of the machine with the largest output. Assuming all goes well, one machine will have an output equal to 1, meaning that its class is the desired output, and all other machines will have an output equal to −1, meaning that their class is not the desired output. This approach has been widely used to address multi-class problems. It is convenient in that it allows recycling of binary classifiers that are commonly used. However, its indirect approach assumes that all problems are independent, which is not the case, thus providing sub-optimal solutions.
The second family of approaches is formed by direct methods. In the case of SVMs, it tries to solve the multi-class problem directly without any decomposition scheme. Note that this approach is rarely used in the literature and is computationally more difficult to implement.
To discriminate between both approach families as applied to SVMs, the first approach is referred to as “one-against-the-rest”, or “bin-SVM”, and the second approach “multi-class” or M-SVM.
Table 16 provides the results obtained using a linear M-SVM. The errors are computed for different binary problems and for the overall multi-class problem. Each of these errors represents the fraction of mistakes on the test set and on the training set. Note that one mistake of the test set represents 1.7% errors. On the training set, there are no errors since the system was designed to learn perfectly these points.











TABLE 16








Test Error
Learning Error









Overall Error
26.7%
0%



D + A vs. B + C
16.7%
0%



D vs. A + B + C
 5.0%
0%



A vs. B + C
20.0%
0%



D vs. B + C
 4.4%
0%










The results for the bin-SVM, shown in Table 17, appear to be better than those obtained with M-SVM.











TABLE 17








Test Error
Learning Error









Overall Error
20.0%
0%



D + A vs. B + C
10.0%
0%



D vs. A + B + C
 5.0%
0%



A vs. B + C
11.1%
0%



D vs. B + C
 4.4%
0%










The same SVMs were applied with different polynomial kernels having the form:




  
    
      
        
          k
          ⁡
          
            (
            
              
                x
                i
              
              ,
              
                x
                j
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  〈
                  
                    
                      x
                      i
                    
                    ,
                    
                      x
                      j
                    
                  
                  〉
                
                
                  
                    
                      
                      
                        x
                        i
                      
                      
                    
                    2
                  
                  ⁢
                  
                    
                      
                      
                        x
                        j
                      
                      
                    
                    2
                  
                
              
              +
              1
            
            )
          
          d
        
      
    
    
      
        (
        109
        )
      
    
  




Here, each input, e.g. each spectrum, is normalized by its l2 norm and a constant equal to 1 is added. Normalization avoids the effects of variations in the total amount of proteins that may result from different experimental protocols.
The results obtained using polynomial kernels for M-SVM and bin-SVM are summarized in Tables 18 and 19, respectively.










TABLE 18







Test Error
Learning Error




















Overall Error - degree 2
15.0%
0%



D + A vs. B + C
 6.7%
0%



D vs. A + B + C
 1.7%
0%



A vs. B + C
11.0%
0%



D vs. B + C
  0%
0%



Overall Error - degree 3
15.0%
0%



D + A vs. B + C
10.0%
0%



D vs. A + B + C
 1.7%
0%



A vs. B + C
15.6%
0%



D vs. B + C
  0%
0%



Overall Error - degree 4
18.3%
0%



D + A vs. B + C
11.7%
0%



D vs. A + B + C
 3.3%
0%



A vs. B + C
15.6%
0%



D vs. B + C
 2.2%
0%




















TABLE 19







Test Error
Learning Error




















Overall Error - degree 2
20.0%
0%



D + A vs. B + C
10.0%
0%



D vs. A + B + C
 5.0%
0%



A vs. B + C
13.3%
0%



D vs. B + C
  0%
0%



Overall Error - degree 3
23.3%
0%



D + A vs. B + C
13.3%
0%



D vs. A + B + C
 5.0%
0%



A vs. B + C
17.8%
0%



D vs. B + C
  0%
0%



Overall Error - degree 4
21.7%
0%



D + A vs. B + C
13.3%
0%



D vs. A + B + C
 5.0%
0%



A vs. B + C
15.6%
0%



D vs. B + C
 4.4%
0%










As can be seen by comparing the table, the best results are obtained with a M-SVM with a polynomial kernel of degree 2. The selection of one algorithm versus another should not be based on the test set since the latter must be used at the very end of the learning process. This independent property between the test set and the training method is fundamental in order to obtain good accuracy on the generalization error estimated on this test set.
A 10-fold cross validation procedure is applied to select the final algorithm. The cross validation method comprises cutting the training set into 10 pieces, then considering ten learning problems, each of which has one piece is removed, in a leave-one-out (LOO) scheme. The learning algorithm is trained on the nine others and tested on the removed piece to provide an estimation of the generalization error, then all the estimates are averaged to provide a score. The model selection procedure involves selecting the algorithm that has the lowest score given by the cross validation method. Table 20 presents the cross validation (CV) errors/scores for different models. The value in parenthesis is the standard deviation of all the generalization errors estimated on the 10 test sets used during cross validation.









TABLE 20







10-CV errors (std)



















Linear M-SVM
31.9% (±13.0%)



Poly.2 M-SVM
30.6% (±12.0%)



Poly.3 M-SVM
31.5% (±11.6%)



Linear Bin-SVM
33.4% (±10.9%)



Poly.2 Bin-SVM
30.9% (±13.0%)



Poly.3 Bin-SVM
31.6% (±13.3%)










As becomes apparent when comparing the preceding tables, the cross validation errors are quite different from the errors on the test set. One possible explanation is that the test set does not correctly represent the problem and gives a biased estimate of the generalization error. Thus, the errors estimated on the test set should be viewed with caution.
Looking at Table 20, one of the two best methods is the M-SVM with a polynomial kernel of degree 2. (The other best method which could not be differentiated is a bin-SVM with the same kernel.) The specificity/sensitivity values for the Poly.2 M-SVM are provided in Table 21.










TABLE 21







Sensitivity
Specificity




















A vs. B + C
0.87
0.9



D + A vs. B + C
0.97
0.9










Note that the value of the sensitivity of A vs B+C depends on the way the classification is done from the multi-class system: either class D is considered to be part of class A, or, as it has been done here, all the learning points of class D are discarded and the value on the remaining 45 testing points is computed. In the first case, the same value is obtained as for the line D+A vs B+C.
The detailed output on the test set run on the Poly.2 M-SVM is provided in Table 22, which presents the confusion matrix on the test set for the final classifier.












TABLE 22







True A
True B
True C
True D






















Predicted A
13
3
0
0



Predicted B
0
10
2
0



Predicted C
1
2
13
0



Predicted D
1
0
0
15










The magnitude of the classifier output is often considered a level of confidence, where an output with a large magnitude is considered more trustworthy than an output close to zero. Thus, the distribution of the outputs of a classifier is a measure of how the classifier behaves on a new set it has never seen and how confident it is. This distribution is represented in FIG. 17. Note that for multi-class classification systems, the output magnitude is actually given by what is called the multi-class margin, which is the difference between the largest output and the second largest output. The fact that most of the points in the test set have a multi-class margin greater than 0.3 (negative margins correspond to errors) is quite encouraging.
In addition to the dataset described above, a fifth class, called the “E class”, was formed of 172 spectra that come from the same sample (normal sample) run across a number of different chips and spot positions. If the multi-class system is as good as it is expected, then it should classify all these elements in class D. FIG. 18 represents the output distribution of the M-SVM selected previously, and shows that out of 172 points, 115 were classified as normal by the M-SVM, 45 were considered as benign (class A), 8 as class B and 4 as class C. This result indicates certain weaknesses in the present system that seems to mix a little class A with class D. Note, however, that class A corresponds to a benign condition. On the other hand, the M-SVM discriminate very well between D+A versus B+C. This result is confirmed on the class E set where almost all spectra are classified as D+A, which is the correct answer.
A feature selection process was applied to select those components from the inputs vectors that are relevant for the classification task of interest. In this case, feature selection is used to select peaks that are relevant to discriminate the four different prostate cancer levels. Since these peaks correspond to proteins, such feature selection method is of utmost importance and could potentially point out interesting molecules. A detailed discussion of feature selection methods is provided in co-pending International application No. PCT/US02/16012, filed May 20, 2002, the 371 national stage filing of which has now issued as U.S. Pat. No. 7,318,051. This application shares common inventorship with the present application and is incorporated herein by reference in its entirety.
Briefly, appropriate feature selection methods include recursive feature elimination using SVMs (RFE-SVM or, alternatively, SVM-RFE) and minimizing l0-norm.
RFE methods comprise iteratively 1) training the classifier, 2) computing the ranking criterion for all features, and 3) removing the feature having the smallest ranking criterion. This iterative procedure is an example of backward feature elimination. For computational reasons, it may be more efficient to remove several features at a time at the expense of possible classification performance degradation. In such a case, the method produces a “feature subset ranking”, as opposed to a “feature ranking”. Feature subsets are nested, e.g., F1⊂F2⊂ . . . ⊂F.
If features are removed one at a time, this results in a corresponding feature ranking. However, the features that are top ranked, i.e., eliminated last, are not necessarily the ones that are individually most relevant. It may be the case that the features of a subset Fm are optimal in some sense only when taken in some combination. RFE has no effect on correlation methods since the ranking criterion is computed using information about a single feature.
In the present embodiment, the weights of a classifier are used to produce a feature ranking with a SVM (Support Vector Machine). RFE-SVM contemplates use of for both linear and non-linear decision boundaries of arbitrary complexity. Linear SVMs are particular linear discriminant classifiers. If the training set is linearly separable, a linear SVM is a maximum margin classifier. The decision boundary (a straight line in the case of a two-dimension separation) is positioned to leave the largest possible margin on either side. One quality of SVMs is that the weights wi of the decision function D(x) are a function only of a small subset of the training examples, i.e., “support vectors”. Support vectors are the examples that are closest to the decision boundary and lie on the margin. The existence of such support vectors is at the origin of the computational properties of SVM and its competitive classification performance.
A second method of feature selection according to the present invention comprises minimizing the l0-norm of parameter vectors. Such a procedure is central to many tasks in machine learning, including feature selection, vector quantization and compression methods. This method constructs a classifier which separates data using the smallest possible number of features. Specifically, the l0-norm of w is minimized by solving the optimization problem




  
    
      
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where ∥w∥0=card{wi|wi≠0}. In other words, the goal is to find the fewest non-zero elements in the vector of coefficients w. However, because this problem is combinatorially hard, the following approximation is used:





  
    
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where ε<<1 has been introduced in order to avoid ill-posed problems where one of the wj is zero. There may, however, be many local minima.

A method known as Franke and Wolfe's method is proved to converge to a local minimum. For the problem of interest, it takes the following form, which can be defined as an “Approximation of the l0-norm Minimization”, or “AL0M”:

    
    
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AL0M solves a succession of linear optimization problems with non-sparse constraints. Sometimes, it may be more advantageous to have a quadratic programming formulation of these problems since the dual may have simple constraints and may then become easy to solve. As a generalization of the previous method, a procedure referred to as “l2-AL0M” can be used to minimize the l0 norm as follows:

    
    
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        4. Repeat steps 2 and 3 until convergence.
    
    


To perform a l0-norm minimization for the multi-class problems, the above-described l2 approximation method is used with different constraints and a different system:

    
    
        1. Start with wc=(1, . . . , 1), for c=1, . . . , K
        2. Assume Wk=(w1k, . . . , wck, . . . wKk) is given. Solve
    
    






  
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subject to: wc(i),(xi*wc)i)k)−wc,(xi*wck)+bc(i)−bc≧1 for c=1, . . . , K.

    
    
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        4. Repeat steps 2 and 3 until convergence.
    
    


AL0M SVMs slightly outperform RFE-SVM, whereas conventional SVM tends to overfit. The l0 approximation compared to RFE also has a lower computational cost. In the RFE approach, n iterations are typically performed by removing one feature per iteration, where n is the number of input dimensions, however, RFE can be sped up by removing more than one feature at a time. In addition, in tests run using multi-class SVM, the l2-AL0M multi-class SVM outperforms the classical multi-class SVM.
The following are some preliminary results of feature selection as applied to the present example dataset.
A feature selection algorithm adapted to multi-class classification was used after which a M-SVM with a polynomial kernel of degree 2 and a linear kernel were applied. Note that for the polynomial kernel of degree 2, the data was re-normalized so that the norm of each input vector with the selected feature is equal to one. Fifty-six features on the training set were selected, representing less than 20% of the total number of features. The results on the test set are reported in Table 23, where the results improve for the linear M-SVM but worsen for the polynomial SVM. One mistake corresponds to 1.66% such that the 11.7% difference between polynomial SVM with and without feature selection corresponds to seven mistakes.










TABLE 23







Test Error
Learning Error




















Overall Error - linear 2
23.3%
0%



D + A vs. B + C
15.0%
0%



D vs. A + B + C
3.3%
0%



A vs. B + C
15.7%
0%



D vs. B + C
4.4%
0%



Overall Error - degree 2
26.7%
0%



D + A vs. B + C
16.7%
0%



D vs. A + B + C
5.0%
0%



A vs. B + C
17.8%
0%



D vs. B + C
4.4%
0%










Using the pre-processed data, the results appear to show improvement over the previous study on this dataset. However, it may be possible to obtain even better results by improving the manual pre-processing step during the generation of the data. To evaluate this possibility, the raw data was used directly. The raw data are vectors of around 48,000 dimension. In classical statistics such a number is considered unreasonable and no known technique can handle such a large value. One strength of the SVM, however, is the ability to handle very large input dimensionality. Thus, by dealing directly with the raw data, the curse of dimensionality can be avoided.
It should be pointed out that the use of raw data does not mean that no pre-processing was performed. Rather, different automated methods were applied to address the noise of the experimental process. The first pre-processing algorithm consists of aligning the spectra.
The basic approach of alignment is to slide each example along different feature values (one feature at a time) against a baseline example and rate the similarity between the two examples. Then, each example is offset using the baseline example according to the best score achieved for each. The baseline examples are derived from raw data after applying a baseline subtraction algorithm, which tries to find a smooth line that follows the bottom of the curve without rising into the peaks.
The scoring function used in this example is:




  
    
      
        
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where x1 and xo are two feature vectors and ∥xi∥ is the 1 norm, e.g.,





  
    
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The absolute values are placed in the denominator because sometimes the values go negative. (Intensities are supposed to be positive, however, this may be due to error in measurement and in the baseline subtraction stage). The algorithm is thus:

Take the first example as a baseline, call it x0.
For each example i

    
    
        Denote xi(+j) as example i shifted by j features to the right. (The scoring function S(,) is computed only for the features I1, . . . , n−10 (where n is the dimensionality) to compare vectors of the same size after offsetting them.)
        off(i)=arg minj=−10, . . . , 10 S(xi(+j), x0)
    
    


End For
Now one has the aligned data xi (+off(i)) for all i.
Other scoring methods, e.g., S(xi, x0)=∥xi, =x0∥22, appeared initially to be slightly worse, however, upon review, most examples were assigned the same offsets by the different scoring functions, so it does not appear to be very significant.
Alignment experiments were performed using various pre-processing steps prior to computing the alignment scores:

    
    
        1. Smoothing of the signal was performed by averaging over a window where points in the window are given different weights according to their distance to the center d using an exponential exp(−σ*d2). A value of 0.06 was used for σ.
        2. Alignment scores were computed only for masses larger than 1000, thus avoiding the “noise” at the start of the signal which tends to have very large values and may dominate the alignment score.
        3. Normalization of the example,
    
    






  
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Note that compared to previous normalization, the l1 norm has been used here.
FIG. 19 provides an example of the effect of the alignment algorithm comparing the two random samples before alignment and after alignment. This effect is global across the entire signal, although some error remains.
To illustrate the techniques describe herein, the binary problem A versus B and the binary problem A versus B and C were considered. The binary problems were considered first because binary classification is easier than multi-class classification. However, the most difficult binary classification task according to the previous study on the pre-processed data was considered.
The following procedure also differs from those described above in that only the second copy of the replicates was used.
In each of the following experiments, a linear SVM with C=∞ is used. Results are provided for 10 fold cross validation (10-CV) and an approximation of the leave-one-out (LOO) error, which is an m-fold cross validation where m is the number of points in the training set, using the span (smoothed with a sigmoid to take into account variance). Because the 10-CV has rather large variance, in some experiments results are provided for repeated runs of CV with different random permutations of the data, running 10 or 100 times (10×10CV or 100×10CV respectively). The same random splits were used across all algorithms.










TABLE 24





Alignment Algorithm
10 × 10 CV
100 × 10 CV
SPAN


















A vs. B





none
9.63 ± 6.03
9.77 ± 6.42
10.63


align w/smoothing, norm
9.35 ± 6.72
9.45 ± 6.60
9.50


align w/large mass, norm
8.83 ± 6.01
9.03 ± 6.59
8.53


align w/norm
8.42 ± 6.69
8.42 ± 6.34
8.40


align w/raw data
8.21 ± 5.62
8.44 ± 6.11
9.04


A vs. B + C


none
9.42 ± 5.16
9.13 ± 4.95
11.43


align w/smoothing, norm
8.83 ± 5.43
7.94 ± 4.86
9.46


align w/large mass, norm
8.14 ± 5.19
7.54 ± 4.76
9.62


align w/norm
7.41 ± 4.93
7.13 ± 4.67
9.20


align w/raw data
7.97 ± 4.80
7.37 ± 4.62
9.07









The error rates for the two binary problems are provided in Table 24. Based on these results, it appears that alignment of the data can help improve performance, with the best method being to align after normalizing the data (“ALIGN WITH NORM”).
The effect of smoothing of the raw data on the classification rate of a linear SVM was also examined. Smoothing is considered to be distinct from alignment of the data, which is described above. The results on A versus B+C are reported, using a linear kernel and averaging the two replicates for each training sample.
The smoothing of the raw data depends on an integer scaling factor a and is defined as the following convolution:

ŝ(i)=s*N(0,σ2/4)(σ)i 

where s is the original signal, ŝ is the smoothed signal and N(0,σ2/4) is a gaussian of zero mean and standard deviation σ/2.

Generally, smoothing convolves the original signal with a Gaussian whose standard deviation is σ/2, then sub-samples the signal by a factor σ. The result is that the length of the signal has been divided by σ.
Note that this is almost equivalent to make a wavelet transform of the signal and keep the low frequencies information. The parameter σ controls where is the cut-off of the high frequencies.
Results are plotted in FIG. 20, which shows the test error against the value of the smoothing parameter. It appears that the smoothing does not provide improvement relative to taking the raw data (σ=1).
Another method evaluated for its ability to improve performance involved attempting to detect the portion of the signal corresponding to peaks, and then use the peak data instead. It can be presumed that this information is discriminative and acts as prior knowledge to code a feature selection algorithm so that the classifier does not overfit on other (noisy) data.
FIG. 21 provides an example of the peak detection comparing the peaks detected with two random samples (FIG. 21a) where the vertical lines are the detected peaks over the two random samples. FIG. 21b shows how the algorithm works, in which a baseline signal is smoothed with a windowing technique and all the features of the smoothed signal where the derivative switches sign are detected as peaks.
The peak detection algorithm is as follows: (1) select a signal from which peaks are to be detected; (2) smooth the signal by averaging over a window where points in the window are given different weights according to their distance to the center d using an exponential exp(−σ*d2); (3) take all the points in the signal where the derivative changes from positive to negative (the intensity is larger than the intensity of the features to the left and right). Controlling the value of σ controls the number of peaks detected. Two different signals were selected for detection of peaks: a single random signal from the dataset (assume that this signal is representative) and the mean of all signals.
Using the same protocol as the previous experiments and looking at the binary problem A versus B+C, the second copy of the replicates in all these experiments were taken. A linear SVM with C=∞ was used. The same random splits in cross validation were used across all algorithms. The data after alignment using the algorithm “ALIGN WITH NORM” from the previous section was used, so these results should be compared to the baseline (of using all the data) with 1×10CV (on the same splits). The result for this is: 7.9%±5.6% and 9.2% with the span.
In a first set of experiments peaks with varying values of σ were detected. The intensities at these peak indexes were taken using two different methods: taking a random signal and the mean signal. The peak indexes were then used for classification. The error rates for intensity at peaks after peak detection on problem A versus B are shown in Tables 25 and 26 for a random signal and mean signal, respectively. No improvement was obtained relative to not doing the peak detection, and it is not clear which algorithm is better.












TABLE 25







σ
peaks
1 × 10 CV error
span





















0.01
847
14.20 ± 7.20 
14.17



0.03
1366
15.22 ± 5.67 
14.06



0.06
1762
12.44 ± 10.03
14.69



0.1
1998
11.75 ± 10.67
13.02



0.3
2388
11.05 ± 9.30 
12.25



0.5
2555
11.74 ± 8.77 
11.35



1
2763
9.67 ± 8.25
11.45



2
3065
12.43 ± 7.47 
12.25



10
3276
12.43 ± 9.21 
12.94






















TABLE 26







σ
peaks
1 × 10 CV error
span





















0.01
930
12.81 ± 5.65
14.45



0.03
1500
16.60 ± 6.02
17.41



0.06
1884
14.20 ± 5.04
15.46



0.1
2050
14.19 ± 5.50
15.89



1
2757
11.07 ± 8.25
12.67



10
3277
10.38 ± 7.09
12.63










The random signal peak detection algorithm was selected, then, for each value of σ, the features selected were those features falling within a radius r of any of the peaks. The error rate varying both σ and r was recorded. The results are provided in the tables shown in FIGS. 22 and 23. These results appear to indicate that peak detection as a means for feature selection provides no improvement over using all of the data since the results only improve when the radius is so large that nearly all of the features are selected.
All experiments thus far were performed using all of the data, including the data in the low input range. The intensities in this region contain a lot of chemical noise. Further, the values obtained in this region may be experimental “artifacts” that are biased from the original protocol of the experiments and are not i.i.d. (independent and identically distributed). If these values are biased according to the classes (e.g., if each class was measured by a different instrument or scientist), then it may appear that good generalization ability is obtained using this data which will not be true for future examples.
A few experiments were performed on the (aligned) data to show the effect of the lower region on test error. These experiments were performed with and without normalization, with and without the lower region and just the lower region on its own. The results are shown in Table 27.










TABLE 27







DATA TYPE
1 × 10 CV



















all data, original (aligned) form
7.94



all data, normalized
14.22



only non-lower region
22.14



only non-lower region, normalized
17.02



non-lower region normalized, lower
7.27



region in original form



only the lower region (original form)
10.71










These experiments clearly show the lower region has a large impact in generalization ability and affects the selection of preprocessing procedures. As can be seen from the results, a global normalization increases the error rate when the lower region is not discarded, while, when the lower region is discarded, the error rate decreases.
Control data was grouped according to plate identity, which fell into three groups corresponding to the plates of classes A, B and C respectively.
A multi-class system was constructed to discriminate on the original four classes A,B,C,D and then the labels given to the three types of control data (viewing them as test data) were reviewed. The control example experiments were performed using different types of features: (1) all data; (2) all data except the lower region; (3) only the lower region; and (4) only large mass (>20000). The plate ID prediction is the percentage of samples for which the plate ID is correctly predicted. The results are shown in Table 28.













TABLE 28






predicted to







DATA TYPE
be in class:
A
B
C
D
plate ID prediction





















all

30
18
25
6
67%


all non-lower

55
1
13
10
25%


only lower

16
30
31
2
87%


only large mass

28
15
20
16
38%









The results show that one can predict the plate identity with good accuracy by using the lower region data. This helps predict the labels of interest since they were measured on the same plates. This means that the lower region information in this data cannot be used as it is not possible to assess the true generalization ability of the methods. Use of only the non- lower region data is questionable.
In summary, use of the lower region allows accurate discrimination between benign and malignant disease (e.g. class A vs. B+C). This becomes much more difficult when removing this lower region. The reason for this has been identified and comes from the plates that have been used during the experiment: the same plate has been used for the same class. Using a control set with the plate identities it was possible to predict the plate identities quite well with only the matrix region. However, removing the lower region data, the plate identities could not be predicted, indicating that the information about the plate identity in the non- lower region may not be used by the learning system to predict the class, and that predicting the grade cancer with the non- lower region may be valuable.
Using pre-processed data, the inventive method was able to discriminate the 4 classes with 85% percent of success rate. The quality of the classifier was confirmed by its output on a control class that was taken from normal patients. When the discrimination concerns only benign disease against malignant disease (A+D vs. B+C), the learning system is even better and reaches around 95% of success with values of specificity and sensitivity around 0.9.
Example 6
Neuropsychiatric Illness Profiling
A test was designed to determine which fraction provided a means for classifying patients with bipolar disorder. The dataset had three groups of patients: Group A: a subset of patients from Group B that had been treated and obtained functional recovery; Group B: patient group; and Group C: control group. Three types of protein chips with different active surfaces were used in the experiments: IMAC: immobilized metal affinity chromatography loaded with copper; WCX4: weak cation exchange pH 4; and WCX7: weak cation exchange pH 7.
Samples were processed by placing them in a centrifuge to coarsely separate the proteins according to charge, and divided into four fractions. This reduced the number of proteins per sample, thereby improving the ability to discriminate between proteins. There were “borderline” proteins that appeared to some degree in more than one fraction.
Each fraction was applied to the three types of protein chips in duplicate, producing 4×3×2=24 spectra per sample. There were many missing spectra. The chips came in groups of twelve, together forming a plate that undergoes the rest of the process together. The data files were structure in three categories, each containing data from one type of protein chip.
Four files (one for each of the four conditions) for each of the three chip types in EXCEL® spreadsheet format were supplied. These spreadsheets contained peaks that were extracted manually. A correspondence between the peaks in the spectra of a particular condition was found, also manually. Peaks are numbered 1−n, and for each peak number, the intensity and the mass at which the peak intensity was found are noted. The simplest representation of a spectrum was thus as a vector x of fixed length, with Xn as the intensity of the nth peak. The intensity of missing peaks is set to zero. The standard Euclidean dot product was then applied to this vector.
The following additional information was available for each sample: sample ID; plate ID; batch ID; replicate; spectrum number; the target variable; total signal (useful for normalization). The data was extracted into a MATLAB® (The MathWorks, Inc., Natick, Mass.) environment.
Most of the results shown in the following refer to the IMACfx1 condition. Similar results were found to hold for the other conditions as well. The analysis began by performing principal components analysis (PCA) on the data and displaying it in terms of the class variable in order to determine whether there was some discernable structure in terms of this variable. (See FIG. 24.) While there seemed to be no apparent structure in terms of the class variable, as shown in FIG. 25, a plot according to the plate ID provides an entirely different picture.
To determine the level of plate ID dependence, each spectrum was normalized according to the total signal variable by dividing the intensity of each peak by the total ion current. Two normalizations were tried: 1) divide peak intensities by the total signal of that particular spectrum; and 2) divide peak intensities by the median signal level computed over the plate to which the sample belongs. Using a third approach, the variables in each plate were standardized separately.
The principal components in the first two normalizations are shown in FIGS. 26 and 27. The symbols “×”, “ ”, and “+” represent plate ID. Here, separation according to plate ID is still visible. Standardizing according to plate ID is shown in FIGS. 28a and 28b. Structure according to plate ID is no longer apparent, but no other grouping appears either.
The next stage was to attempt to assess the information content in the dataset relative to the class variable. Two classes were considered: the patient (Group B) and normal (control Group C). The following statistic was computed for each variable (peak):




  g
  =
  
    
      
        μ
        1
      
      -
      
        μ
        2
      
    
    
      
        σ
        1
      
      +
      
        σ
        2
      
    
  




where μi is the mean peak intensity in class i and σi is the standard deviation. The distribution of this statistic is plotted in FIG. 29. These correlations were not significantly different from correlations on random data (FIG. 30). In view of the above results on plate ID dependence, the Golub statistic was computed for each of Plates 1, 2 and 3, and the values were significantly higher, as shown in FIGS. 31a-31c. 

The values of the Golub statistic on normalized data (using normalization (1) above) gave values similar to those obtained for the non-normalized data. These results suggest that the two normalizations are not effective in eliminating plate variability.
The plate dependence of the data together with poor results on the whole dataset led to experiments run on each plate separately, which can also serve as a benchmark for results on the whole dataset. The peaks extracted manually were used with a Euclidean dot product on the intensity vector. A linear SVM with C=100 was used. The error rates resulting from a 10×10 fold cross validation are provided in Table 29 and show that there is a difference in performance across plates. Standard deviation is shown in parenthesis.











TABLE 29









Plate 1
Plate 2
Plate 3













Condition
A − B
B − C
A − B
B − C
A − B
B − C





Imacfx1
41 (4)
33 (4)
30 (3)
24 (3)
15 (3)
26 (3)


W4fx2
36 (4)
39 (5)
22 (4)
31 (2)
33 (4)
32 (5)


W4fx3
35 (5)
43 (3)
34 (2)
42 (2)
26 (3)
32 (3)


W7fx3
48 (4)
35 (3)
41 (4)
39 (3)
36 (4)
36 (4)









Plate 3 achieved consistently lower error rates, which plate 1 was usually the worst. Plate 1 is the outlier plate in the PCA plots shown. For comparison, the error rates on the whole data set (non-normalized, or normalized by standardizing in each plate) are shown. Error rates on all plates were similar to those on the worst plate, with a small improvement with normalization. Error rates using a quadratic kernel were higher—23% and 32% in plate 3 for the Imacfx1 condition.
In another test, twelve dataset were used with the same targets for all datasets. Preliminary experiments have shown that training on only one dataset is difficult. To improve performance, different systems trained on each of the datasets separately were combined. Table 30 provides the 10×10 fold cross validation results of different classifiers that are trained with only one dataset at a time. A linear SVM with C=100 was used. The format of the datasets was the same as in the previous test. Error rates are shown for all plates, not normalized or normalized by standardizing each plate. Standard deviation is provide in parenthesis.










TABLE 30









A − B
B − C












not normalized
normalized
not normalized
normalized















Imacfx1
38 (2)
36 (4)
38 (2)
36 (2)


W4fx2
37 (4)
35 (2)
39 (2)
40 (2)


W4fx3
40 (4)
44 (2)
41 (3)
36 (2)


W7fx3
48 (2)
40 (3)
44 (2)
45 (3)









The methods described herein provide means for identifying features in a spectrogram generated during analysis of biological samples, such as on DNA microarray and protein chips, and discriminating between different classes of samples. The samples are pre-processed by normalization and/or alignment so that there is agreement regarding the position of peaks across the spectra. A similarity measure is created to compare pairs of samples and an SVM is trained to discriminate between the different classes of samples. Because of the SVM's ability to process large datasets, the entire spectrogram may be used to identify the most predictive features, which can provide greater accuracy than prior art methods that use only the highest peaks of a spectrogram.
Although the foregoing example relate to analysis of spectrograms taken on biological samples, the methods disclosed herein are application to other types of spectral data, such as elemental analysis by measurement of gamma energy spectra, and other types of structured data, such as strings, e.g., DNA sequences, ECG signals and microarray expression profiles; spectra; images; spatio-temporal data; and relational data.
Alternative embodiments of the present invention will become apparent to those having ordinary skill in the art to which the present invention pertains. Such alternate embodiments are considered to be encompassed within the spirit and scope of the present invention. Accordingly, the scope of the present invention is described by the appended claims and is supported by the foregoing description.

## Claims

1. A method for analysis of data contained in a plurality of spectra generated from mass spectrographic measurement of protein samples corresponding to different biological conditions, wherein the different biological conditions have associated different levels of protein expression, the method comprising:
downloading the plurality of spectra into a computer system comprising a processor and a storage device, wherein the processor is programmed to execute at least one support vector machine and performs the steps of:
aligning the plurality of spectra, comprising:
selecting a first example spectrum as a baseline example;
sliding each spectral peak of a second example spectrum one at a time along a plurality of peaks within the baseline example;
applying a scoring function to obtain a similarity score between each spectral peak of the second example spectrum and the peaks within the baseline example, the similarity score being determined according to the relationship






  
    
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where xi and x0 are feature vectors corresponding to peaks of an ith example spectrum and the baseline spectrum, respectively, and ∥xi∥ is the l1 norm,




  
    
      
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offsetting the second example spectrum relative to the baseline example according to the similarity score achieved for the second example spectrum;
repeating the step of aligning the spectra for at least one additional example spectrum to create a set of aligned spectra;
applying a feature selection algorithm to the set of aligned spectra to select a subset of spectral peaks that discriminate between the different biological conditions;

training and testing the at least one support vector machine using the set of aligned spectra to provide a trained at least one support vector machine for processing of live data;
inputting live spectral data from live protein samples to be analyzed, selecting the subset of spectral peaks from the live spectral data, aligning the live spectral data and processing the live spectral data using the trained at least one support vector machine; and
displaying a listing comprising two or more subsets of live protein samples wherein each subset is identified as belonging to a group possessing one of the different biological conditions.

2. The method of claim 1, wherein the proteins are extracted from serum samples and the different biological conditions comprise normal, benign and prostate cancer.

3. The method of claim 1, further comprising, before or after the step of aligning, normalizing each of the plurality of spectra.

4. The method of claim 1, wherein the kernel comprises a polynomial kernel of degree 2.

5. The method of claim 1, wherein the at least one support vector machine comprises a multi-class support vector machine.

6. The method of claim 1, further comprising, before or after the step of aligning, smoothing the plurality of spectra by averaging over a window.

7. The method of claim 1, further comprising, before or after the step of aligning, extracting pre-determined peaks from the plurality of spectra.

8. The method of claim 1, wherein the scoring of similarity is made noise invariant by discarding a lower portion of the plurality of spectra.

9. The method of claim 1, wherein the scoring of similarity is made noise invariant by using repeated measurements of the protein samples.

10. The method of claim 1, further comprising storing the listing in the storage device.

11. The method of claim 1, wherein the feature selection algorithm is selected from SVM-recursive feature elimination and l0-norm Minimization.

12. A method for analysis of mass spectrographic data generated from protein samples comprising different biological conditions, the method comprising:
downloading a plurality of spectra comprising the mass spectrographic data into a computer system comprising a processor and a storage device, wherein the processor is programmed to execute at least one support vector machine and performs the steps of:
aligning spectra within the mass spectrographic data, comprising:
selecting a first example spectrum as a baseline example;
sliding each spectral peak of a second example spectrum one at a time along a plurality of peaks within the baseline example;
applying a scoring function to obtain a similarity score between the second example spectrum and the baseline example, the similarity score being determined according to the relationship






  
    
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where xi and x0 are feature vectors corresponding to peaks of an ith example spectrum and the baseline spectrum, respectively, and ∥xi∥ is the li norm,




  
    
      
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offsetting the second example spectrum relative to the baseline example according to the similarity score achieved for the second example spectrum;
repeating the step of aligning the spectra for at least one additional example spectrum to create a set of aligned spectra;

applying a feature selection algorithm to the set of aligned spectra to select a subset of spectral peaks that discriminate between the different biological conditions;
training the at least one support vector machine using a first portion of the set of aligned spectra to discriminate between the plurality of different biological conditions, wherein the at least one support vector machine comprises a kernel;
testing the trained at least one support vector machine using a second portion of the set of aligned spectra;
if the trained at least one support vector machine has produced an optimal solution, inputting live spectral data from live protein samples to be analyzed, selecting the subset of spectral peaks from the live spectral data, aligning the live spectral data and processing the live spectral data using the trained at least one support vector machine;
if the trained at least one support vector machine has not produced an optimal solution, repeating the steps of training and testing with a different kernel until an optimal solution is obtained; and
displaying on a display device a listing comprising two or more subsets of live protein samples, wherein each subset is identified as belonging to a group possessing one of the different biological conditions.

13. The method of claim 12, wherein the mass spectrographic data are generated by a surface emitting laser desorption ionization system.

14. The method of claim 12, further comprising, before or after the step of aligning, normalizing each spectrum.

15. The method of claim 12, wherein the kernel comprises a polynomial kernel of degree 2.

16. The method of claim 12, wherein the at least one support vector machine comprises a multi-class support vector machine.

17. The method of claim 12, further comprising, before or after the step of aligning, smoothing the plurality of spectra by averaging over a window.

18. The method of claim 12, further comprising, before or after the step of aligning, extracting pre-determined peaks from the plurality of spectra.

19. The method of claim 12, wherein the scoring of similarity is made noise invariant by discarding a lower portion of the plurality of spectra.

20. The method of claim 12, wherein the scoring of similarity is made noise invariant by using repeated measurements of the protein samples.

21. The method of claim 12, further comprising storing the listing in the storage device.

22. The method of claim 12, wherein the feature selection algorithm is selected from SVM-recursive feature elimination and l0-norm Minimization.

23. A method for analysis of protein mass spectrographic data contained in a plurality of spectra generated from a plurality of protein samples comprising a plurality of different medical conditions, the data residing in a storage device, the method comprising:
aligning the plurality of spectra using a processor by constructing a similarity measure for comparing pairs of spectra with a baseline spectrum, the similarity measure being determined according to the relationship





  
    
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where xi and x0 are feature vectors corresponding to peaks of an ith example spectrum and the baseline spectrum, respectively, and ∥xi∥ is the l1 norm,




  
    
      
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based upon the similarity measure, offsetting each example spectrum from the baseline to provide a set of aligned spectra;
applying a feature selection algorithm to the set of aligned spectra to select a subset of spectral peaks that discriminate between the different biological conditions;
training at least one support vector machine with a first portion of the subset of spectral peaks to discriminate between the plurality of different sample classes, wherein the at least one support vector machine comprises a kernel;
testing the trained at least one support vector machine using a second portion of the subset of spectral peaks;
if the trained at least one support vector machine has produced an optimal solution based on known classifications of the training and test spectra, inputting live spectral data from live protein samples to be analyzed, aligning the live spectral data, selecting the subset of spectral peaks from the live spectral data, and processing the live spectral data using the trained at least one support vector machine;
if the trained at least one support vector machine has not produced an optimal solution, repeating the steps of training and testing with a different kernel until an optimal solution is obtained; and
displaying on a display device a listing comprising two or more subsets of live protein samples, wherein each subset is associated with one of the different medical conditions.

24. The method of claim 23, further comprising storing the listing in the storage device.

25. The method of claim 23, wherein the feature selection algorithm is selected from SVM-recursive feature elimination and l0-norm Minimization.