Source: http://www.google.com/patents/US6571227?dq=5,960,411
Timestamp: 2015-02-01 09:43:28
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Patent US6571227 - Method, system and computer program product for non-linear mapping of multi ... - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Sign inAdvanced Patent SearchPatentsA method, system and computer program product for scaling, or dimensionally reducing, multi-dimensional data sets, that scales well for large data sets. The invention scales multi-dimensional data sets by determining one or more non-linear functions between a sample of points from the multi-dimensional...http://www.google.com/patents/US6571227?utm_source=gb-gplus-sharePatent US6571227 - Method, system and computer program product for non-linear mapping of multi-dimensional dataAdvanced Patent SearchPublication numberUS6571227 B1Publication typeGrantApplication numberUS 09/303,671Publication dateMay 27, 2003Filing dateMay 3, 1999Priority dateNov 4, 1996Fee statusPaidAlso published asCA2371649A1, EP1175648A1, US7117187, US20030195897, WO2000067148A1Publication number09303671, 303671, US 6571227 B1, US 6571227B1, US-B1-6571227, US6571227 B1, US6571227B1InventorsDimitris K. Agrafiotis, Victor S. Lobanov, Francis R. SalemmeOriginal Assignee3-Dimensional Pharmaceuticals, Inc.Export CitationBiBTeX, EndNote, RefManPatent Citations (80), Non-Patent Citations (170), Referenced by (52), Classifications (16), Legal Events (5) External Links: USPTO, USPTO Assignment, EspacenetMethod, system and computer program product for non-linear mapping of multi-dimensional dataUS 6571227 B1Abstract A method, system and computer program product for scaling, or dimensionally reducing, multi-dimensional data sets, that scales well for large data sets. The invention scales multi-dimensional data sets by determining one or more non-linear functions between a sample of points from the multi-dimensional data set and a corresponding set of dimensionally reduced points, and thereafter using the non-linear function to non-linearly map additional points. The additional points may be members of the original multi-dimensional data set or may be new, previously unseen points. In an embodiment, the invention begins with a sample of points from an n-dimensional data set and a corresponding set of m-dimensional points. Alternatively, the invention selects a sample of points from an n-dimensional data set and non-linearly maps the sample of points to obtain the corresponding set of m-dimensional points.
CROSS REFERENCE TO RELATED APPLICATIONS This application is a continuation -in-part of Ser. No. 09/073,845 filed May 7, 1998 now U.S. Pat. No. 6,453,246, Sep. 17, 2002 titled, �System, Method and Computer Program Product for Representing Proximity Data in a Multi-Dimensional Space,� which is a continuation-in-part of Ser. No. 08/963,872 filed Nov. 11, 1997, now U.S. Pat. No. 6,295,514, titled �System, Method and Computer Program Product for Representing Similarly/Dissimilarly Between Current Compounds� which claims priority to U.S. Provisional Patent Application Ser. No. 60/030,187, filed Nov. 4, 1996, titled �Stochastic Algorithms for Maximizing Molecular Diversity.�
SUMMARY OF THE INVENTION A method, system and computer program product for scaling, or dimensionally reducing, multi-dimensional data sets, that scales well for large data sets. The invention scales multi-dimensional data sets by determining one or more non-linear functions between a sample of points from the multi-dimensional data set and a corresponding set of dimensionally reduced points, and thereafter using the non-linear function to non-linearly map additional points. The additional points may be members of the original multi-dimensional data set or may be new, previously unseen points. In an embodiment, the invention begins with a sample of points from an n-dimensional data set and a corresponding set of m-dimensional points. Alternatively, the invention selects a sample of points from an n-dimensional data set and non-linearly maps the sample of points to obtain the corresponding set of m-dimensional points. Any suitable non-linear mapping or multi-dimensional scaling technique can be employed. The process then trains a system (e.g., a neural network), using the corresponding sets of points. During, or at the conclusion of the training process, the system develops or determines a relationship between the two sets of points. In an embodiment, the relationship is in the form of one or more non-linear functions. The one or more non-linear functions are then implemented in a system. Thereafter, additional n-dimensional points are provided to the system, which maps the additional points using the one or more non-linear functions, which is much faster than using conventional multi-dimensional scaling techniques. In an embodiment, the determination of the non-linear relationship is performed by a self-learning system such as a neural network. The additional points are then be mapped using the self-learning system in a feed-forward manner.
Non-Linear Mapping Networks A. Introduction Among the many dimensionality reduction techniques that have appeared in the statistical literature, multi-dimensional scaling and non-linear mapping stand alone for their conceptual elegance and ability to reproduce the topology and structure of the data space in a faithful and unbiased manner. Unfortunately, all known algorithms exhibit quadratic time complexity which imposes severe limitations on the size of data sets that can be effectively analyzed using existing hardware. This specification describes a novel approach that combines �conventional� non-linear mapping techniques with feed-forward neural networks, and allows the processing of data sets orders of magnitude larger than those accessible using conventional methodologies. Rooted on the principle of probability sampling, the method employs an algorithm to multi-dimensionally scale a small random sample, and then �learns� the underlying non-linear transform using a multi-layer perceptron trained with the back-propagation algorithm. Once trained, the neural network can be used in a feed-forward manner to project new, unseen samples with minimal distortion. Using examples from the fields of combinatorial chemistry and computer vision, we demonstrate that this method can generate projections that are virtually indistinguishable from those derived by classical methodologies, and can do so at a fraction of the time required by these techniques. The ability to encode the non-linear transform in the form of a neural network opens new possibilities and makes non-linear mapping applicable in situations where its use has been hitherto unthinkable.
B. Dimensionality Reduction Dimensionality reduction and visualization is of paramount importance in scientific data analysis. Of particular importance is the ability to understand the structure and topology of the data, and the interrelationships and associations between the subjects of our study. Such relationships are often described by means of a similarity index derived either through direct observation, or through the measurement of a set of characteristic features which are subsequently combined in some form of dissimilarity or distance measure. Indeed, distance is an ubiquitous concept, and represents one of the most reliable guiding principles for understanding our universe, one that we can comprehend, feel comfortable with, and navigate with ease and confidence.
This specification describes a new approach to an old-fashioned way of looking at high-dimensional data. High-dimensional spaces possess properties that challenge and often contradict the intuition that we have developed from our experience with 2- or 3-dimensional geometry. This complexity has often been referred to as the �curse of dimensionality�, a term originally introduced by Bellman to describe the complexity of combinatorial optimization over many dimensions. (See Bellman, R. E., Adaptive Control Processes, Princeton University Press, 1961, incorporated herein by reference in its entirety). In statistics, this expression is used to describe the sparsity of data in higher dimensions. As the classic examples of the effect of dimensionality on the volumes of a hypersphere and a hypercube illustrate, (see for example, Wegman, E. J. Ann. Statist., 1970, and Scott, D. W., Multivariate Density Estimation. Theory, Practice and Visualization, Wiley, N.Y., 1992, both of which are incorporated herein by reference in their entireties), most of the density of a high-dimensional space is concentrated near its boundaries, leaving the �core� virtually empty. Indeed, the concept of �neighborhoods� in higher dimensions is somewhat distorted; if the neighborhoods are �local�, they are virtually empty; if they are not empty, then they are not �local�. This has important consequences in many statistical applications.
C. Non-Linear Mapping Multi-dimensional scaling (MDS), non-linear mapping (NLM) and Kohonen networks represent alternative dimensionality reduction techniques that deal specifically with non-linear spaces. See for example: Borg, I., Groenen, P., Modern Multidimensional Scaling, Springer-Verlag, N.Y., 1997; Sammon, J. W. IEEE Trans. Comp., 1969; and Kohonen, T. Self-Organizing Maps, Springer-Verlag, Heidelberg, 1996; respectively, all of which are incorporated by reference in their entirety.
Non-linear mapping (NLM) is a closely related technique proposed by Sammon in 1969. See for example, Sammon, J. W. IEEE Trans. Comp., 1969, discussed above. Just like MDS, NLM attempts to approximate local geometric relationships on a 2- or 3-dimensional plot. Although an �exact� projection is only possible when the distance matrix is positive definite, meaningful projections can be obtained even when this criterion is not satisfied. As in MDS, the process starts with a finite set of samples {xi, i=1, 2, . . . , k}, a symmetric dissimilarity matrix dij, and a set of images of xi on a display plane {ξi, i=1, 2, . . . k; ξi∈ d}, and attempts to place ξi onto the plane in such a way that their Euclidean distances δij=||ξi−ξj|| approximate as closely as possible the corresponding values dij. The embedding (which can only be made approximately) is carried out in an iterative fashion by minimizing an error function, E, which measures the difference between the distance matrices of the original and projected vector sets: E  ( m ) = ∑ k i < j  [ d ij - δ ij  ( m ) ] 2 d ij ∑ k i < j  d ij Eq.��2 E is minimized using a steepest-descent algorithm. The initial coordinates, ξi, are determined at random or by some other projection technique such as PCA, and are updated using Eq. 3:
ξpq(m+1)=ξpq(m)−λΔpq(m) Eq. 3 where m is the iteration number and λ is the learning rate parameter, and Δ pq  ( m ) = ∂ E  ( m ) ∂ ξ pq  ( m ) /  ∂ 2  E  ( m ) ∂ ξ pq  ( m ) 2  Eq.��4 The advantage of non-linear maps compared to Kohonen networks is that they provide much greater individual detail and lend themselves beautifully for interactive analysis and visual inspection. By preserving the distances of the original samples on the projected map, MDS and NLM are able to represent the topology and structural relationships in the data set in a unique and faithful manner. Although in most cases projection does lead to some loss of information, the amount of distortion induced by NLM and MDS is minimal compared to other dimensionality reduction techniques. Unfortunately, despite these advantages, all known non-linear mapping algorithms exhibit quadratic time complexity and scale adversely with the size of the data set.
D. Non-Linear Mapping Networks The method described herein is rooted in the principle of probability sampling, i.e. the notion that a small number of randomly chosen members of a given population will tend to have the same characteristics, and in the same proportion, with the population as a whole. Our approach is to employ an algorithm to: multi-dimensionally scale a small random sample which reflects the overall structure of the data, and then �learn� the underlying non-linear transform using a multi-layer perceptron trained with the back-propagation algorithm. See for example, Haykin, S. Neural Networks. A Comprehensive Foundation. Prentice-Hall, 1998.
Let us first examine the characteristics of the Kruskal stress function which measures the quality of the fit between the distance matrices of the original and projected vector sets. As is evident from Eq. 1, this function scales to the square of the number of items in the collection, and is impossible to compute for data sets containing hundreds of thousands to millions of items. However, like many quantities of this kind, stress has well-defined statistical properties: one can obtain reliable estimates of that quantity by examining only a small fraction of the total number of pair-wise distances in the data. FIG. 2 illustrates stochastic stress of a non-linear map as a function of sample size for the face data set. The first four columns and their respective error bars represent the mean and standard deviation of the stress of the NLM projection computed using 100, 1,000, 10,000 and 100,000 randomly selected distances, respectively. The last two columns represent the true stress of the NLM and PC projections, respectively, computed by evaluating all 3,457,135 pair-wise distances. FIG. 2 shows the dependence of stress on the size of the sample used to derive it for the 2-dimensional AA non-linear map of the face data set. The numbers reported rip were derived by selecting pairs of points at random, measuring their pair-wise distances in the original and projected vector spaces, and accumulating the error in Eq. 1. For each sample size, n, 100 stress evaluations were carried out, each using a different randomly chosen set of n pair-wise distances. The mean and standard deviation of the resulting distributions are plotted in FIG. 2. It is clear that the �stochastic� stress obtained by this method shows negligible variance for all but the smallest samples, and asymptotically approaches the true stress; indeed, by sampling a mere one thousandth of the total number of distances we obtain a stress that is within 4 decimal places to the true value. This turns out to be true for every data set that we studied, regardless of dimensionality, structure and origin.
We now turn our attention to the effect of sampling in deriving the non-linear map itself. The 2-D PCA and NLM projections of the face data are shown in FIG. 3a and FIG. 3b, respectively. For this and all other data sets used in this study, the PCA projection was derived from the first 2 principal components that accounted for most of the variance in the data, while the non-linear map was obtained with a variant of Sammon's original algorithm developed by our group. See, for example, U.S. Patent Application Ser. No. 09/073,845, filed May 7, 1998, now U.S. Pat. No. 6,453,246 titled, �System, Method and Computer Program Product for Representing Proximity Data in a Multi-Dimensional Space,� incorporated in its entirety above by reference. In general, the two projections are very similar, but differ in one important aspect: in the principal component projection, one dimension is completely suppressed and all characteristics of the man's profile are virtually lost (see FIG. 3a). In contrast, the non-linear map represents a �hybrid� view that combines important, distinctive features of the entire object. While the general shape is still dominated by the head-on view, one can clearly recognize key elements of the facial profile such as the nose, the lips and the chin, as well as a detectable protrusion in the occipotal area of the skull (FIG. 3b). In terms of distortion, NLM does a much better job in preserving the distance matrix than PCA, as manifested by a Kruskal stress of 0.152 and 0.218 for the NLM and PCA projections, respectively.
The present invention captures, or determines, one or more non-linear transforms produced by the classical iterative algorithm, or equivalents thereof, in the form of an analytical function. In order to determine how many points are needed to extract such a relationship, we used a multi-layered perceptron as the non-linear mapping device, and carried out an extensive set of simulations using several sample sizes ranging from 100 to 1,600 points. The experiment consisted of the following steps. For each sample size, n, 100 different random subsets of n points were extracted from the original 3-dimensional object, and were independently mapped using our �classical� non-linear mapping algorithm. The 3 D input and 2 D output coordinates obtained from the NLM were then used to train 100 separate neural networks with 3 input, 10 hidden and 2 output neurons having logistic activation functions. All networks were trained for 10,000 epochs with a linearly decreasing learning rate from 0.5 to 0.01, and a momentum of 0.8. Once the networks were trained, the entire data set of 2,630 points was presented to each one, and 100 new sets of 2-D coordinates were obtained. As stated above, this procedure was repeated for 5 different sample sizes containing 100, 200, 400, 800 and 1,600 points, respectively. To simplify the notation, we will refer to each of these 100 subsets and all of its associated data as a separate �run�.
To get a better appreciation of what the stress values really mean in the context of structure, FIG. 3c shows the non-linear map obtained from a neural network trained with an �average� 400-point sample (400-point training set) and having a stress of 0.158. The map is virtually identical to that obtained by NLM, revealing the same characteristic mix of features from the frontal and lateral views, albeit in a more �regular� form. Indeed, for the purposes of exploratory data analysis, the two images are virtually indistinguishable.
E. Combinatorial Chemistry Although the face data provides a useful test case, it pales in comparison to the data sets for which this method was intended, both in terms of size and dimensionality. One area of particular interest to us, and one in which large data sets are commonplace, is combinatorial chemistry. In recent years, the pharmaceutical and chemical industry have embraced a new set of technologies that allow the simultaneous synthesis and biological evaluation of large chemical libraries containing hundreds to hundreds of thousands or even millions of molecules. See for example, Thompson, L. A., Ellman, J. A. Chem. Rev., 1996, incorporated herein by reference in its entirety. Also see U.S. Pat. No. 5,463,564, incorporated by reference in its entirety.
The results are summarized in FIG. 11. FIG. 11 illustrates stress as a function of sample size for the diamine data set. The two columns and their respective error bars represent the mean and standard deviation of the stress of the NLM projection for the training set and the NN projection for the entire data set over 100 runs for 5 different sample sizes. Each run represents a different set of points comprising the training set. The last two columns represent the stress of the NLM and the PC projections of the entire data set, respectively. The network had 16 input, 10 hidden and 2 output neurons (i.e. a total of 180 synaptic weights) and was trained for 10,000 epochs for training sets containing 100 and 200 points, and for 5,000, 2,000 and 1,000 epochs for training sets containing 400, 800 and 1,600 points, respectively. Surprisingly enough, the network does not overfit even with training sets of 100 points (less than 0.2% of the entire library) where there are nearly 2 synapses per training case. Indeed, every single network that we trained outperformed PCA by a wide margin. As for the overall trends, they are no different than those observed in the two previous examples: increase in sample size leads to better approximations and less variability across different samples. A random sample of 400 points (0.7% of the entire library) leads to a neural map with an average stress of 0.193�0.006, while increase in sample size to 1,600 points (2.8% of the entire library) improves the stress to 0.183�0.002, close to the actual NLM stress of 0.169. The resulting maps (FIG. 12) confirm the close agreement between the conventional and neural non-linear mapping algorithms, and the substantial differences between them and PCA which had a stress of 0.332. A look at the variances of the principal components in Table 1, (FIG. 23) reveals why PCA is such a poor method in this case. The first 2 PC's account for only 69% of the total variance in the data, 10% less than the variance captured by the first 2 PC's in the Gasteiger data set, and 14% less than that of the respective components in the face data set. This unaccounted �residual� variance leads to significant distortion in the principal component map, which in the case of the diamine library provides a mere hint of the true structure of the data, evidenced only by the presence of the two disproportionately populated clusters. Finally, the distance plots (FIG. 13) reveal that short distances are distorted much more severely by PCA, a problem that was also encountered to a lesser extent in the face data set (FIG. 6). Conversely, the non-linear maps distribute the error more evenly across the board, and they show little difference between them.
II. Implementation in a Process The present invention is now described in terms of a series of steps which may be performed manually or automatically (or a combination thereof) by an apparatus utilizing hardware, software, firmware, or any combination thereof. FIG. 14 illustrates a process flow chart 1402 implementing the present invention.
A. Selecting a Sample of Points The process begins at step 1404 which includes selecting a sample of points from an n-dimensional data set. The n-dimensional data set is referred to interchangeably herein as a multi-dimensional data set.
B. Non-Linearly Mapping the Sample of Points Step 1406 includes non-linearly mapping the n-dimensional sample of points to an m-dimensional space, where n and m can be any values, so long as n is greater than m. Step 1406 is also referred to as dimensionality reduction because it scales or reduces the number of dimensions associated with the n-dimensional sample of points. The sample of points are non-linearly mapped using any of a variety of suitable conventional or yet to be developed mapping techniques, or combinations thereof, including, but not limited to techniques disclosed in U.S. Patent Application Ser. No. 09/073,845, filed May 7, 1998, titled, �System, Method and Computer Program Product for Representing Proximity Data in Multi-Dimensional Space.�
C. Determining One or More Non-Linear Functions Step 1408 includes determining one or more non-linear functions underlying the mapped sample of points. In other words, Step 1408 determines one or more non-linear functions that correlate the n-dimensional sample of points from step 1404 with the corresponding m-dimensional points from step 1406. Step 1408 can be performed using any of a variety of techniques, including self learning or organizing techniques such as, but not limited to, multilayer neural networks, as well as other search and/or optimization techniques including, but not limited to, Monte Carlo/random sampling, greedy search algorithms, simulated annealing, evolutionary programming, genetic algorithms, genetic programming, gradient minimization techniques, and combinations thereof.
D. Mapping Additional Points Using the Non-Linear Functions Step 1410 includes mapping additional points using the one or more non-linear functions determined in step 1408. In other words, the one or more non-linear functions are then used to map additional n-dimensional points to the m-dimensional space. The additional points can include remaining members of the original n-dimensional data-set and/or new, previously unseen points, which are not part of the original n-dimensional data-set.
III. Implementation in a System The present invention can be implemented in hardware, software, firmware or combinations thereof. FIG. 15A illustrates an exemplary block diagram of modules and data flow that can be included in a system 1502 that implements the present invention. The block diagram of FIG. 15A is intended to aid in the understanding of the present invention. The present invention is not limited to the exemplary embodiment illustrated in the block diagram of FIG. 15A.
IV. Additional Features of the Invention Additional features of the invention and optional implementation enhancements are now described.
A. Multiple Non-Linear Functions In an embodiment, multiple non-linear functions are determined or derived for the n-dimensional points. In an embodiment, non-linear functions are determined for different sub-sets of points. In an embodiment, one or more of the sub-sets of points overlap. In an alternative embodiment, the multiple sub-sets of point are mutually exclusive.
V. Implementation in a Computer Program Product The present invention can be implemented using one or more computers. Referring to FIG. 22, an exemplary computer 2202 includes one or more processors, such as processor 2204. Processor 2204 is connected to a communication bus 2206. Various software embodiments are described in terms of this example computer system. After reading this description, it will become apparent to a person skilled in the relevant art(s) how to implement the invention using other computer systems and/or computer architectures.
In this document, the term �computer program product� is used to generally refer to media such as removable storage units 2216, 2220, a hard drive 2212 that can be removed from the computer 2202, and signals carrying software received by the communications interface 2222. These computer program products are means for providing software to the computer 2202.
VI. Conclusions This specification describes new non-linear mapping systems, methods, and computer program products, designed specifically for very large data sets, and useful for smaller data sets as well. In an embodiment, the invention combines �conventional� non-linear mapping techniques with feed-forward neural networks, and allows the processing of data sets orders of magnitude larger than those accessible using conventional methodologies. Embodiments use one or more classical techniques, or equivalents thereof, to multi-dimensionally scale a sample, which may be a small random sample, and then �learn� one or more underlying non-linear transforms using, for example, a multi-layer perceptron. Once trained, the neural network can be used in a feed-forward manner to project the remaining members of the population as well as new, unseen samples with minimal distortion. This method is rooted on the principle of probability sampling, and works extremely well across a wide variety of data sets of diverse origin, structure, and dimensionality. This approach makes possible the multi-dimensional scaling of very large data sets, and, by capturing non-linear transforms in the form of analytical functions, opens new possibilities for the application of this invaluable statistical technique.
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