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f14f85c294864e7cd75baa4cdbe3114d46cfa4ae

Invariances in Classification: an efficient SVM implementation

Often, in pattern recognition, complementary knowledge is available. This could be useful to improve the performance of the recognition system. Part of this knowledge regards invariances, in particular when treating images or voice data. Many approaches have been proposed to incorporate invariances in pattern recognition systems. Some of these approaches require a pre-processing phase, others integrate the invariances in the algorithms. We present a unifying formulation of the problem of incorporating invariances into a pattern recognition classifier and we extend the SimpleSVM algorithm [Vishwanathan et al., 2003] to handle invariances efficiently.

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A tutorial on support vector regression

In this tutorial we give an overview of the basic ideas underlying Support Vector (SV) machines for function estimation. Furthermore, we include a summary of currently used algorithms for training SV machines, covering both the quadratic (or convex) programming part and advanced methods for dealing with large datasets. Finally, we mention some modifications and extensions that have been applied to the standard SV algorithm, and discuss the aspect of regularization from a SV perspective.

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Online learning with kernels

Kernel-based algorithms such as support vector machines have achieved considerable success in various problems in batch setting, where all of the training data is available in advance. Support vector machines combine the so-called kernel trick with the large margin idea. There has been little use of these methods in an online setting suitable for real-time applications. In this paper, we consider online learning in a reproducing kernel Hilbert space. By considering classical stochastic gradient descent within a feature space and the use of some straightforward tricks, we develop simple and computationally efficient algorithms for a wide range of problems such as classification, regression, and novelty detection. In addition to allowing the exploitation of the kernel trick in an online setting, we examine the value of large margins for classification in the online setting with a drifting target. We derive worst-case loss bounds, and moreover, we show the convergence of the hypothesis to the minimizer of the regularized risk functional. We present some experimental results that support the theory as well as illustrating the power of the new algorithms for online novelty detection.

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Une boîte à outils rapide et simple pour les SVM

Resume : Si les SVM (Support Vector Machines, ou Separateurs a Vaste Marge) sont aujourd’hui reconnus comme l’une des meilleures methodes d’apprentissage, ils restent consideres comme lents. Nous proposons ici une boite a outils Matlab permettant d’utiliser simplement et rapidement les SVM grâce a une methode de gradient projete particulierement bien adaptee au probleme : SimpleSVM (Vishwanathan et al., 2003). Nous avons choisi de coder cet algorithme dans l’environnement Matlab afin de profiter de sa convivialite tout en s’assurant une bonne efficacite. La comparaison de notre solution avec l’etat de l’art dans le domaine SMO (Sequential Minimal Optimization), montre qu’il s’agit la d’une solution dans certains cas plus rapide et d’une complexite moindre. Pour illustrer la simplicite et la rapidite de notre methode, nous montrons enfin que sur la base de donnees MNIST, il a ete possible d’obtenir des resultats satisfaisants en un temps relativement court (une heure et demi de calcul sur un PC sous linux pour construire 45 classifieurs binaires sur 60.000 exemples en dimension 576). Mots-cles : Support Vector Machine, Separateur a Vaste Marge, SVM, Apprentissage, Boite a outils Matlab, Contraintes actives, Gradient projete, MNIST.

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Binet-Cauchy Kernels

We propose a family of kernels based on the Binet-Cauchy theorem and its extension to Fredholm operators. This includes as special cases all currently known kernels derived from the behavioral framework, diffusion processes, marginalized kernels, kernels on graphs, and the kernels on sets arising from the subspace angle approach. Many of these kernels can be seen as the extrema of a new continuum of kernel functions, which leads to numerous new special cases. As an application, we apply the new class of kernels to the problem of clustering of video sequences with encouraging results.

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Experimentally optimal v in support vector regression for different noise models and parameter settings

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Sample based generalisation bounds

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Learning with non-positive kernels

In this paper we show that many kernel methods can be adapted to deal with indefinite kernels, that is, kernels which are not positive semidefinite. They do not satisfy Mercer's condition and they induce associated functional spaces called Reproducing Kernel Kreĭn Spaces (RKKS), a generalization of Reproducing Kernel Hilbert Spaces (RKHS).Machine learning in RKKS shares many "nice" properties of learning in RKHS, such as orthogonality and projection. However, since the kernels are indefinite, we can no longer minimize the loss, instead we stabilize it. We show a general representer theorem for constrained stabilization and prove generalization bounds by computing the Rademacher averages of the kernel class. We list several examples of indefinite kernels and investigate regularization methods to solve spline interpolation. Some preliminary experiments with indefinite kernels for spline smoothing are reported for truncated spectral factorization, Landweber-Fridman iterations, and MR-II.

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Gaussian process classification for segmenting and annotating sequences

Many real-world classification tasks involve the prediction of multiple, inter-dependent class labels. A prototypical case of this sort deals with prediction of a sequence of labels for a sequence of observations. Such problems arise naturally in the context of annotating and segmenting observation sequences. This paper generalizes Gaussian Process classification to predict multiple labels by taking dependencies between neighboring labels into account. Our approach is motivated by the desire to retain rigorous probabilistic semantics, while overcoming limitations of parametric methods like Conditional Random Fields, which exhibit conceptual and computational difficulties in high-dimensional input spaces. Experiments on named entity recognition and pitch accent prediction tasks demonstrate the competitiveness of our approach.

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Exponential Families for Conditional Random Fields

In this paper we define conditional random fields in reproducing kernel Hilbert spaces and show connections to Gaussian Process classification. More specifically, we prove decomposition results for undirected graphical models and we give constructions for kernels. Finally we present efficient means of solving the optimization problem using reduced rank decompositions and we show how stationarity can be exploited efficiently in the optimization process.

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A Second Order Cone programming Formulation for Classifying Missing Data

We propose a convex optimization based strategy to deal with uncertainty in the observations of a classification problem. We assume that instead of a sample (xi, yi) a distribution over (xi, yi) is specified. In particular, we derive a robust formulation when the distribution is given by a normal distribution. It leads to Second Order Cone Programming formulation. Our method is applied to the problem of missing data, where it outperforms direct imputation.

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Kernel Extrapolations for Enzyme Classification

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Exponential Families and Kernels

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Laplace Propagation

We present a novel method for approximate inference in Bayesian models and regularized risk functionals. It is based on the propagation of mean and variance derived from the Laplace approximation of conditional probabilities in factorizing distributions, much akin to Minka's Expectation Propagation. In the jointly normal case, it coincides with the latter and belief propagation, whereas in the general case, it provides an optimization strategy containing Support Vector chunking, the Bayes Committee Machine, and Gaussian Process chunking as special cases.

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Logic, Trees and Kernels

Kernel based methods achieved much of their initial success on problems with real valued attributes. There are many problems with discrete attributes (including Boolean) and in this paper we present a number of results concerning the kernelisation of Boolean and discrete problems. We give results about the learnability and required complexity of logical formulae to solve classification problems. These results are obtained by linking propositional logic with kernel machines. In particular we show that decision trees and disjunctive normal forms (DNF) can be represented via of a special kernel, which connects the regularised risk to the margin of separation. Subsequently we derive a number of lower bounds on the required complexity of logical formulae using properties of algorithms for generation of linear machines machines. An interesting side eect of the development is a number of connections between machine learning algorithms that utilize discrete structures (such as a trees) and kernel machines. We also present some more general kernel constructions on discrete sets using the machinery of frames. These can be used to progressively penalize higher order interactions by explicitly constructing reproducing kernel Hilbert spaces, their associated kernels and the concomitant use of norm-based used regularization.

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Classification in a normalized feature space using support vector machines

This paper discusses classification using support vector machines in a normalized feature space. We consider both normalization in input space and in feature space. Exploiting the fact that in this setting all points lie on the surface of a unit hypersphere we replace the optimal separating hyperplane by one that is symmetric in its angles, leading to an improved estimator. Evaluation of these considerations is done in numerical experiments on two real-world datasets. The stability to noise of this offset correction is subsequently investigated as well as its optimality.

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Kernels and Regularization on Graphs

We introduce a family of kernels on graphs based on the notion of regularization operators. This generalizes in a natural way the notion of regularization and Greens functions, as commonly used for real valued functions, to graphs. It turns out that diffusion kernels can be found as a special case of our reasoning. We show that the class of positive, monotonically decreasing functions on the unit interval leads to kernels and corresponding regularization operators.

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Machine Learning with Hyperkernels

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Advanced Lectures on Machine Learning

Advanced lectures on machine learning , Advanced lectures on machine learning , کتابخانه دیجیتال جندی شاپور اهواز

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Machine Learning Program , National ICT for Australia , Canberra , ACT 0200 , Australia

We present a fast iterative support vector training algorithm for a large variety of different formulations. It works by incrementally changing a candidate support vector set using a greedy approach, until the supporting hyperplane is found within a finite number of iterations. It is derived from a simple active set method which sweeps through the set of Lagrange multipliers and keeps optimality in the unconstrained variables, while discarding large amounts of bound-constrained variables. The hard-margin version can be viewed as a simple (yet computationally crucial) modification of the incremental SVM training algorithms of Cauwenberghs and Poggio. Experimental results for various settings are reported. In all cases our algorithm is considerably faster than competing methods such as Sequential Minimal Optimization or the Nearest Point Algorithm.

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Kernel Methods and Support Vector Machines

Over the past ten years kernel methods such as Support Vector Machines and Gaussian Processes have become a staple for modern statistical estimation and machine learning. The groundwork for this field was laid in the second half of the 20th century by Vapnik and Chervonenkis (geometrical formulation of an optimal separating hyperplane, capacity measures for margin classifiers), Mangasarian (linear separation by a convex function class), Aronszajn (Reproducing Kernel Hilbert Spaces), Aizerman, Braverman, and Rozonoer (nonlinearity via kernel feature spaces), Arsenin and Tikhonov (regularization and ill-posed problems), and Wahba (regularization in Reproducing Kernel Hilbert Spaces). However, it took until the early 90s until positive definite kernels became a popular and viable means of estimation. Firstly this was due to the lack of sufficiently powerful hardware, since kernel methods require the computation of the socalled kernel matrix, which requires quadratic storage in the number of data points (a computer of at least a few megabytes of memory is required to deal with 1000+ points). Secondly, many of the previously mentioned techniques lay dormant or existed independently and only recently the (in hindsight obvious) connections were made to turn this into a practical estimation tool. Nowadays, a variety of good reference books exist and anyone serious about dealing with kernel methods is recommended to consult one of the following works for further information [15, 5, 8, 12]. Below, we will summarize the main ideas of kernel method and support vector machines, building on the summary given in [13].

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The kernel mutual information

We introduce a new contrast function, the kernel mutual information (KMI), to measure the degree of independence of continuous random variables. This contrast function provides an approximate upper bound on the mutual information, as measured near independence, and is based on a kernel density estimate of the mutual information between a discretised approximation of the continuous random variables. We show that the kernel generalised variance (KGV) of F. Bach and M. Jordan (see JMLR, vol.3, p.1-48, 2002) is also an upper bound on the same kernel density estimate, but is looser. Finally, we suggest that the addition of a regularising term in the KGV causes it to approach the KMI, which motivates the introduction of this regularisation.

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svlab - A Kernel Methods Package

Abstractsvlab is an extensible, object oriented, package for kernel based learningin R. Its main objective is to provide a tool kit consisting of basic kernelfunctionality, optimizers and high level algorithms such as Support VectorMachines and Kernel Principal Component Analysis which can be extendedby the user in a very modular way. Based on this infrastructure kernel-basedmethods can be easily be constructed and developed. 1 Introduction It is often difficult to solve problems like classification, regression and clustering—ormore generally: supervised and unsupervised learning—in the space in which theunderlying observations have been made. One way out is to project the observationsinto a higher-dimensional feature space where these problems are easier to solve,e.g., by using simple linear methods. If the methods applied in the feature spaceare only based on dot or inner products the projection does not have to be carriedout explicitely but only implicitely using kernel functions. This is often referred toas the “kernel trick”. More precisely, if a projection Φ : X → H is used the dotproduct hΦ(x),Φ(y)i can be represented by a kernel function kk(x,y) = hΦ(x),Φ(y)i, (1)

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Hilbert space embeddings in dynamical systems

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Constructing Descriptive and Discriminative Nonlinear Features: Rayleigh Coefficients in Kernel Feature Spaces

We incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination. Employing a unified framework in terms of a nonlinearized variant of the Rayleigh coefficient, we propose nonlinear generalizations of Fisher's discriminant and oriented PCA using support vector kernel functions. Extensive simulations show the utility of our approach.

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Machine learning using hyperkernels

We expand on the problem of learning a kernel via a RKHS on the space of kernels itself. The resulting optimization problem is shown to have a semidefinite programming solution. We demonstrate that it is possible to learn the kernel for various formulations of machine learning problems. Specifically, we provide mathematical programming formulations and experimental results for the C-SVM, ν-SVM and Lagrangian SVM for classification on UCI data, and novelty detection.

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Bayesian Kernel Models

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Advanced Lectures on Machine Learning: Machine Learning Summer School 2002, Canberra, Australia, February 11-22, 2002, Revised Lectures

A Few Notes on Statistical Learning Theory.- A Short Introduction to Learning with Kernels.- Bayesian Kernel Methods.- An Introduction to Boosting and Leveraging.- An Introduction to Reinforcement Learning Theory: Value Function Methods.- Learning Comprehensible Theories from Structured Data.- Algorithms for Association Rules.- Online Learning of Linear Classifiers.

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Multi-Instance Kernels

Learning from structured data is becoming increasingly important. However, most prior work on kernel methods has focused on learning from attribute-value data. Only recently, research started investigating kernels for structured data. This paper considers kernels for multi-instance problems a class of concepts on individuals represented by sets. The main result of this paper is a kernel on multi-instance data that can be shown to separate positive and negative sets under natural assumptions. This kernel compares favorably with state of the art multi-instance learning algorithms in an empirical study. Finally, we give some concluding remarks and propose future work that might further improve the results.

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Large Margin Classification for Moving Targets

We consider using online large margin classification algorithms in a setting where the target classifier may change over time. The algorithms we consider are Gentile's ALMA, and an algorithm we call NORMA which performs a modified online gradient descent with respect to a regularised risk. The update rule of ALMA includes a projection-based regularisation step, whereas NORMA has a weight decay type of regularisation. For ALMA we can prove mistake bounds in terms of the total distance the target moves during the trial sequence. For NORMA, we need the additional assumption that the movement rate stays sufficiently low uniformly over time. In addition to the movement of the target, the mistake bounds for both algorithms depend on the hinge loss of the target. Both algorithms use a margin parameter which can be tuned to make them mistake-driven (update only when classification error occurs) or more aggressive (update when the confidence of the classification is below the margin). We get similar mistake bounds both for the mistake-driven and a suitable aggressive tuning. Experiments on artificial data confirm that an aggressive tuning is often useful even if the goal is just to minimise the number of mistakes.

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Kernel Ma hines and Boolean Fun tions

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Support Vector Machines and Kernel Algorithms

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Minimal Kernel Classifiers

A finite concave minimization algorithm is proposed for constructing kernel classifiers that use a minimal number of data points both in generating and characterizing a classifier. The algorithm is theoretically justified on the basis of linear programming perturbation theory and a leave-one-out error bound as well as effective computational results on seven real world datasets. A nonlinear rectangular kernel is generated by systematically utilizing as few of the data as possible both in training and in characterizing a nonlinear separating surface. This can result in substantial reduction in kernel data-dependence (over 94% in six of the seven public datasets tested on) and with test set correctness equal to that obtained by using a conventional support vector machine classifier that depends on many more data points. This reduction in data dependence results in a much faster classifier that requires less storage. To eliminate data points, the proposed approach makes use of a novel loss function, the "pound" function (·)#, which is a linear combination of the 1-norm and the step function that measures both the magnitude and the presence of any error.

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Adapting Codes and Embeddings for Polychotomies

In this paper we consider formulations of multi-class problems based on a generalized notion of a margin and using output coding. This includes, but is not restricted to, standard multi-class SVM formulations. Differently from many previous approaches we learn the code as well as the embedding function. We illustrate how this can lead to a formulation that allows for solving a wider range of problems with for instance many classes or even "missing classes". To keep our optimization problems tractable we propose an algorithm capable of solving them using two-class classifiers, similar in spirit to Boosting.

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Learning with Kernels: support vector machines, regularization, optimization, and beyond

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Fast Kernels for String and Tree Matching

In this paper we present a new algorithm suitable for matching discrete objects such as strings and trees in linear time, thus obviating dynamic programming with quadratic time complexity. Furthermore, prediction cost in many cases can be reduced to linear cost in the length of the sequence to be classified, regardless of the number of support vectors. This improvement on the currently available algorithms makes string kernels a viable alternative for the practitioner.

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A Short Introduction to Learning with Kernels

We briefly describe the main ideas of statistical learning theory, support vector machines, and kernel feature spaces. This includes a derivation of the support vector optimization problem for classification and regression, the v-trick, various kernels and an overview over applications of kernel methods.

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Bayesian Kernel Methods

Bayesian methods allow for a simple and intuitive representation of the function spaces used by kernel methods. This chapter describes the basic principles of Gaussian Processes, their implementation and their connection to other kernel-based Bayesian estimation methods, such as the Relevance Vector Machine.

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Sparse Kernel Feature Analysis

Kernel Principal Component Analysis (KPCA) has proven to be a versatile tool for unsupervised learning, however at a high computational cost due to the dense expansions in terms of kernel functions. We overcome this problem by proposing a new class of feature extractors employing l1 norms in coefficient space instead of the Reproducing Kernel Hilbert Space in which KPCA was originally formulated in. Moreover, the modified setting allows us to efficiently extract features which maximize criteria other than the variance in a way similar to projection pursuit.

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Hyperkernels

We consider the problem of choosing a kernel suitable for est imation using a Gaussian Process estimator or a Support Vector Machi ne. A novel solution is presented which involves defining a Reprod ucing Kernel Hilbert Space on the space of kernels itself. By utilizin g an analog of the classical representer theorem, the problem of choosi ng a kernel from a parameterized family of kernels (e.g. of varying widt h) is reduced to a statistical estimation problem akin to the problem of mi ni izing a regularized risk functional. Various classical settings f or model or kernel selection are special cases of our framework.

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A Tutorial Introduction

This chapter contains sections titled: Data Representation and Similarity, A Simple Pattern Recognition Algorithm, Some Insights From Statistical Learning Theory, Hyperplane Classifiers, Support Vector Classification, Support Vector Regression, Kernel Principal Component Analysis, Empirical Results and Implementations

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A Generalized Representer Theorem

Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empirical risk terms, and give a self-contained proof utilizing the feature space associated with a kernel. The result shows that a wide range of problems have optimal solutions that live in the finite dimensional span of the training examples mapped into feature space, thus enabling us to carry out kernel algorithms independent of the (potentially infinite) dimensionality of the feature space.

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Sparse GreedyGaussian Pro ess Regression

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Elements of Statistical Learning Theory

This chapter contains sections titled: Introduction, The Law of Large Numbers, When Does LearningWork: the Question of Consistency, Uniform Convergence and Consistency, How to Derive a VC Bound, A Model Selection Example, Summary, Problems

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Notation and Symbols

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Regularized Principal Manifolds

Many settings of unsupervised learning can be viewed as quantization problems -- the minimization of the expected quantization error subject to some restrictions. This allows the use of tools such as regularization from the theory of (supervised) risk minimization for unsupervised settings. Moreover, this setting is very closely related to both principal curves and the generative topographic map. We explore this connection in two ways: 1) we propose an algorithm for finding principal manifolds that can be regularized in a variety of ways. Experimental results demonstrate the feasibility of the approach. 2) We derive uniform convergence bounds and hence bounds on the learning rates of the algorithm. In particular, we give good bounds on the covering numbers which allows us to obtain a nearly optimal learning rate of order O(m-1/2+α) for certain types of regularization operators, where m is the sample size and ff an arbitrary positive constant.

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Kernel Fisher Discriminant

This chapter contains sections titled: Introduction, Fisher's Discriminant in Feature Space, Efficient Training of Kernel Fisher Discriminants, Probabilistic Outputs, Experiments, Summary, Problems

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Pre-Images and Reduced Set Methods

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Single-Class Problems: Quantile Estimation and Novelty Detection

This chapter contains sections titled: Introduction, A Distribution's Support and Quantiles, Algorithms, Optimization, Theory, Discussion, Experiments, Summary, Problems

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Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond

From the Publisher: In the 1990s, a new type of learning algorithm was developed, based on results from statistical learning theory: the Support Vector Machine (SVM). This gave rise to a new class of theoretically elegant learning machines that use a central concept of SVMs—-kernels--for a number of learning tasks. Kernel machines provide a modular framework that can be adapted to different tasks and domains by the choice of the kernel function and the base algorithm. They are replacing neural networks in a variety of fields, including engineering, information retrieval, and bioinformatics. Learning with Kernels provides an introduction to SVMs and related kernel methods. Although the book begins with the basics, it also includes the latest research. It provides all of the concepts necessary to enable a reader equipped with some basic mathematical knowledge to enter the world of machine learning using theoretically well-founded yet easy-to-use kernel algorithms and to understand and apply the powerful algorithms that have been developed over the last few years.

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Regularized principal manifolds

Many settings of unsupervised learning can be viewed as quantization problems - the minimization of the expected quantization error subject to some restrictions. This allows the use of tools such as regularization from the theory of (supervised) risk minimization for unsupervised learning. This setting turns out to be closely related to principal curves, the generative topographic map, and robust coding.We explore this connection in two ways: (1) we propose an algorithm for finding principal manifolds that can be regularized in a variety of ways; and (2) we derive uniform convergence bounds and hence bounds on the learning rates of the algorithm. In particular, we give bounds on the covering numbers which allows us to obtain nearly optimal learning rates for certain types of regularization operators. Experimental results demonstrate the feasibility of the approach.

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Concepts and Tools

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An improved training algorithm for kernel Fisher discriminants

We present a fast training algorithm for the kernel Fisher discriminant classifier. It uses a greedy approximation technique and has an empirical scaling behavior which improves upon the state of the art by more than an order of magnitude, thus rendering the kernel Fisher algorithm a viable option also for large datasets.

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Estimating the Support of a High-Dimensional Distribution

Suppose you are given some data set drawn from an underlying probability distribution P and you want to estimate a simple subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified value between 0 and 1. We propose a method to approach this problem by trying to estimate a function f that is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabeled data.

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Kernel Feature Extraction

This chapter contains sections titled: Introduction, Kernel PCA, Kernel PCA Experiments, A Framework for Feature Extraction, Algorithms for Sparse KFA, KFA Experiments, Summary, Problems

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Bound on the Leave-One-Out Error for Density Support Estimation using nu-SVMs

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Support vector machine learning

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Learning Theory Revisited

This chapter contains sections titled: Concentration of Measure Inequalities, Leave-One-Out Estimates, PAC-Bayesian Bounds, Operator-Theoretic Methods in Learning Theory, Summary, Problems

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Risk and Loss Functions

This chapter contains sections titled: Loss Functions, Test Error and Expected Risk, A Statistical Perspective, Robust Estimators, Summary, Problems

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Kernel Machines and Boolean Functions

We give results about the learnability and required complexity of logical formulae to solve classification problems. These results are obtained by linking propositional logic with kernel machines. In particular we show that decision trees and disjunctive normal forms (DNF) can be represented by the help of a special kernel, linking regularized risk to separation margin. Subsequently we derive a number of lower bounds on the required complexity of logic formulae using properties of algorithms for generation of linear estimators, such as perceptron and maximal perceptron learning.

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Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators

We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinite-dimensional unit ball in feature space into a finite-dimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence, we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines.

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Query Learning with Large Margin Classifiers

The active selection of instances can significantly improve the generalisation performance of a learning machine. Large margin classifiers such as support vector machines classify data using the most informative instances (the support vectors). This makes them natural candidates for instance selection strategies. In this paper we propose an algorithm for the training of support vector machines using instance selection. We give a theoretical justification for the strategy and experimental results on real and artificial data demonstrating its effectiveness. The technique is most efficient when the data set can be learnt using few support vectors.

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GACV for Support Vector Machines

This chapter contains sections titled: Introduction, The SVM Variational Problem, The Dual Problem, The Generalized Comparative Kullback-Leibler Distance, Leaving-out-one and the GACV, Numerical Results, Acknowledgments

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Adaptive Margin Support Vector Machines

This chapter contains sections titled: Introduction, Leave-One-Out Support Vector Machines, Adaptive Margin SVMs, Relationship of AM-SVMs to Other SVMs, Theoretical Analysis, Experiments, Discussion

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Regularization with Dot-Product Kernels

In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) = k(x ċ y) satisfy Mercer's condition and thus may be used in Support Vector Machines (SVM), Regularization Networks (RN) or Gaussian Processes (GP). In particular, we show that if the kernel is analytic (i.e. can be expanded in a Taylor series), all expansion coefficients have to be nonnegative. We give an explicit functional form for the feature map by calculating its eigenfunctions and eigenvalues.

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Towards a Strategy for Boosting Regressors

This chapter contains sections titled: Introduction, Background and Results, Top Level Description of the Boosting Strategy, Generation of Weak Learners, Overall Algorithm, Experiments, Conclusions

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Large Margin Bank Boundaries for Ordinal Regression

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Linear Discriminant and Support Vector Classifiers

This chapter contains sections titled: Introduction, What is a Linear Discriminant?, Formulation of the Linear Discriminant Training Problem, Training Algorithms, Which Linear Discriminant?, Conclusion, Acknowledgments

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Margin Distribution and Soft Margin

This chapter contains sections titled: Introduction, Margin Distribution Bound on Generalization, An Explanation for the Soft Margin Algorithm, Related Techniques, Conclusion, Acknowledgments

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Support Vectors and Statistical Mechanics

This chapter contains sections titled: Introduction, The Basic SVM Setting, The Learning Problem, The Approach of Statistical Mechanics, Results I: General, Results II: Overfitting, Results III: Dependence on the Input Density, Discussion and Outlook, Acknowledgments

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Computing the Bayes Kernel Classifier

This chapter contains sections titled: Introduction, A Simple Geometric Problem, The Maximal Margin Perceptron, The Bayes Perceptron, The Kernel-Billiard, Numerical Tests, Conclusions, Appendix

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Bounds on Error Expectation for SVM

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Maximal Margin Perception

This chapter contains sections titled: Introduction, Basic Approximation Steps, Basic Algorithms, Kernel Machine Extension, Soft Margin Extension, Experimental Results, Discussion, Conclusions, Appendix: Details of comparison against six other methods for iterative generation of support vector machines

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Invariant Feature Extra tion andClassi ation in Kernel Spa

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Choosing /spl nu/ in support vector regression with different noise models-theory and experiments

In support vector (SV) regression, a parameter /spl nu/ controls the number of support vectors and the number of points that come to lie outside of the so-called /spl epsi/-insensitive tube. For various noise models and SV parameter settings, we experimentally determine the values of /spl nu/ that lead to the lowest generalization error. We find good agreement with the values that had previously been predicted by a theoretical argument based on the asymptotic efficiency of a simplified model of SV regression.

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Regularization Networks and Support Vector Machines

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New Support Vector Algorithms

We propose a new class of support vector algorithms for regression and classification. In these algorithms, a parameter lets one effectively control the number of support vectors. While this can be useful in its own right, the parameterization has the additional benefit of enabling us to eliminate one of the other free parameters of the algorithm: the accuracy parameter in the regression case, and the regularization constant C in the classification case. We describe the algorithms, give some theoretical results concerning the meaning and the choice of , and report experimental results.

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Sparse Greedy Gaussian Process Regression

We present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m. In particular, computational requirements are O(n2m), storage is O(nm), the cost for prediction is O(n) and the cost to compute confidence bounds is O(nm), where n ≪ m. We show how to compute a stopping criterion, give bounds on the approximation error, and show applications to large scale problems.

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Robust Ensemble Learning for Data Mining

We propose a new boosting algorithm which similarly to v- Support-Vector Classification allows for the possibility of a pre-specified fraction v of points to lie in the margin area or even on the wrong side of the decision boundary. It gives a nicely interpretable way of controlling the trade-off between minimizing training error and capacity. Furthermore, it can act as a filter for finding and selecting informative patterns from a database.

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Kernel method for percentile feature extraction

A method is proposed which computes a direction in a dataset such that a speci ed fraction of a particular class of all examples is separated from the overall mean by a maximal margin. The projector onto that direction can be used for class-speci c feature extraction. The algorithm is carried out in a feature space associated with a support vector kernel function, hence it can be used to construct a large class of nonlinear feature extractors. In the particular case where there exists only one class, the method can be thought of as a robust form of principal component analysis, where instead of variance we maximize percentile thresholds. Finally, we generalize it to also include the possibility of specifying negative examples.

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Sparse Greedy Matrix Approximation for Machine Learning

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Beyond the Margin

The concept of large margins is a unifying principle for the analysis of many different approaches to the classification of data from examples, including boosting, mathematical programming, neural networks, and support vector machines. The fact that it is the margin, or confidence level, of a classification--that is, a scale parameter--rather than a raw training error that matters has become a key tool for dealing with classifiers. This book shows how this idea applies to both the theoretical analysis and the design of algorithms.The book provides an overview of recent developments in large margin classifiers, examines connections with other methods (e.g., Bayesian inference), and identifies strengths and weaknesses of the method, as well as directions for future research. Among the contributors are Manfred Opper, Vladimir Vapnik, and Grace Wahba.

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Generalized Support Vector Machines

This chapter contains sections titled: Introduction, GSVM: The General Support Vector Machine, Quadratic Programming Support Vector Machines, Linear Programming Support Vector Machines, A Simple Illustrative Example, Conclusion, Acknowledgments

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Proc.17th Int Conference on Machine Learning

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Natural Regularization from Generative Models

This chapter contains sections titled: Introduction, Natural Kernels, The Natural Regularization Operator, The Feature Map of Natural Kernel, Experiments, Discussion

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Probabilities for SV Machines

This chapter contains sections titled: Introduction, Fitting a Sigmoid After the SVM, Empirical Tests, Conclusions, Appendix: Pseudo-code for the Sigmoid Training

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Introduction to Large Margin Classifiers

This chapter contains sections titled: A Simple Classification Problem, Theory, Support Vector Machines, Boosting, Empirical Results, Implementations, and Further Developments, Notations

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[Anastomosis using the Valtrac ring--pro and con].

Authors present their experiences with the gastrointestinal anastomosis construction with the biofragmentable ring (Valtrac). Between May 1995 and June 1999 they used it in the group of 75 patients with mean age 58.4 (range 19-59) years. They used Valtrac most often--in 32 patients--to construct the anastomosis between small and large bowel after right hemicolectomy. One enterocutaneous fistula and one intestinal obstruction due to adhesions occurred in this group. Two patients had the signs of fecal impaction in postoperative period, which disappeared after the ring fragmentation. According to their experiences authors claim, that the anastomosis with Valtrac is a safe procedure with a few number of postoperative complications. Disadvantage in our conditions is its high price.

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Robust Ensemble Learning for Data

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Advances in Large Margin Classifiers

From the Publisher: The concept of large margins is a unifying principle for the analysis of many different approaches to the classification of data from examples, including boosting, mathematical programming, neural networks, and support vector machines. The fact that it is the margin, or confidence level, of a classification--that is, a scale parameter--rather than a raw training error that matters has become a key tool for dealing with classifiers. This book shows how this idea applies to both the theoretical analysis and the design of algorithms. The book provides an overview of recent developments in large margin classifiers, examines connections with other methods (e.g., Bayesian inference), and identifies strengths and weaknesses of the method, as well as directions for future research. Among the contributors are Manfred Opper, Vladimir Vapnik, and Grace Wahba.

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The Entropy Regularisation Information Criterion

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Leave-One-Out Methods

The concept of large margins is a unifying principle for the analysis of many different approaches to the classification of data from examples, including boosting, mathematical programming, neural networks, and support vector machines. The fact that it is the margin, or confidence level, of a classification--that is, a scale parameter--rather than a raw training error that matters has become a key tool for dealing with classifiers. This book shows how this idea applies to both the theoretical analysis and the design of algorithms.The book provides an overview of recent developments in large margin classifiers, examines connections with other methods (e.g., Bayesian inference), and identifies strengths and weaknesses of the method, as well as directions for future research. Among the contributors are Manfred Opper, Vladimir Vapnik, and Grace Wahba.

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Dynamic Alignment Kernels

This chapter contains sections titled: Introduction: Linear Methods using Kernel function, Applying Linear Methods to Structured Objects, Conditional Symmetric Independence Kernels, Pair Hidden Markov Models, Conditionally Symmetrically Independent PHMMs, Conclusion

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Gaussian Processes and SVM: Mean Field and Leave-One-Out

This chapter contains sections titled: Introduction, Gaussian Process Classification, Modeling the Noise, From Gaussian Processes to SVM, Leave-One-Out Estimator, Naive Mean Field Algorithm, Simulation Results, Conclusion

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Entropy Numbers of Linear Function Classes

This paper collects together a miscellany of results originally motivated by the analysis of the generalization performance of the “maximum-margin” algorithm due to Vapnik and others. The key feature of the paper is its operator-theoretic viewpoint. New bounds on covering numbers for classes related to Maximum Margin classes are derived directly without making use of a combinatorial dimension such as the VC-dimension. Specific contents of the paper include: a new and self-contained proof of Maurey’s theorem and some generalizations with small explicit values of constants; bounds on the covering numbers of maximum margin classes suitable for the analysis of their generalization performance; the extension of such classes to those induced by balls in quasi-Banach spaces (such as norms with ). extension of results on the covering numbers of convex hulls of basis functions to -convex hulls ( ); an appendix containing the tightest known bounds on the entropy numbers of the identity operator between and ( ).

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Entropy Numbers for Convex Combinations and MLPs

This chapter contains sections titled: Introduction, Tools from Functional Analysis, Convex Combinations of Parametric Families, Convex Combinations of Kernels, Multilayer Networks, Discussion, Appendix: A Remark on Traditional Weight Decay, Appendix: Proofs

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Lernen mit Kernen

Zusammenfassung. Dieser Beitrag erläutert neue Ansätze und Ergebnisse der statistischen Lerntheorie. Nach einer Einleitung wird zunächst das Lernen aus Beispielen vorgestellt und erklärt, dass neben dem Erklären der Trainingdaten die Komplexität von Lernmaschinen wesentlich für den Lernerfolg ist. Weiterhin werden Kern-Algorithmen in Merkmalsräumen eingeführt, die eine elegante und effiziente Methode darstellen, verschiedene Lernmaschinen mit kontrollierbarer Komplexität durch Kernfunktionen zu realisieren. Beispiele für solche Algorithmen sind Support-Vektor-Maschinen(SVM), die Kernfunktionen zur Schätzung von Funktionen verwenden, oder Kern-PCA (principal component analysis), die Kernfunktionen zur Extraktion von nichtlinearen Merkmalen aus Datensätzen verwendet. Viel wichtiger als jedes einzelne Beispiel ist jedoch die Einsicht, dass jeder Algorithmus, der sich anhand von Skalarprodukten formulieren lässt, durch Verwendung von Kernfunktionen nichtlinear verallgemeinert werden kann.Die Signifikanz der Kernalgorithmen soll durch einen kurzen Abriss einiger industrieller und akademischer Anwendungen unterstrichen werden. Hier konnten wir Rekordergebnisse auf wichtigen praktisch relevanten Benchmarks erzielen.Abstract. We describe recent developments and results of statistical learning theory. In the framework of learning from examples, two factors control generalization ability: explaining the training data by a learning machine of a suitable complexity. We describe kernel algorithms in feature spaces as elegant and efficient methods of realizing such machines. Examples thereof are Support Vector Machines (SVM) and Kernel PCA (Principal Component Analysis). More important than any individual example of a kernel algorithm, however, is the insight that any algorithm that can be cast in terms of dot products can be generalized to a nonlinear setting using kernels.Finally, we illustrate the significance of kernel algorithms by briefly describing industrial and academic applications, including ones where we obtained benchmark record results.

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Invariant Feature Extraction and Classification in Kernel Spaces

We incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination. Employing a unified framework in terms of a nonlinear variant of the Rayleigh coefficient, we propose non-linear generalizations of Fisher's discriminant and oriented PCA using Support Vector kernel functions. Extensive simulations show the utility of our approach.

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Kernel principal component analysis

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Classification on proximity data with LP-machines

We provide a new linear program to deal with classification of data in the case of data given in terms of pairwise proximities. This allows to avoid the problems inherent in using feature spaces with indefinite metric in support vector machines, since the notion of a margin is purely needed in input space where the classification actually occurs. Moreover in our approach we can enforce sparsity in the proximity representation by sacrificing training error. This turns out to be favorable for proximity data. Similar to /spl nu/-SV methods, the only parameter needed in the algorithm is the (asymptotical) number of data points being classified with a margin. Finally, the algorithm is successfully compared with /spl nu/-SV learning in proximity space and K-nearest-neighbors on real world data from neuroscience and molecular biology.