Notes on PCA, Regularization, Sparsity and Support Vector Machines

We derive a new representation for a function as a linear combination of local correlation kernels at optimal sparse locations and discuss its relation to PCA, regularization, sparsity principles and Support Vector Machines. We first review previous results for the approximation of a function from discrete data (Girosi, 1998) in the context of Vapnik"s feature space and dual representation (Vapnik, 1995). We apply them to show 1) that a standard regularization functional with a stabilizer defined in terms of the correlation function induces a regression function in the span of the feature space of classical Principal Components and 2) that there exist a dual representations of the regression function in terms of a regularization network with a kernel equal to a generalized correlation function. We then describe the main observation of the paper: the dual representation in terms of the correlation function can be sparsified using the Support Vector Machines (Vapnik, 1982) technique and this operation is equivalent to sparsify a large dictionary of basis functions adapted to the task, using a variation of Basis Pursuit De-Noising (Chen, Donoho and Saunders, 1995; see also related work by Donahue and Geiger, 1994; Olshausen and Field, 1995; Lewicki and Sejnowski, 1998). In addition to extending the close relations between regularization, Support Vector Machines and sparsity, our work also illuminates and formalizes the LFA concept of Penev and Atick (1996). We discuss the relation between our results, which are about regression, and the different problem of pattern classification.

From Regression to Classification in Support Vector Machines

We study the relation between support vector machines (SVMs) for regression (SVMR) and SVM for classification (SVMC). We show that for a given SVMC solution there exists a SVMR solution which is equivalent for a certain choice of the parameters. In particular our result is that for $epsilon$ sufficiently close to one, the optimal hyperplane and threshold for the SVMC problem with regularization parameter C_c are equal to (1-epsilon)^{- 1} times the optimal hyperplane and threshold for SVMR with regularization parameter C_r = (1-epsilon)C_c. A direct consequence of this result is that SVMC can be seen as a special case of SVMR.

A Unified Framework for Regularization Networks and Support Vector Machines

Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples -- in particular the regression problem of approximating a multivariate function from sparse data. We present both formulations in a unified framework, namely in the context of Vapnik's theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics.

An Equivalence Between Sparse Approximation and Support Vector Machines

In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT&T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.

Support Vector Machines: Training and Applications

The Support Vector Machine (SVM) is a new and very promising classification technique developed by Vapnik and his group at AT&T Bell Labs. This new learning algorithm can be seen as an alternative training technique for Polynomial, Radial Basis Function and Multi-Layer Perceptron classifiers. An interesting property of this approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle. The derivation of Support Vector Machines, its relationship with SRM, and its geometrical insight, are discussed in this paper. Training a SVM is equivalent to solve a quadratic programming problem with linear and box constraints in a number of variables equal to the number of data points. When the number of data points exceeds few thousands the problem is very challenging, because the quadratic form is completely dense, so the memory needed to store the problem grows with the square of the number of data points. Therefore, training problems arising in some real applications with large data sets are impossible to load into memory, and cannot be solved using standard non-linear constrained optimization algorithms. We present a decomposition algorithm that can be used to train SVM's over large data sets. The main idea behind the decomposition is the iterative solution of sub-problems and the evaluation of, and also establish the stopping criteria for the algorithm. We present previous approaches, as well as results and important details of our implementation of the algorithm using a second-order variant of the Reduced Gradient Method as the solver of the sub-problems. As an application of SVM's, we present preliminary results we obtained applying SVM to the problem of detecting frontal human faces in real images.

A Note on Support Vector Machines Degeneracy

When training Support Vector Machines (SVMs) over non-separable data sets, one sets the threshold $b$ using any dual cost coefficient that is strictly between the bounds of $0$ and $C$. We show that there exist SVM training problems with dual optimal solutions with all coefficients at bounds, but that all such problems are degenerate in the sense that the "optimal separating hyperplane" is given by ${f w} = {f 0}$, and the resulting (degenerate) SVM will classify all future points identically (to the class that supplies more training data). We also derive necessary and sufficient conditions on the input data for this to occur. Finally, we show that an SVM training problem can always be made degenerate by the addition of a single data point belonging to a certain unboundedspolyhedron, which we characterize in terms of its extreme points and rays.

Characterization and classification of tumor lesions using computerized fractal-based texture analysis and support vector machines in digital mammograms

Objective: This paper presents a detailed study of fractal-based methods for texture characterization of mammographic mass lesions and architectural distortion. The purpose of this study is to explore the use of fractal and lacunarity analysis for the characterization and classification of both tumor lesions and normal breast parenchyma in mammography. Materials and methods: We conducted comparative evaluations of five popular fractal dimension estimation methods for the characterization of the texture of mass lesions and architectural distortion. We applied the concept of lacunarity to the description of the spatial distribution of the pixel intensities in mammographic images. These methods were tested with a set of 57 breast masses and 60 normal breast parenchyma (dataset1), and with another set of 19 architectural distortions and 41 normal breast parenchyma (dataset2). Support vector machines (SVM) were used as a pattern classification method for tumor classification. Results: Experimental results showed that the fractal dimension of region of interest (ROIs) depicting mass lesions and architectural distortion was statistically significantly lower than that of normal breast parenchyma for all five methods. Receiver operating characteristic (ROC) analysis showed that fractional Brownian motion (FBM) method generated the highest area under ROC curve (A z = 0.839 for dataset1, 0.828 for dataset2, respectively) among five methods for both datasets. Lacunarity analysis showed that the ROIs depicting mass lesions and architectural distortion had higher lacunarities than those of ROIs depicting normal breast parenchyma. The combination of FBM fractal dimension and lacunarity yielded the highest A z value (0.903 and 0.875, respectively) than those based on single feature alone for both given datasets. The application of the SVM improved the performance of the fractal-based features in differentiating tumor lesions from normal breast parenchyma by generating higher A z value. Conclusion: FBM texture model is the most appropriate model for characterizing mammographic images due to self-affinity assumption of the method being a better approximation. Lacunarity is an effective counterpart measure of the fractal dimension in texture feature extraction in mammographic images. The classification results obtained in this work suggest that the SVM is an effective method with great potential for classification in mammographic image analysis.

Secondary structure prediction with support vector machines

Motivation: A new method that uses support vector machines (SVMs) to predict protein secondary structure is described and evaluated. The study is designed to develop a reliable prediction method using an alternative technique and to investigate the applicability of SVMs to this type of bioinformatics problem. Methods: Binary SVMs are trained to discriminate between two structural classes. The binary classifiers are combined in several ways to predict multi-class secondary structure. Results: The average three-state prediction accuracy per protein (Q3) is estimated by cross-validation to be 77.07 ± 0.26% with a segment overlap (Sov) score of 73.32 ± 0.39%. The SVM performs similarly to the 'state-of-the-art' PSIPRED prediction method on a non-homologous test set of 121 proteins despite being trained on substantially fewer examples. A simple consensus of the SVM, PSIPRED and PROFsec achieves significantly higher prediction accuracy than the individual methods. Availability: The SVM classifier is available from the authors. Work is in progress to make the method available on-line and to integrate the SVM predictions into the PSIPRED server.

Parkinson’s Disease tremor classification – a comparison between Support Vector Machines and neural networks

Deep Brain Stimulation has been used in the study of and for treating Parkinson’s Disease (PD) tremor symptoms since the 1980s. In the research reported here we have carried out a comparative analysis to classify tremor onset based on intraoperative microelectrode recordings of a PD patient’s brain Local Field Potential (LFP) signals. In particular, we compared the performance of a Support Vector Machine (SVM) with two well known artificial neural network classifiers, namely a Multiple Layer Perceptron (MLP) and a Radial Basis Function Network (RBN). The results show that in this study, using specifically PD data, the SVM provided an overall better classification rate achieving an accuracy of 81% recognition.

A meta-learning approach to automatic kernel selection for support vector machines

Appropriate choice of a kernel is the most important ingredient of the kernel-based learning methods such as support vector machine (SVM). Automatic kernel selection is a key issue given the number of kernels available, and the current trial-and-error nature of selecting the best kernel for a given problem. This paper introduces a new method for automatic kernel selection, with empirical results based on classification. The empirical study has been conducted among five kernels with 112 different classification problems, using the popular kernel based statistical learning algorithm SVM. We evaluate the kernels’ performance in terms of accuracy measures. We then focus on answering the question: which kernel is best suited to which type of classification problem? Our meta-learning methodology involves measuring the problem characteristics using classical, distance and distribution-based statistical information. We then combine these measures with the empirical results to present a rule-based method to select the most appropriate kernel for a classification problem. The rules are generated by the decision tree algorithm C5.0 and are evaluated with 10 fold cross validation. All generated rules offer high accuracy ratings.

Prediction of wool knitwear pilling propensity using support vector machines

The propensity of wool knitwear to form entangled fiber balls, known as pills, on the surface is affected by a large number of factors. This study examines, for the first time, the application of the support vector machine (SVM) data mining tool to the pilling propensity prediction of wool knitwear. The results indicate that by using the binary classification method and the radial basis function (RBF) kernel function, the SVM is able to give high pilling propensity prediction accuracy for wool knitwear without data over-fitting. The study also found that the number of records available for each pill rating greatly affects the learning and prediction capability of SVM models.