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.

A Trainable Object Detection System: Car Detection in Static Images

This paper describes a general, trainable architecture for object detection that has previously been applied to face and peoplesdetection with a new application to car detection in static images. Our technique is a learning based approach that uses a set of labeled training data from which an implicit model of an object class -- here, cars -- is learned. Instead of pixel representations that may be noisy and therefore not provide a compact representation for learning, our training images are transformed from pixel space to that of Haar wavelets that respond to local, oriented, multiscale intensity differences. These feature vectors are then used to train a support vector machine classifier. The detection of cars in images is an important step in applications such as traffic monitoring, driver assistance systems, and surveillance, among others. We show several examples of car detection on out-of-sample images and show an ROC curve that highlights the performance of our system.

Vector-Based Integration of Local and Long-Range Information in Visual Cortex

Integration of inputs by cortical neurons provides the basis for the complex information processing performed in the cerebral cortex. Here, we propose a new analytic framework for understanding integration within cortical neuronal receptive fields. Based on the synaptic organization of cortex, we argue that neuronal integration is a systems--level process better studied in terms of local cortical circuitry than at the level of single neurons, and we present a method for constructing self-contained modules which capture (nonlinear) local circuit interactions. In this framework, receptive field elements naturally have dual (rather than the traditional unitary influence since they drive both excitatory and inhibitory cortical neurons. This vector-based analysis, in contrast to scalarsapproaches, greatly simplifies integration by permitting linear summation of inputs from both "classical" and "extraclassical" receptive field regions. We illustrate this by explaining two complex visual cortical phenomena, which are incompatible with scalar notions of neuronal integration.

Feature Selection for Face Detection

We present a new method to select features for a face detection system using Support Vector Machines (SVMs). In the first step we reduce the dimensionality of the input space by projecting the data into a subset of eigenvectors. The dimension of the subset is determined by a classification criterion based on minimizing a bound on the expected error probability of an SVM. In the second step we select features from the SVM feature space by removing those that have low contributions to the decision function of the SVM.

Improving Multiclass Text Classification with the Support Vector Machine

We compare Naive Bayes and Support Vector Machines on the task of multiclass text classification. Using a variety of approaches to combine the underlying binary classifiers, we find that SVMs substantially outperform Naive Bayes. We present full multiclass results on two well-known text data sets, including the lowest error to date on both data sets. We develop a new indicator of binary performance to show that the SVM's lower multiclass error is a result of its improved binary performance. Furthermore, we demonstrate and explore the surprising result that one-vs-all classification performs favorably compared to other approaches even though it has no error-correcting properties.

Properties of Support Vector Machines

Support Vector Machines (SVMs) perform pattern recognition between two point classes by finding a decision surface determined by certain points of the training set, termed Support Vectors (SV). This surface, which in some feature space of possibly infinite dimension can be regarded as a hyperplane, is obtained from the solution of a problem of quadratic programming that depends on a regularization parameter. In this paper we study some mathematical properties of support vectors and show that the decision surface can be written as the sum of two orthogonal terms, the first depending only on the margin vectors (which are SVs lying on the margin), the second proportional to the regularization parameter. For almost all values of the parameter, this enables us to predict how the decision surface varies for small parameter changes. In the special but important case of feature space of finite dimension m, we also show that there are at most m+1 margin vectors and observe that m+1 SVs are usually sufficient to fully determine the decision surface. For relatively small m this latter result leads to a consistent reduction of the SV number.

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.