Comparing Support Vector Machines with Gaussian Kernels to Radial Basis Function Classifiers

The Support Vector (SV) machine is a novel type of learning machine, based on statistical learning theory, which contains polynomial classifiers, neural networks, and radial basis function (RBF) networks as special cases. In the RBF case, the SV algorithm automatically determines centers, weights and threshold such as to minimize an upper bound on the expected test error. The present study is devoted to an experimental comparison of these machines with a classical approach, where the centers are determined by $k$--means clustering and the weights are found using error backpropagation. We consider three machines, namely a classical RBF machine, an SV machine with Gaussian kernel, and a hybrid system with the centers determined by the SV method and the weights trained by error backpropagation. Our results show that on the US postal service database of handwritten digits, the SV machine achieves the highest test accuracy, followed by the hybrid approach. The SV approach is thus not only theoretically well--founded, but also superior in a practical application.

Perceptually-based Comparison of Image Similarity Metrics

The image comparison operation ??sessing how well one image matches another ??rms a critical component of many image analysis systems and models of human visual processing. Two norms used commonly for this purpose are L1 and L2, which are specific instances of the Minkowski metric. However, there is often not a principled reason for selecting one norm over the other. One way to address this problem is by examining whether one metric better captures the perceptual notion of image similarity than the other. With this goal, we examined perceptual preferences for images retrieved on the basis of the L1 versus the L2 norm. These images were either small fragments without recognizable content, or larger patterns with recognizable content created via vector quantization. In both conditions the subjects showed a consistent preference for images matched using the L1 metric. These results suggest that, in the domain of natural images of the kind we have used, the L1 metric may better capture human notions of image similarity.

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.

On the difficulty of feature-based attentional modulations in visual object recognition: A modeling study.

Numerous psychophysical experiments have shown an important role for attentional modulations in vision. Behaviorally, allocation of attention can improve performance in object detection and recognition tasks. At the neural level, attention increases firing rates of neurons in visual cortex whose preferred stimulus is currently attended to. However, it is not yet known how these two phenomena are linked, i.e., how the visual system could be "tuned" in a task-dependent fashion to improve task performance. To answer this question, we performed simulations with the HMAX model of object recognition in cortex [45]. We modulated firing rates of model neurons in accordance with experimental results about effects of feature-based attention on single neurons and measured changes in the model's performance in a variety of object recognition tasks. It turned out that recognition performance could only be improved under very limited circumstances and that attentional influences on the process of object recognition per se tend to display a lack of specificity or raise false alarm rates. These observations lead us to postulate a new role for the observed attention-related neural response modulations.

A Vector Signal Processing Approach to Color

Surface (Lambertain) color is a useful visual cue for analyzing material composition of scenes. This thesis adopts a signal processing approach to color vision. It represents color images as fields of 3D vectors, from which we extract region and boundary information. The first problem we face is one of secondary imaging effects that makes image color different from surface color. We demonstrate a simple but effective polarization based technique that corrects for these effects. We then propose a systematic approach of scalarizing color, that allows us to augment classical image processing tools and concepts for multi-dimensional color signals.

An Analysis of the Effect of Gaussian Error in Object Recognition

Object recognition is complicated by clutter, occlusion, and sensor error. Since pose hypotheses are based on image feature locations, these effects can lead to false negatives and positives. In a typical recognition algorithm, pose hypotheses are tested against the image, and a score is assigned to each hypothesis. We use a statistical model to determine the score distribution associated with correct and incorrect pose hypotheses, and use binary hypothesis testing techniques to distinguish between them. Using this approach we can compare algorithms and noise models, and automatically choose values for internal system thresholds to minimize the probability of making a mistake.

Understanding Subsystems in Biology through Dimensionality Reduction, Graph Partitioning and Analytical Modeling

Biological systems exhibit rich and complex behavior through the orchestrated interplay of a large array of components. It is hypothesized that separable subsystems with some degree of functional autonomy exist; deciphering their independent behavior and functionality would greatly facilitate understanding the system as a whole. Discovering and analyzing such subsystems are hence pivotal problems in the quest to gain a quantitative understanding of complex biological systems. In this work, using approaches from machine learning, physics and graph theory, methods for the identification and analysis of such subsystems were developed. A novel methodology, based on a recent machine learning algorithm known as non-negative matrix factorization (NMF), was developed to discover such subsystems in a set of large-scale gene expression data. This set of subsystems was then used to predict functional relationships between genes, and this approach was shown to score significantly higher than conventional methods when benchmarking them against existing databases. Moreover, a mathematical treatment was developed to treat simple network subsystems based only on their topology (independent of particular parameter values). Application to a problem of experimental interest demonstrated the need for extentions to the conventional model to fully explain the experimental data. Finally, the notion of a subsystem was evaluated from a topological perspective. A number of different protein networks were examined to analyze their topological properties with respect to separability, seeking to find separable subsystems. These networks were shown to exhibit separability in a nonintuitive fashion, while the separable subsystems were of strong biological significance. It was demonstrated that the separability property found was not due to incomplete or biased data, but is likely to reflect biological structure.

Learning-Based Approach to Real Time Tracking and Analysis of Faces

This paper describes a trainable system capable of tracking faces and facialsfeatures like eyes and nostrils and estimating basic mouth features such as sdegrees of openness and smile in real time. In developing this system, we have addressed the twin issues of image representation and algorithms for learning. We have used the invariance properties of image representations based on Haar wavelets to robustly capture various facial features. Similarly, unlike previous approaches this system is entirely trained using examples and does not rely on a priori (hand-crafted) models of facial features based on optical flow or facial musculature. The system works in several stages that begin with face detection, followed by localization of facial features and estimation of mouth parameters. Each of these stages is formulated as a problem in supervised learning from examples. We apply the new and robust technique of support vector machines (SVM) for classification in the stage of skin segmentation, face detection and eye detection. Estimation of mouth parameters is modeled as a regression from a sparse subset of coefficients (basis functions) of an overcomplete dictionary of Haar wavelets.

Image-Based View Synthesis

We present a new method for rendering novel images of flexible 3D objects from a small number of example images in correspondence. The strength of the method is the ability to synthesize images whose viewing position is significantly far away from the viewing cone of the example images ("view extrapolation"), yet without ever modeling the 3D structure of the scene. The method relies on synthesizing a chain of "trilinear tensors" that governs the warping function from the example images to the novel image, together with a multi-dimensional interpolation function that synthesizes the non-rigid motions of the viewed object from the virtual camera position. We show that two closely spaced example images alone are sufficient in practice to synthesize a significant viewing cone, thus demonstrating the ability of representing an object by a relatively small number of model images --- for the purpose of cheap and fast viewers that can run on standard hardware.

Model-Based Matching by Linear Combinations of Prototypes

We describe a method for modeling object classes (such as faces) using 2D example images and an algorithm for matching a model to a novel image. The object class models are "learned'' from example images that we call prototypes. In addition to the images, the pixelwise correspondences between a reference prototype and each of the other prototypes must also be provided. Thus a model consists of a linear combination of prototypical shapes and textures. A stochastic gradient descent algorithm is used to match a model to a novel image by minimizing the error between the model and the novel image. Example models are shown as well as example matches to novel images. The robustness of the matching algorithm is also evaluated. The technique can be used for a number of applications including the computation of correspondence between novel images of a certain known class, object recognition, image synthesis and image compression.

On Convergence Properties of the EM Algorithm for Gaussian Mixtures

"Expectation-Maximization'' (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix $P$, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of $P$ and provide new results analyzing the effect that $P$ has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models.

Example Based Image Analysis and Synthesis

Image analysis and graphics synthesis can be achieved with learning techniques using directly image examples without physically-based, 3D models. In our technique: -- the mapping from novel images to a vector of "pose" and "expression" parameters can be learned from a small set of example images using a function approximation technique that we call an analysis network; -- the inverse mapping from input "pose" and "expression" parameters to output images can be synthesized from a small set of example images and used to produce new images using a similar synthesis network. The techniques described here have several applications in computer graphics, special effects, interactive multimedia and very low bandwidth teleconferencing.

The Individual is Nothing, the Class Everything: Psychophysics and Modeling of Recognition in Obect Classes

Most psychophysical studies of object recognition have focussed on the recognition and representation of individual objects subjects had previously explicitely been trained on. Correspondingly, modeling studies have often employed a 'grandmother'-type representation where the objects to be recognized were represented by individual units. However, objects in the natural world are commonly members of a class containing a number of visually similar objects, such as faces, for which physiology studies have provided support for a representation based on a sparse population code, which permits generalization from the learned exemplars to novel objects of that class. In this paper, we present results from psychophysical and modeling studies intended to investigate object recognition in natural ('continuous') object classes. In two experiments, subjects were trained to perform subordinate level discrimination in a continuous object class - images of computer-rendered cars - created using a 3D morphing system. By comparing the recognition performance of trained and untrained subjects we could estimate the effects of viewpoint-specific training and infer properties of the object class-specific representation learned as a result of training. We then compared the experimental findings to simulations, building on our recently presented HMAX model of object recognition in cortex, to investigate the computational properties of a population-based object class representation as outlined above. We find experimental evidence, supported by modeling results, that training builds a viewpoint- and class-specific representation that supplements a pre-existing repre-sentation with lower shape discriminability but possibly greater viewpoint invariance.

On the Noise Model of Support Vector Machine Regression

Support Vector Machines Regression (SVMR) is a regression technique which has been recently introduced by V. Vapnik and his collaborators (Vapnik, 1995; Vapnik, Golowich and Smola, 1996). In SVMR the goodness of fit is measured not by the usual quadratic loss function (the mean square error), but by a different loss function called Vapnik"s $epsilon$- insensitive loss function, which is similar to the "robust" loss functions introduced by Huber (Huber, 1981). The quadratic loss function is well justified under the assumption of Gaussian additive noise. However, the noise model underlying the choice of Vapnik's loss function is less clear. In this paper the use of Vapnik's loss function is shown to be equivalent to a model of additive and Gaussian noise, where the variance and mean of the Gaussian are random variables. The probability distributions for the variance and mean will be stated explicitly. While this work is presented in the framework of SVMR, it can be extended to justify non-quadratic loss functions in any Maximum Likelihood or Maximum A Posteriori approach. It applies not only to Vapnik's loss function, but to a much broader class of loss functions.

Multivariate Density Estimation: An SVM Approach

We formulate density estimation as an inverse operator problem. We then use convergence results of empirical distribution functions to true distribution functions to develop an algorithm for multivariate density estimation. The algorithm is based upon a Support Vector Machine (SVM) approach to solving inverse operator problems. The algorithm is implemented and tested on simulated data from different distributions and different dimensionalities, gaussians and laplacians in $R^2$ and $R^{12}$. A comparison in performance is made with Gaussian Mixture Models (GMMs). Our algorithm does as well or better than the GMMs for the simulations tested and has the added advantage of being automated with respect to parameters.