Optimal Unsupervised Learning in Feedforward Neural Networks

We investigate the properties of feedforward neural networks trained with Hebbian learning algorithms. A new unsupervised algorithm is proposed which produces statistically uncorrelated outputs. The algorithm causes the weights of the network to converge to the eigenvectors of the input correlation with largest eigenvalues. The algorithm is closely related to the technique of Self-supervised Backpropagation, as well as other algorithms for unsupervised learning. Applications of the algorithm to texture processing, image coding, and stereo depth edge detection are given. We show that the algorithm can lead to the development of filters qualitatively similar to those found in primate visual cortex.

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.

An O(N) Algorithm for Three-Dimensional N-Body Simulations

We develop an algorithm that computes the gravitational potentials and forces on N point-masses interacting in three-dimensional space. The algorithm, based on analytical techniques developed by Rokhlin and Greengard, runs in order N time. In contrast to other fast N-body methods such as tree codes, which only approximate the interaction potentials and forces, this method is exact ?? computes the potentials and forces to within any prespecified tolerance up to machine precision. We present an implementation of the algorithm for a sequential machine. We numerically verify the algorithm, and compare its speed with that of an O(N2) direct force computation. We also describe a parallel version of the algorithm that runs on the Connection Machine in order 0(logN) time. We compare experimental results with those of the sequential implementation and discuss how to minimize communication overhead on the parallel machine.

A Note on the Generalization Performance of Kernel Classifiers with Margin

We present distribution independent bounds on the generalization misclassification performance of a family of kernel classifiers with margin. Support Vector Machine classifiers (SVM) stem out of this class of machines. The bounds are derived through computations of the $V_gamma$ dimension of a family of loss functions where the SVM one belongs to. Bounds that use functions of margin distributions (i.e. functions of the slack variables of SVM) are derived.

A Detailed Look at Scale and Translation Invariance in a Hierarchical Neural Model of Visual Object Recognition

The HMAX model has recently been proposed by Riesenhuber & Poggio as a hierarchical model of position- and size-invariant object recognition in visual cortex. It has also turned out to model successfully a number of other properties of the ventral visual stream (the visual pathway thought to be crucial for object recognition in cortex), and particularly of (view-tuned) neurons in macaque inferotemporal cortex, the brain area at the top of the ventral stream. The original modeling study only used ``paperclip'' stimuli, as in the corresponding physiology experiment, and did not explore systematically how model units' invariance properties depended on model parameters. In this study, we aimed at a deeper understanding of the inner workings of HMAX and its performance for various parameter settings and ``natural'' stimulus classes. We examined HMAX responses for different stimulus sizes and positions systematically and found a dependence of model units' responses on stimulus position for which a quantitative description is offered. Interestingly, we find that scale invariance properties of hierarchical neural models are not independent of stimulus class, as opposed to translation invariance, even though both are affine transformations within the image plane.

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.

Neural Networks

We present an overview of current research on artificial neural networks, emphasizing a statistical perspective. We view neural networks as parameterized graphs that make probabilistic assumptions about data, and view learning algorithms as methods for finding parameter values that look probable in the light of the data. We discuss basic issues in representation and learning, and treat some of the practical issues that arise in fitting networks to data. We also discuss links between neural networks and the general formalism of graphical models.

Fast Learning by Bounding Likelihoods in Sigmoid Type Belief Networks

Sigmoid type belief networks, a class of probabilistic neural networks, provide a natural framework for compactly representing probabilistic information in a variety of unsupervised and supervised learning problems. Often the parameters used in these networks need to be learned from examples. Unfortunately, estimating the parameters via exact probabilistic calculations (i.e, the EM-algorithm) is intractable even for networks with fairly small numbers of hidden units. We propose to avoid the infeasibility of the E step by bounding likelihoods instead of computing them exactly. We introduce extended and complementary representations for these networks and show that the estimation of the network parameters can be made fast (reduced to quadratic optimization) by performing the estimation in either of the alternative domains. The complementary networks can be used for continuous density estimation as well.

Learning from Incomplete Data

Real-world learning tasks often involve high-dimensional data sets with complex patterns of missing features. In this paper we review the problem of learning from incomplete data from two statistical perspectives---the likelihood-based and the Bayesian. The goal is two-fold: to place current neural network approaches to missing data within a statistical framework, and to describe a set of algorithms, derived from the likelihood-based framework, that handle clustering, classification, and function approximation from incomplete data in a principled and efficient manner. These algorithms are based on mixture modeling and make two distinct appeals to the Expectation-Maximization (EM) principle (Dempster, Laird, and Rubin 1977)---both for the estimation of mixture components and for coping with the missing data.

Hierarchical Mixtures of Experts and the EM Algorithm

We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM's). Learning is treated as a maximum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parameters of the architecture. We also develop an on-line learning algorithm in which the parameters are updated incrementally. Comparative simulation results are presented in the robot dynamics domain.

Forecasting Global Temperature Variations by Neural Networks

Global temperature variations between 1861 and 1984 are forecast usingsregularization networks, multilayer perceptrons and linearsautoregression. The regularization network, optimized by stochasticsgradient descent associated with colored noise, gives the bestsforecasts. For all the models, prediction errors noticeably increasesafter 1965. These results are consistent with the hypothesis that thesclimate dynamics is characterized by low-dimensional chaos and thatsthe it may have changed at some point after 1965, which is alsosconsistent with the recent idea of climate change.s

Biologically Plausible Neural Circuits for Realization of Maximum Operations

Object recognition in the visual cortex is based on a hierarchical architecture, in which specialized brain regions along the ventral pathway extract object features of increasing levels of complexity, accompanied by greater invariance in stimulus size, position, and orientation. Recent theoretical studies postulate a non-linear pooling function, such as the maximum (MAX) operation could be fundamental in achieving such invariance. In this paper, we are concerned with neurally plausible mechanisms that may be involved in realizing the MAX operation. Four canonical circuits are proposed, each based on neural mechanisms that have been previously discussed in the context of cortical processing. Through simulations and mathematical analysis, we examine the relative performance and robustness of these mechanisms. We derive experimentally verifiable predictions for each circuit and discuss their respective physiological considerations.

Biologically Plausible Neural Model for the Recognition of Biological Motion and Actions

The visual recognition of complex movements and actions is crucial for communication and survival in many species. Remarkable sensitivity and robustness of biological motion perception have been demonstrated in psychophysical experiments. In recent years, neurons and cortical areas involved in action recognition have been identified in neurophysiological and imaging studies. However, the detailed neural mechanisms that underlie the recognition of such complex movement patterns remain largely unknown. This paper reviews the experimental results and summarizes them in terms of a biologically plausible neural model. The model is based on the key assumption that action recognition is based on learned prototypical patterns and exploits information from the ventral and the dorsal pathway. The model makes specific predictions that motivate new experiments.

Exact Solution of the Nonlinear Dynamics of Recurrent Neural Mechanisms for Direction Selectivity

Different theoretical models have tried to investigate the feasibility of recurrent neural mechanisms for achieving direction selectivity in the visual cortex. The mathematical analysis of such models has been restricted so far to the case of purely linear networks. We present an exact analytical solution of the nonlinear dynamics of a class of direction selective recurrent neural models with threshold nonlinearity. Our mathematical analysis shows that such networks have form-stable stimulus-locked traveling pulse solutions that are appropriate for modeling the responses of direction selective cortical neurons. Our analysis shows also that the stability of such solutions can break down giving raise to a different class of solutions ("lurching activity waves") that are characterized by a specific spatio-temporal periodicity. These solutions cannot arise in models for direction selectivity with purely linear spatio-temporal filtering.

A Conservative Front Tracking Algorithm

The discontinuities in the solutions of systems of conservation laws are widely considered as one of the difficulties in numerical simulation. A numerical method is proposed for solving these partial differential equations with discontinuities in the solution. The method is able to track these sharp discontinuities or interfaces while still fully maintain the conservation property. The motion of the front is obtained by solving a Riemann problem based on the state values at its both sides which are reconstructed by using weighted essentially non oscillatory (WENO) scheme. The propagation of the front is coupled with the evaluation of "dynamic" numerical fluxes. Some numerical tests in 1D and preliminary results in 2D are presented.