Sparse coding is a method for finding a representation of data in which each of the components of the representation is only rarely significantly active. Such a representation is closely related to re(cid:173) dundancy reduction and independent component analysis, and has some neurophysiological plausibility. In this paper, we show how sparse coding can be used for denoising. Using maximum likelihood estimation of nongaussian variables corrupted by gaussian noise, we show how to apply a shrinkage nonlinearity on the components of sparse coding so as to reduce noise. Furthermore, we show how to choose the optimal sparse coding basis for denoising.

We present a probabilistic method for fusion of images produced by multiple sensors. The approach is based on an image formation model in which the sensor images are noisy, locally linear functions of an underlying, true scene. A Bayesian framework then provides for maximum likelihood or maximum a posteriori estimates of the true scene from the sensor images. Maximum likelihood estimates of the parameters of the image formation model involve (local) second order image statistics, and thus are related to local principal component analysis. We demonstrate the efficacy of the method on images from visible-band and infrared sensors.

We introduce a novel framework for simultaneous structure and parameter learning in hidden-variable conditional probability models, based on an en tropic prior and a solution for its maximum a posteriori (MAP) estimator. The MAP estimate minimizes uncertainty in all respects: cross-entropy between model and data; entropy of the model; entropy of the data's descriptive statistics. Iterative estimation extinguishes weakly supported parameters, compressing and sparsifying the model. Trimming operators accelerate this process by removing excess parameters and, unlike most pruning schemes, guarantee an increase in posterior probability. Entropic estimation takes a overcomplete random model and simplifies it, inducing the structure of relations between hidden and observed variables.

This paper presents probabilistic modeling methods to solve the problem of dis(cid:173) criminating between five facial orientations with very little labeled data. The first model maintains no inter-pixel dependencies, the second model is capable of modeling a set of arbitrary pair-wise dependencies, and the last model allows dependencies only between neighboring pixels. We show that for all three of these models, the accuracy of the learned models can be greatly improved by augmenting a small number of labeled training images with a large set of unlabeled images using Expectation-Maximization. This is important because it is often difficult to obtain image labels, while many unla(cid:173) beled images are readily available. Through a large set of empirical tests, we examine the benefits of unlabeled data for each of the models.

The difficulties lie in the Monte-Carlo E-step which consists of sampling from the posterior distribution of the hidden variables given the observations. The new idea presented in this paper is to generate samples from a Gaussian approximation to the true posterior from which it is easy to obtain independent samples. The parameters of the Gaussian approximation are either derived from the extended Kalman filter or the Fisher scoring algorithm. In case the posterior density is mul(cid:173) timodal we propose to approximate the posterior by a sum of Gaussians (mixture of modes approach). We show that sampling from the approxi(cid:173) mate posterior densities obtained by the above algorithms leads to better models than using point estimates for the hidden states.

Local "belief propagation" rules of the sort proposed by Pearl [15] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have em(cid:173) pirically demonstrated good performance of "loopy belief propagation"(cid:173) using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannon-limit performance of "Turbo codes", whose decoding algorithm is equivalent to loopy belief propagation. Except for the case of graphs with a single loop, there has been little theo(cid:173) retical understanding of the performance of loopy propagation. Here we analyze belief propagation in networks with arbitrary topologies when the nodes in the graph describe jointly Gaussian random variables.

We propose a new Markov Chain Monte Carlo algorithm which is a gen(cid:173) eralization of the stochastic dynamics method. The algorithm performs exploration of the state space using its intrinsic geometric structure, facil(cid:173) itating efficient sampling of complex distributions. Applied to Bayesian learning in neural networks, our algorithm was found to perform at least as well as the best state-of-the-art method while consuming considerably less time.

In this paper, we propose a full Bayesian model for neural networks. This model treats the model dimension (number of neurons), model parameters, regularisation parameters and noise parameters as ran(cid:173) dom variables that need to be estimated. We then propose a re(cid:173) versible jump Markov chain Monte Carlo (MCMC) method to per(cid:173) form the necessary computations. We find that the results are not only better than the previously reported ones, but also appear to be robust with respect to the prior specification. Moreover, we present a geometric convergence theorem for the algorithm.

Gaussian mixtures (or so-called radial basis function networks) for density estimation provide a natural counterpart to sigmoidal neu(cid:173) ral networks for function fitting and approximation. In both cases, it is possible to give simple expressions for the iterative improve(cid:173) ment of performance as components of the network are introduced one at a time. In particular, for mixture density estimation we show that a k-component mixture estimated by maximum likelihood (or by an iterative likelihood improvement that we introduce) achieves log-likelihood within order 1/k of the log-likelihood achievable by any convex combination. Consequences for approximation and es(cid:173) timation using Kullback-Leibler risk are also given. A Minimum Description Length principle selects the optimal number of compo(cid:173) nents k that minimizes the risk bound.

We consider the problem of reconstructing a temporal discrete sequence of multidimensional real vectors when part of the data is missing, under the assumption that the sequence was generated by a continuous pro(cid:173) cess. A particular case of this problem is multivariate regression, which is very difficult when the underlying mapping is one-to-many. We pro(cid:173) pose an algorithm based on a joint probability model of the variables of interest, implemented using a nonlinear latent variable model. Each point in the sequence is potentially reconstructed as any of the modes of the conditional distribution of the missing variables given the present variables (computed using an exhaustive mode search in a Gaussian mix(cid:173) ture). Mode selection is determined by a dynamic programming search that minimises a geometric measure of the reconstructed sequence, de(cid:173) rived from continuity constraints.