Optimal Brain Damage (OBD) is a method for reducing the number of weights in a neural network. OBD estimates the increase in cost function if weights are pruned and is a valid approximation if the learning algorithm has converged into a local minimum. On the other hand it is often desirable to terminate the learning process before a local minimum is reached (early stopping). In this paper we show that OBD estimates the increase in cost function incorrectly if the network is not in a local minimum. We also show how OBD can be extended such that it can be used in connection with early stopping. We call this new approach Early Brain Damage, EBD. EBD also allows to revive already pruned weights. We demonstrate the improvements achieved by EBD using three publicly available data sets.

Unsupervised learning algorithms based on convex and conic encoders are proposed. The encoders find the closest convex or conic combination of basis vectors to the input. The learning algorithms produce basis vectors that minimize the reconstruction error of the encoders. The convex algorithm develops locally linear models of the input, while the conic algorithm discovers features. Both algorithms are used to model handwritten digits and compared with vector quantization and principal component analysis.

In the present paper, we propose a method to unify information maximization and minimization in hidden units. The information maximization and minimization are performed on two different levels: collective and individual level. Thus, two kinds of information: collective and individual information are defined. By maximizing collective information and by minimizing individual information, simple networks can be generated in terms of the number of connections and the number of hidden units. Obtained networks are expected to give better generalization and improved interpretation of internal representations.

The essential property of BBNs is summarized by the Markov condition, which asserts that each variable is independent of its non-descendants given its parents.

Smoothing regularizers for radial basis functions have been studied extensively, but no general smoothing regularizers for projective basis junctions (PBFs), such as the widely-used sigmoidal PBFs, have heretofore been proposed. We derive new classes of algebraically-simple mH'-order smoothing regularizers for networks of the form f(W, x)

Field (1994) has suggested that neurons with line and edge selectivities found in primary visual cortex of cats and monkeys form a sparse, distributed representation of natural scenes, and Barlow (1989) has reasoned that such responses should emerge from an unsupervised learning algorithm that attempts to find a factorial code of independent visual features. We show here that nonlinear'infomax', when applied to an ensemble of natural scenes, produces sets of visual filters that are localised and oriented. Some of these filters are Gabor-like and resemble those produced by the sparseness-maximisation network of Olshausen & Field (1996). In addition, the outputs of these filters are as independent as possible, since the infomax network is able to perform Independent Components Analysis (ICA). We compare the resulting ICA filters and their associated basis functions, with other decorrelating filters produced by Principal Components Analysis (PCA) and zero-phase whitening filters (ZCA).

When triangulating a belief network we aim to obtain a junction tree of minimum state space. According to (Rose, 1970), searching for the optimal triangulation can be cast as a search over all the permutations of the graph's vertices. Our approach is to embed the discrete set of permutations in a convex continuous domain D. By suitably extending the cost function over D and solving the continous nonlinear optimization task we hope to obtain a good triangulation with respect to the aformentioned cost. This paper presents two ways of embedding the triangulation problem into continuous domain and shows that they perform well compared to the best known heuristic.

By adapting a framework used for supervised learning, we construct iterative algorithms that maximize the likelihood of the observations while also attempting to stay "close" to the current estimated parameters. We use a bound on the relative entropy between the two HMMs as a distance measure between them. The result is new iterative training algorithms which are similar to the EM (Baum-Welch) algorithm for training HMMs. The proposed algorithms are composed of a step similar to the expectation step of Baum-Welch and a new update of the parameters which replaces the maximization (re-estimation) step. The algorithm takes only negligibly more time per iteration and an approximated version uses the same expectation step as Baum-Welch.

Many real world patterns have a hierarchical underlying structure in which simple features have a higher order structure among themselves. Because these relationships are often statistical in nature, it is natural to view the process of discovering such structures as a statistical inference problem in which a hierarchical model is fit to data. Hierarchical statistical structure can be conveniently represented with Bayesian belief networks (Pearl, 1988; Lauritzen and Spiegelhalter, 1988; Neal, 1992). These 530 M. S. Lewicki and T. 1. Sejnowski models are powerful, because they can capture complex statistical relationships among the data variables, and also mathematically convenient, because they allow efficient computation of the joint probability for any given set of model parameters.

We have investigated the possibility that rapid processing in the visual system could be achieved by using the order of firing in different neurones as a code, rather than more conventional firing rate schemes. Using SPIKENET, a neural net simulator based on integrate-and-fire neurones and in which neurones in the input layer function as analogto-delay converters, we have modeled the initial stages of visual processing. Initial results are extremely promising. Even with activity in retinal output cells limited to one spike per neuron per image (effectively ruling out any form of rate coding), sophisticated processing based on asynchronous activation was nonetheless possible.