The inverse of the Fisher information matrix is used in the natural gradientdescent algorithm to train single-layer and multi-layer perceptrons. We have discovered a new scheme to represent the Fisher information matrix of a stochastic multi-layer perceptron. Based on this scheme, we have designed an algorithm to compute the natural gradient. When the input dimension n is much larger than the number of hidden neurons, the complexity of this algorithm isof order O(n). It is confirmed by simulations that the natural gradient descent learning rule is not only efficient but also robust. 1 INTRODUCTION The inverse of the Fisher information matrix is required to find the Cramer-Rae lower bound to analyze the performance of an unbiased estimator. It is also needed in the natural gradient learning framework (Amari, 1997) to design statistically efficient algorithms for estimating parameters in general and for training neural networks in particular. In this paper, we assume a stochastic model for multilayer perceptrons.Considering a Riemannian parameter space in which the Fisher information matrix is a metric tensor, we apply the natural gradient learning rule to train single-layer and multi-layer perceptrons. The main difficulty encountered is to compute the inverse of the Fisher information matrix of large dimensions when the input dimension is high. By exploring the structure of the Fisher information matrix and its inverse, we design a fast algorithm with lower complexity to implement the natural gradient learning algorithm.

Using displays of line orientations taken from Wolfe's experiments [1992], we study the hypothesis that the distinction between parallel versus serial processes arises from the availability of global information in the internal representations of the visual scene. The model operates in two phases. First, the visual displays are compressed via principal-component-analysis. Second, the compressed data is processed by a target detector module inorder to identify the existence of a target in the display. Our main finding is that targets in displays which were found experimentally tobe processed in parallel can be detected by the system, while targets in experimentally-serial displays cannot. This fundamental difference is explained via variance analysis of the compressed representations, providing a numerical criterion distinguishing parallelfrom serial displays. Our model yields a mapping of response-time slopes that is similar to Duncan and Humphreys's "search surface" [1989], providing an explicit formulation of their intuitive notion of feature similarity. It presents a neural realization ofthe processing that may underlie the classical metaphorical explanations of visual search.

A globally convergent homotopy method is defined that is capable of sequentially producing large numbers of stationary points of the multi-layer perceptron mean-squared error surface. Using this algorithm large subsets of the stationary points of two test problems are found. It is shown empirically that the MLP neural network appears to have an extreme ratio of saddle points compared to local minima, and that even small neural network problems have extremely large numbers of solutions.

To reduce the computational complexity of classification systems using tangent distance, Hastie et al. (HSS) developed an algorithm to devise rich models for representing large subsets of the data which computes automatically the "best" associated tangent subspace. Schwenk & Milgram proposed a discriminant modular classification system (Diabolo) based on several autoassociative multilayer perceptrons which use tangent distance as error reconstruction measure. We propose a gradient based constructive learning algorithm for building a tangent subspace model with discriminant capabilities which combines several of the the advantages of both HSS and Diabolo: devised tangent models hold discriminant capabilities, space requirements are improved with respect to HSS since our algorithm is discriminant and thus it needs fewer prototype models, dimension of the tangent subspace is determined automatically by the constructive algorithm, and our algorithm is able to learn new transformations.

A globally convergent homotopy method is defined that is capable of sequentially producing large numbers of stationary points of the multi-layer perceptron mean-squared error surface. Using this algorithm large subsets of the stationary points of two test problems are found. It is shown empirically that the MLP neural network appears to have an extreme ratio of saddle points compared to local minima, and that even small neural network problems have extremely large numbers of solutions.

The paper is developed in two parts where we discuss a new approach to self-organization in a single-layer linear feed-forward network. First, two novel algorithms for self-organization are derived from a two-layer linear hetero-associative network performing a one-of-m classification, and trained with the constrained least-mean-squared classification error criterion. Second, two adaptive algorithms are derived from these selforganizing procedures to compute the principal generalized eigenvectors of two correlation matrices from two sequences of random vectors. These novel adaptive algorithms can be implemented in a single-layer linear feed-forward network. We give a rigorous convergence analysis of the adaptive algorithms by using stochastic approximation theory. As an example, we consider a problem of online signal detection in digital mobile communications.

We consider the microscopic equations for learning problems in neural networks. The aligning fields of an example are obtained from the cavity fields, which are the fields if that example were absent in the learning process. In a rough energy landscape, we assume that the density of the local minima obey an exponential distribution, yielding macroscopic properties agreeing with the first step replica symmetry breaking solution. Iterating the microscopic equations provide a learning algorithm, which results in a higher stability than conventional algorithms. 1 INTRODUCTION Most neural networks learn iteratively by gradient descent. As a result, closed expressions for the final network state after learning are rarely known. This precludes further analysis of their properties, and insights into the design of learning algorithms.

It is also shown that bounds following the true learning curve can be derived from a formalism based on the density of error patterns.

The mixture of experts [1, 2] is a well known example which implements the philosophy of divide-and-conquer elegantly. Whereas this model are gaining more popularity in various applications, there have been little efforts to evaluate generalization capability of these modular approaches theoretically. Here we present the first analytic study of generalization in the mixture of experts from the statistical 184 K. Kang and 1. Oh physics perspective. Use of statistical mechanics formulation have been focused on the study of feedforward neural network architectures close to the multilayer perceptron[5, 6], together with the VC theory[8]. We expect that the statistical mechanics approach can also be effectively used to evaluate more advanced architectures including mixture models.

The parameter space of neural networks has a Riemannian metric structure. The natural Riemannian gradient should be used instead of the conventional gradient, since the former denotes the true steepest descent direction of a loss function in the Riemannian space. The behavior of the stochastic gradient learning algorithm is much more effective if the natural gradient is used. The present paper studies the information-geometrical structure of perceptrons and other networks, and prove that the online learning method based on the natural gradient is asymptotically as efficient as the optimal batch algorithm. Adaptive modification of the learning constant is proposed and analyzed in terms of the Riemannian measure and is shown to be efficient.