Consistency problems arise if the discretization needs to be refined, e.g. for more accuracy, application of multi-grid iteration or better starting values for the iteration of the approximate optimal value function. In [7] it was shown, that for diffusion dominated problems, a state to time discretization ratio k/ h of Ch'r, I

We derive a first-order approximation of the density of maximum entropy for a continuous 1-D random variable, given a number of simple constraints. This results in a density expansion which is somewhat similar to the classical polynomial density expansions by Gram-Charlier and Edgeworth. Using this approximation of density, an approximation of 1-D differential entropy is derived. The approximation of entropy is both more exact and more robust against outliers than the classical approximation based on the polynomial density expansions, without being computationally more expensive. The approximation has applications, for example, in independent component analysis and projection pursuit. 1 Introduction The basic information-theoretic quantity for continuous one-dimensional random variables is differential entropy. The differential entropy H of a scalar random variable X with density f(x) is defined as H(X) - / f(x) log f(x)dx.

We apply a general algorithm for merging prediction strategies (the Aggregating Algorithm) to the problem of linear regression with the square loss; our main assumption is that the response variable is bounded. It turns out that for this particular problem the Aggregating Algorithm resembles, but is slightly different from, the wellknown ridge estimation procedure. From general results about the Aggregating Algorithm we deduce a guaranteed bound on the difference between our algorithm's performance and the best, in some sense, linear regression function's performance. We show that the AA attains the optimal constant in our bound, whereas the constant attained by the ridge regression procedure in general can be 4 times worse. 1 INTRODUCTION The usual approach to regression problems is to assume that the data are generated by some stochastic mechanism and make some, typically very restrictive, assumptions about that stochastic mechanism. In recent years, however, a different approach to this kind of problems was developed (see, e.g., DeSantis et al. [2], Littlestone and Warmuth [7]): in our context, that approach sets the goal of finding an online algorithm that performs not much worse than the best regression function found off-line; in other words, it replaces the usual statistical analyses by the competitive analysis of online algorithms. DeSantis et al. [2] performed a competitive analysis of the Bayesian merging scheme for the log-loss prediction game; later Littlestone and Warmuth [7] and Vovk [10] introduced an online algorithm (called the Weighted Majority Algorithm by the Competitive Online Linear Regression 365 former authors) for the simple binary prediction game. These two algorithms (the Bayesian merging scheme and the Weighted Majority Algorithm) are special cases of the Aggregating Algorithm (AA) proposed in [9, 11]. The AA is a member of a wide family of algorithms called "multiplicative weight" or "exponential weight" algorithms. Closer to the topic of this paper, Cesa-Bianchi et al. [1) performed a competitive analysis, under the square loss, of the standard Gradient Descent Algorithm and Kivinen and Warmuth [6] complemented it by a competitive analysis of a modification of the Gradient Descent, which they call the Exponentiated Gradient Algorithm.

One frequently estimates density functions for which there is little prior knowledge on the shape of the density and for which one wants a flexible and robust estimator (allowing multimodality if it exists). In this context, the methods of choice tend to be finite mixture models and kernel density estimation methods. For mixture modeling, mixtures of Gaussian components are frequently assumed and model choice reduces to the problem of choosing the number k of Gaussian components in the model (Titterington, Smith and Makov, 1986). For kernel density estimation, kernel shapes are typically chosen from a selection of simple unimodal densities such as Gaussian, triangular, or Cauchy densities, and kernel bandwidths are selected in a data-driven manner (Silverman 1986; Scott 1994). As argued by Draper (1996), model uncertainty can contribute significantly to pre- - Also with the Jet Propulsion Laboratory 525-3660, California Institute of Technology, Pasadena, CA 91109 Stacked Density Estimation 669 dictive error in estimation. While usually considered in the context of supervised learning, model uncertainty is also important in unsupervised learning applications such as density estimation. Even when the model class under consideration contains the true density, if we are only given a finite data set, then there is always a chance of selecting the wrong model. Moreover, even if the correct model is selected, there will typically be estimation error in the parameters of that model.

Our model consists of a classical energy mechanism whose output is divided by nonclassical gain control and texture contrast mechanisms. We apply this model to review movies, a stimulus sequence that replicates the stimulation a cell receives during free viewing of natural images. Data were collected from three cells using five different review movies, and the model was fit separately to the data from each movie. For the energy mechanism alone we find modest but significant correlations (rE 0.41, 0.43, 0.59, 0.35) between model and data. These correlations are improved somewhat when we allow for suppressive surround effects (rE G 0.42, 0.56, 0.60, 0.37). In one case the inclusion of a delayed suppressive surround dramatically improves the fit to the data by modifying the time course of the model's response.

Singular value decomposition (SVD) can be viewed as a method for unsupervised training of a network that associates two classes of events reciprocally by linear connections through a single hidden layer. SVD was used to learn and represent relations among very large numbers of words (20k-60k) and very large numbers of natural text passages (lk-70k) in which they occurred. The result was 100-350 dimensional "semantic spaces" in which any trained or newly aibl word or passage could be represented as a vector, and similarities were measured by the cosine of the contained angle between vectors. Good accmacy in simulating human judgments and behaviors has been demonstrated by performance on multiple-choice vocabulary and domain knowledge tests, emulation of expert essay evaluations, and in several other ways. Examples are also given of how the kind of knowledge extracted by this method can be applied.

Here we sudy the continuous time, continuous state-space stochastic case, which covers a wide variety of control problems including target, viability, optimization problems (see [FS93], [KP95])}or which a formalism is the following.

The S-Map is a network with a simple learning algorithm that combines the self-organization capability of the Self-Organizing Map (SOM) and the probabilistic interpretability of the Generative Topographic Mapping (GTM). The simulations suggest that the S Map algorithm has a stronger tendency to self-organize from random initial configuration than the GTM. The S-Map algorithm can be further simplified to employ pure Hebbian learning, without changing the qualitative behaviour of the network. 1 Introduction The self-organizing map (SOM; for a review, see [1]) forms a topographic mapping from the data space onto a (usually two-dimensional) output space. The SOM has been succesfully used in a large number of applications [2]; nevertheless, there are some open theoretical questions, as discussed in [1, 3]. Most of these questions arise because of the following two facts: the SOM is not a generative model, i.e. it does not generate a density in the data space, and it does not have a well-defined objective function that the training process would strictly minimize.

In normal vision, the inputs from the two eyes are integrated into a single percept. When dissimilar images are presented to the two eyes, however, perceptual integration gives way to alternation between monocular inputs, a phenomenon called binocular rivalry. Although recent evidence indicates that binocular rivalry involves a modulation of neuronal responses in extrastriate cortex, the basic mechanisms responsible for differential processing of con:6.icting

We present a method for determining the globally optimal online learning rule for a soft committee machine under a statistical mechanics framework. This work complements previous results on locally optimal rules, where only the rate of change in generalization error was considered. We maximize the total reduction in generalization error over the whole learning process and show how the resulting rule can significantly outperform the locally optimal rule. 1 Introduction We consider a learning scenario in which a feed-forward neural network model (the student) emulates an unknown mapping (the teacher), given a set of training examples produced by the teacher. The performance of the student network is typically measured by its generalization error, which is the expected error on an unseen example. The aim of training is to reduce the generalization error by adapting the student network's parameters appropriately. A common form of training is online learning, where training patterns are presented sequentially and independently to the network at each learning step.