Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 Abstract We propose diffusion networks, a type of recurrent neural network with probabilistic dynamics, as models for learning natural signals that are continuous in time and space. We give a formula for the gradient of the log-likelihood of a path with respect to the drift parameters for a diffusion network. This gradient can be used to optimize diffusion networks in the nonequilibrium regime for a wide variety of problems paralleling techniques which have succeeded in engineering fields such as system identification, state estimation and signal filtering. An aspect of this work which is of particular interestto computational neuroscience and hardware design is that with a suitable choice of activation function, e.g., quasi-linear sigmoidal, the gradient formula is local in space and time. 1 Introduction Many natural signals, like pixel gray-levels, line orientations, object position, velocity andshape parameters, are well described as continuous-time continuous-valued stochastic processes; however, the neural network literature has seldom explored the continuous stochastic case. Since the solutions to many decision theoretic problems of interest are naturally formulated using probability distributions, it is desirable to have a flexible framework for approximating probability distributions on continuous pathspaces.

Next, C1 is used to form the indicator matrix, N. A correspondence analysis is performed on N to derive the scaled space, A

This paper introduces a probability model, the mixture of trees that can account for sparse, dynamically changing dependence relationships. We present a family of efficient algorithms that use EM and the Minimum Spanning Tree algorithm to find the ML and MAP mixture of trees for a variety of priors, including the Dirichlet and the MDL priors.

Multiple-instance learning is a variation on supervised learning, where the task is to learn a concept given positive and negative bags of instances. Each bag may contain many instances, but a bag is labeled positive even if only one of the instances in it falls within the concept. A bag is labeled negative only if all the instances in it are negative. We describe a new general framework, called Diverse Density, for solving multiple-instance learning problems. We apply this framework to learn a simple description of a person from a series of images (bags) containing that person, to a stock selection problem, and to the drug activity prediction problem.

The mathematical framework for factorizing equivalence classes of multivariate functions is formulated in this paper. Independent component analysis is shown to be a special case of this decomposition.

We derive a learning algorithm for inferring an overcomplete basis by viewing it as probabilistic model of the observed data.

The S-Map is a network with a simple learning algorithm that combines theself-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 initialconfiguration than the GTM. The S-Map algorithm can be further simplified to employ pure Hebbian learning, without changingthe 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.

We present a computationally efficient algorithm for function approximation withpiecewise linear sigmoidal nodes. A one hidden layer network is constructed one node at a time using the method of fitting the residual. The task of fitting individual nodes is accomplished usinga new algorithm that searchs for the best fit by solving a sequence of Quadratic Programming problems. This approach offers significantadvantages over derivative-based search algorithms (e.g.

An adaptive online algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images. 1 Introduction Unsupervised learning addresses the problem to detect structure inherent in unlabeled andunclassified data. N. The encoding usually is represented by an assignment matrix M (Mia), where Mia 1 if and only if Xi belongs to cluster L: 1 MiaV (Xi, Ya) measures the quality of a data partition, Le., optimal assignments and prototypes (M,y)OPt argminM,y1i (M,Y) minimize the inhomogeneity of clusters w.r.t. a given distance measure V. For reasons of simplicity we restrict the presentation to the ' sum-of-squared-error criterion V(x, y) To facilitate this minimization a deterministic annealing approach was proposed in [5] which maps the discrete optimization problem, i.e. how to determine the data assignments, viathe Maximum Entropy Principle [2] to a continuous parameter es- Unsupervised Online Learning ofDecision Trees for Data Analysis 515 timation problem.

Some learning techniques for classification tasks work indirectly, by first trying to fit a full probabilistic model to the observed data. Whether this is a good idea or not depends on the robustness with respect to deviations from the postulated model. We study this question experimentally in a restricted, yet nontrivial and interesting case: we consider a conditionally independent attribute (CIA) model which postulates a single binary-valued hidden variable z on which all other attributes (i.e., the target and the observables) depend. In this model, finding the most likely value of anyone variable (given known values for the others) reduces to testing a linear function of the observed values. We learn CIA with two techniques: the standard EM algorithm, and a new algorithm we develop based on covariances. We compare these, in a controlled fashion, against an algorithm (a version of Winnow) that attempts to find a good linear classifier directly. Our conclusions help delimit the fragility of using the CIA model for classification: once the data departs from this model, performance quickly degrades and drops below that of the directly-learned linear classifier.