An approach is presented to learning high dimensional functions in the case where the learning algorithm can affect the generation of new data. A local modeling algorithm, locally weighted regression, is used to represent the learned function. Architectural parameters of the approach, such as distance metrics, are also localized and become a function of the query point instead of being global. Statistical tests are given for when a local model is good enough and sampling should be moved to a new area. Our methods explicitly deal with the case where prediction accuracy requirements exist during exploration: By gradually shifting a "center of exploration" and controlling the speed of the shift with local prediction accuracy, a goal-directed exploration of state space takes place along the fringes of the current data support until the task goal is achieved.

Dimension reduction provides compact representations for storage, transmission, and classification. Dimension reduction algorithms operate by identifying and eliminating statistical redundancies in the data. The optimal linear technique for dimension reduction is principal component analysis (PCA).

The Singular Value Decomposition (SVD) is an important tool for linear algebra and can be used to invert or approximate matrices. Although many authors use "SVD" synonymously with "Eigenvector Decomposition" or "Principal Components Transform", it is important to realize that these other methods apply only to symmetric matrices, while the SVD can be applied to arbitrary nonsquare matrices. This property is important for applications to signal transmission and control. I propose two new algorithms for iterative computation of the SVD given only sample inputs and outputs from a matrix. Although there currently exist many algorithms for Eigenvector Decomposition (Sanger 1989, for example), these are the first true samplebased SVD algorithms. 1 INTRODUCTION The Singular Value Decomposition (SVD) is a method for writing an arbitrary nons quare matrix as the product of two orthogonal matrices and a diagonal matrix.

We represent objects with n-dimensional pattern vectors and consider piecewise-linear classifiers consisting of a collection of (labeled) codebook vectors in the space of the input patterns (See Figure 1). The classification boundaries are gi ven by the voronoi tessellation of the codebook vectors. Patterns are said to belong to the class (given by the label) of the codebook vector to which they are closest.

Data clustering amounts to a combinatorial optimization problem to reduce the complexity of a data representation and to increase its precision. Central and pairwise data clustering are studied in the maximum entropy framework. For central clustering we derive a set of reestimation equations and a minimization procedure which yields an optimal number of clusters, their centers and their cluster probabilities. A meanfield approximation for pairwise clustering is used to estimate assignment probabilities. A se1fconsistent solution to multidimensional scaling and pairwise clustering is derived which yields an optimal embedding and clustering of data points in a d-dimensional Euclidian space. 1 Introduction A central problem in information processing is the reduction of the data complexity with minimal loss in precision to discard noise and to reveal basic structure of data sets. Data clustering addresses this tradeoff by optimizing a cost function which preserves the original data as complete as possible and which simultaneously favors prototypes with minimal complexity (Linde et aI., 1980; Gray, 1984; Chou et aI., 1989; Rose et ai., 1990). We discuss an objective function for the joint optimization of distortion errors and the complexity of a reduced data representation. A maximum entropy estimation of the cluster assignments yields a unifying framework for clustering algorithms with a number of different distortion and complexity measures. The close analogy of complexity optimized clustering with winner-take-all neural networks suggests a neural-like implementation resembling topological feature maps (see Figure 1).

The distance measure and learning problem are formally described as nested objective functions. We derive an efficient algorithm by using optimization techniques that allow us to divide up the objective function into parts which may be minimized in distinct phases. The algorithm has accurately recreated 10 prototypes from a randomly generated sample database of 100 images consisting of 20 points each in 120 experiments. Finally, by incorporating permutation invariance in our distance measure, we have a technique that we may be able to apply to the clustering of graphs. Our goal is to develop measures which will enable the learning of objects with shape or structure. Acknowledgements This work has been supported by AFOSR grant F49620-92-J-0465 and ONR/DARPA grant N00014-92-J-4048.

This paper introduces GNARL, an evolutionary program which induces recurrent neural networks that are structurally unconstrained. In contrast to constructive and destructive algorithms, GNARL employs a population of networks and uses a fitness function's unsupervised feedback to guide search through network space. Annealing is used in generating both gaussian weight changes and structural modifications. Applying GNARL to a complex search and collection task demonstrates that the system is capable of inducing networks with complex internal dynamics.

This paper presents a simple algorithm to learn trajectories with a continuous time, continuous activation version of the Boltzmann machine. The algorithm takes advantage of intrinsic Brownian noise in the network to easily compute gradients using entirely local computations. The algorithm may be ideal for parallel hardware implementations. This paper presents a learning algorithm to train continuous stochastic networks to respond with desired trajectories in the output units to environmental input trajectories. This is a task, with potential applications to a variety of problems such as stochastic modeling of neural processes, artificial motor control, and continuous speech recognition.

Learning to recognize or predict sequences using long-term context has many applications. However, practical and theoretical problems are found in training recurrent neural networks to perform tasks in which input/output dependencies span long intervals. Starting from a mathematical analysis of the problem, we consider and compare alternative algorithms and architectures on tasks for which the span of the input/output dependencies can be controlled. Results on the new algorithms show performance qualitatively superior to that obtained with backpropagation. 1 Introduction Recurrent neural networks have been considered to learn to map input sequences to output sequences. Machines that could efficiently learn such tasks would be useful for many applications involving sequence prediction, recognition or production. However, practical difficulties have been reported in training recurrent neural networks to perform tasks in which the temporal contingencies present in the input/output sequences span long intervals. In fact, we can prove that dynamical systems such as recurrent neural networks will be increasingly difficult to train with gradient descent as the duration of the dependencies to be captured increases. A mathematical analysis of the problem shows that either one of two conditions arises in such systems.

We show how an "Elman" network architecture, constructed from recurrently connected oscillatory associative memory network modules, can employ selective "attentional" control of synchronization to direct the flow of communication and computation within the architecture to solve a grammatical inference problem. Previously we have shown how the discrete time "Elman" network algorithm can be implemented in a network completely described by continuous ordinary differential equations. The time steps (machine cycles) of the system are implemented by rhythmic variation (clocking) of a bifurcation parameter. In this architecture, oscillation amplitude codes the information content or activity of a module (unit), whereas phase and frequency are used to "softwire" the network. Only synchronized modules communicate by exchanging amplitude information; the activity of non-resonating modules contributes incoherent crosstalk noise. Attentional control is modeled as a special subset of the hidden modules with ouputs which affect the resonant frequencies of other hidden modules. They control synchrony among the other modules and direct the flow of computation (attention) to effect transitions between two subgraphs of a thirteen state automaton which the system emulates to generate a Reber grammar. The internal crosstalk noise is used to drive the required random transitions of the automaton.