Many real world learning problems are best characterized by an interaction of multiple independent causes or factors. Discover(cid:173) ing such causal structure from the data is the focus of this paper. Based on Zemel and Hinton's cooperative vector quantizer (CVQ) architecture, an unsupervised learning algorithm is derived from the Expectation-Maximization (EM) framework. Due to the com(cid:173) binatorial nature of the data generation process, the exact E-step is computationally intractable. Two alternative methods for com(cid:173) puting the E-step are proposed: Gibbs sampling and mean-field approximation, and some promising empirical results are presented.

Experiments demonstrated that sigmoid multilayer perceptron (MLP) networks provide slightly better risk prediction than conventional logistic regression when used to predict the risk of death, stroke, and renal failure on 1257 patients who underwent coronary artery bypass operations at the Lahey Clinic. MLP networks with no hidden layer and networks with one hidden layer were trained using stochastic gradient descent with early stopping. MLP networks and logistic regression used the same input features and were evaluated using bootstrap sampling with 50 replications. ROC areas for predicting mortality using preoperative input features were 70.5% for logistic regression and 76.0% for MLP networks. Regularization provided by early stopping was an important component of improved perfonnance.

Visualizing and structuring pairwise dissimilarity data are difficult combinatorial op(cid:173) timization problems known as multidimensional scaling or pairwise data clustering. Algorithms for embedding dissimilarity data set in a Euclidian space, for clustering these data and for actively selecting data to support the clustering process are discussed in the maximum entropy framework. Active data selection provides a strategy to discover structure in a data set efficiently with partially unknown data.

We present a new method for obtaining local error bars for nonlinear regression, i.e., estimates of the confidence in predicted values that de(cid:173) pend on the input. We approach this problem by applying a maximum(cid:173) likelihood framework to an assumed distribution of errors. We demon(cid:173) strate our method first on computer-generated data with locally varying, normally distributed target noise. We then apply it to laser data from the Santa Fe Time Series Competition where the underlying system noise is known quantization error and the error bars give local estimates of model misspecification. In both cases, the method also provides a weighted(cid:173) regression effect that improves generalization performance.

Nearest neighbor classification expects the class conditional prob(cid:173) abilities to be locally constant, and suffers from bias in high di(cid:173) mensions We propose a locally adaptive form of nearest neighbor classification to try to finesse this curse of dimensionality. We use a local linear discriminant analysis to estimate an effective met(cid:173) ric for computing neighborhoods. We determine the local decision boundaries from centroid information, and then shrink neighbor(cid:173) hoods in directions orthogonal to these local decision boundaries, and elongate them parallel to the boundaries. Thereafter, any neighborhood-based classifier can be employed, using the modified neighborhoods. We also propose a method for global dimension reduction, that combines local dimension information.

The process of machine learning can be considered in two stages: model selection and parameter estimation. In this paper a technique is presented for constructing dynamical systems with desired qualitative properties. The approach is based on the fact that an n-dimensional nonlinear dynamical system can be decomposed into one gradient and (n - 1) Hamiltonian sys(cid:173) tems. Thus, the model selection stage consists of choosing the gradient and Hamiltonian portions appropriately so that a certain behavior is obtainable. To estimate the parameters, a stably convergent learning rule is presented.

In this paper we introduce new algorithms for optimizing noisy plants in which each experiment is very expensive. The algorithms build a global non-linear model of the expected output at the same time as using Bayesian linear regression analysis of locally weighted polynomial models. The local model answers queries about confi(cid:173) dence, noise, gradient and Hessians, and use them to make auto(cid:173) mated decisions similar to those made by a practitioner of Response Surface Methodology. The global and local models are combined naturally as a locally weighted regression. We examine the ques(cid:173) tion of whether the global model can really help optimization, and we extend it to the case of time-varying functions.

A new technique, termed soft assign, is applied for the first time to two classic combinatorial optimization problems, the travel(cid:173) ing salesman problem and graph partitioning. Soft assign, which has emerged from the recurrent neural network/statistical physics framework, enforces two-way (assignment) constraints without the use of penalty terms in the energy functions. The soft assign can also be generalized from two-way winner-take-all constraints to multiple membership constraints which are required for graph par(cid:173) titioning. The soft assign technique is compared to the softmax (Potts glass). Within the statistical physics framework, softmax and a penalty term has been a widely used method for enforcing the two-way constraints common within many combinatorial optimiza(cid:173) tion problems.

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is pre(cid:173) sented and used to investigate the role of the learning rate in con(cid:173) trolling the evolution and convergence of the learning process. Learning in layered neural networks refers to the modification of internal parameters {J} which specify the strength of the interneuron couplings, so as to bring the map fJ implemented by the network as close as possible to a desired map 1. The degree of success is monitored through the generalization error, a measure of the dissimilarity between fJ and 1. Consider maps from an N-dimensional input space e onto a scalar (, as arise in the formulation of classification and regression tasks.

A new approach for clustering is proposed. This method is based on an analogy to a physical model; the ferromagnetic Potts model at thermal equilibrium is used as an analog computer for this hard optimization problem . We do not assume any structure of the un(cid:173) derlying distribution of the data. Phase space of the Potts model is divided into three regions; ferromagnetic, super-paramagnetic and paramagnetic phases. The region of interest is that corresponding to the super-paramagnetic one, where domains of aligned spins ap(cid:173) pear.