The adaptation rule for Vector Quantization algorithms, and consequently the convergence of the generated sequence, depends on the existence and properties of a function called the energy function, defined on a topological manifold. Our aim is to investigate the conditions of existence of such a function for a class of algorithms examplified by the initial ''K-means'' and Kohonen algorithms. The results presented here supplement previous studies and show that the energy function is not always a potential but at least the uniform limit of a series of potential functions which we call a pseudo-potential. Our work also shows that a large number of existing vector quantization algorithms developped by the Artificial Neural Networks community fall into this category. The framework we define opens the way to study the convergence of all the corresponding adaptation rules at once, and a theorem gives promising insights in that direction. We also demonstrate that the ''K-means'' energy function is a pseudo-potential but not a potential in general. Consequently, the energy function associated to the ''Neural-Gas'' is not a potential in general.

We present a novel approach to semi-supervised learning which is based on statistical physics. Most of the former work in the field of semi-supervised learning classifies the points by minimizing a certain energy function, which corresponds to a minimal k-way cut solution. In contrast to these methods, we estimate the distribution of classifications, instead of the sole minimal k-way cut, which yields more accurate and robust results. Our approach may be applied to all energy functions used for semi-supervised learning. The method is based on sampling using a Mul-ticanonical Markov chain Monte-Carlo algorithm, and has a straightforward probabilistic interpretation, which allows for soft assignments of points to classes, and also to cope with yet unseen class types. The suggested approach is demonstrated on a toy data set and on two real-life data sets of gene expression.

In this paper, we study clustering with respect to the k-modes objective function, a natural formulation of clustering for categorical data. One of the main contributions of this paper is to establish the connection between k-modes and k-median, i.e., the optimum of k-median is at most twice the optimum of k-modes for the same categorical data clustering problem. Based on this observation, we derive a deterministic algorithm that achieves an approximation factor of 2. Furthermore, we prove that the distance measure in k-modes defines a metric. Hence, we are able to extend existing approximation algorithms for metric k-median to k-modes. Empirical results verify the superiority of our method.

A method of {\it topological grammars} is proposed for multidimensional data approximation. For data with complex topology we define a {\it principal cubic complex} of low dimension and given complexity that gives the best approximation for the dataset. This complex is a generalization of linear and non-linear principal manifolds and includes them as particular cases. The problem of optimal principal complex construction is transformed into a series of minimization problems for quadratic functionals. These quadratic functionals have a physically transparent interpretation in terms of elastic energy. For the energy computation, the whole complex is represented as a system of nodes and springs. Topologically, the principal complex is a product of one-dimensional continuums (represented by graphs), and the grammars describe how these continuums transform during the process of optimal complex construction. This factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allow us to construct efficient algorithms.

Experimental physics, along with many other fields in applied and basic research, uses experiments, physical tests, and observations to gain insight into various phenomena and to validate hypotheses and models. Shock p hysics is a field that explores the response of materials to the extremes of p ressure, deformation, and temperature which are present when shock waves interact with those materials [17]. High explosive (HE) or propellant guns are often used to generate these strong shock waves. Many different diagnostic ap proaches have been used to probe these phenomena [8]. Because of the energetic nature of the shock wave drive, often a large amount of experimental equipment is destroyed during the test.

Given a finite set of words w1,...,wn independently drawn according to a fixed unknown distribution law P called a stochastic language, an usual goal in Grammatical Inference is to infer an estimate of P in some class of probabilistic models, such as Probabilistic Automata (PA). Here, we study the class of rational stochastic languages, which consists in stochastic languages that can be generated by Multiplicity Automata (MA) and which strictly includes the class of stochastic languages generated by PA. Rational stochastic languages have minimal normal representation which may be very concise, and whose parameters can be efficiently estimated from stochastic samples. We design an efficient inference algorithm DEES which aims at building a minimal normal representation of the target. Despite the fact that no recursively enumerable class of MA computes exactly the set of rational stochastic languages over Q, we show that DEES strongly identifies tis set in the limit. We study the intermediary MA output by DEES and show that they compute rational series which converge absolutely to one and which can be used to provide stochastic languages which closely estimate the target.

This paper deals with the problem of classifying signals. The new method for building so called local classifiers and local features is presented. The method is a combination of the lifting scheme and the support vector machines. Its main aim is to produce effective and yet comprehensible classifiers that would help in understanding processes hidden behind classified signals. To illustrate the method we present the results obtained on an artificial and a real dataset.

We consider the problem of on-line prediction competitive with a benchmark class of continuous but highly irregular prediction rules. It is known that if the benchmark class is a reproducing kernel Hilbert space, there exists a prediction algorithm whose average loss over the first N examples does not exceed the average loss of any prediction rule in the class plus a "regret term" of O(N^(-1/2)). The elements of some natural benchmark classes, however, are so irregular that these classes are not Hilbert spaces. In this paper we develop Banach-space methods to construct a prediction algorithm with a regret term of O(N^(-1/p)), where p is in [2,infty) and p-2 reflects the degree to which the benchmark class fails to be a Hilbert space.

We propose a new method for the estimation of parameters of hidden diffusion processes. Based on parametrization of the transition matrix, the Baum-Welch algorithm is improved. The algorithm is compared to the particle filter in application to the noisy periodic systems. It is shown that the modified Baum-Welch algorithm is capable of estimating the system parameters with better accuracy than particle filters.

From a geometric perspective most nonlinear binary classification algorithms, including state of the art versions of Support Vector Machine (SVM) and Radial Basis Function Network (RBFN) classifiers, and are based on the idea of reconstructing indicator functions. We propose instead to use reconstruction of the signed distance function (SDF) as a basis for binary classification. We discuss properties of the signed distance function that can be exploited in classification algorithms. We develop simple versions of such classifiers and test them on several linear and nonlinear problems. On linear tests accuracy of the new algorithm exceeds that of standard SVM methods, with an average of 50% fewer misclassifications. Performance of the new methods also matches or exceeds that of standard methods on several nonlinear problems including classification of benchmark diagnostic micro-array data sets.