## Section: Scientific Foundations

Keywords : probabilistic modeling, statistical estimation, bayesian decision theory gaussian mixture modeling, Hidden Markov Model, adaptive representation, redundant system, sparse decomposition, sparsity criterion, source separation.

### Introduction

Probabilistic approaches offer a general theoretical framework [68] which has yielded considerable progress in various fields of pattern recognition. In speech processing in particular [57] , the probabilistic framework indeed provides a solid formalism which makes it possible to formulate various problems of segmentation, detection and classification. Coupled to statistical approaches, the probabilistic paradigm makes it possible to easily adapt relatively generic tools to various applicative contexts, thanks to estimation techniques for training from examples.

A particularily productive family of probabilistic models is the Hidden Markov Model, either in its general form or under some degenerated variants. The stochastic framework makes it possible to rely on well-known algorithms for the estimation of the model parameters (EM algorithms, ML criteria, MAP techniques, ...) and for the search of the best model in the sense of the exact or approximate maximum likelihood (Viterbi decoding or beam search, for example).

In practice, however, the use of the theoretical tools must be accompanied by a number of adjustments to take into account problems occurring in real contexts of use, such as model inaccuracy, the insufficiency (or even the absence) of training data, their poor statistical coverage, etc...

Another focus of the activities of the METISS research group is dedicated to the adaptive representations of signals in redundant systems [71] . The use of criteria of sparsity or entropy (in place of the criterion of least squares) to force the unicity of the solution of a underdetermined system of equations makes it possible to seek an economical representation (exact or approximate) of a signal in a redundant system, which is better able to account for the diversity of structures within an audio signal.

This topic opens a vast field of scientific investigation : sparse decomposition, sparsity criteria, pursuit algorithms, construction of efficient redundant dictionaries, links with the non-linear approximation theory, probabilistic extensions, etc... The potential applicative outcomes are numerous.

This section briefly exposes these various theoretical elements, which constitute the fundamentals of our activities.