Géométrie et inférence dans l'optimisation et en théorie de l'information

Abstract : Optimization and constraint satisfaction problems on sets of discrete variables are the main object of algorithmic complexity. These problems have recently benefited from the tools and concepts of the physics of disordered systems, both theoretically and algorithmically. In particular, it was suggested that the practical difficulties roused by some hard instances of optimization problems may be related to the clustered organisation of their solution spaces, which is reminiscent of a glassy phase. On the other hand, state-of-the-art error-correcting codes, which can be mapped onto optimimization problems, rely on the geometrical separability of its messages to ensure error-free communication. The object of this thesis is to explore this relation between inference properties and geometrical organization within a common framework, in problems from both computational complexity and information theory.

This thesis first introduces interesting problems and concepts pertaining to information theory and optimization, from a physical perspective. Then message-passing methods based on the Bethe approximation are presented. These methods are useful from a physical standpoint, as they allow us to study thermodynamical properties of random ensembles of instances. They are also useful for inference tasks. The analysis of distance spectra using both combinatorial and message-passing methods is performed, and used to prove the existence of the clustering phenomenon in the satisfiability problem, and to study its salient features.
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Analyse de données, Statistiques et Probabilités [physics.data-an]. Université Paris Sud - Paris XI, 2007. Français
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T. Mora. Géométrie et inférence dans l'optimisation et en théorie de l'information. Analyse de données, Statistiques et Probabilités [physics.data-an]. Université Paris Sud - Paris XI, 2007. Français. 〈tel-00175221〉

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