My name is Juan Pablo Vigneaux, and I am a fourth-year doctoral student at Institut de Mathématiques de Jussieu – Paris Rives Gauche (IMJ-PRG).
My advisor is Daniel Bennequin, and the subject of my thesis is “Geometrical theories of information”.
The results obtained so far can be found in the Research section.
The main focus of my thesis has been the ‘information cohomology’ introduced by Baudot and Bennequin. I’m currently extending the framework to cover the differential entropy and also some discrete quantities, like the (q-)multinomial coefficients.
I established the homological nature of Tsallis q-entropy. When q=2, this function also accepts a combinatorial interpretation, in the spirit of Boltzmann and Shannon. I used this analogy to extend the idea of typical sequences to vector subspaces.
With D. Bennequin, Olivier Peltre and Grégoire Sergeant-Perthuis, I have studied the interactions between information theory, statistical mechanics and machine learning. Graphical models provide a unified language for these theories and some interesting relations: for example, belief propagation -an algorithm introduced in bayesian learning- computes the critical points of the Bethe free energy -an approximation introduced much before in statistical mechanics. Graphical models are also the main family of examples for the “information structures” used in information cohomology.
Lately I have been particularly interested in the combinatorial interpretations of entropy and related concepts, somewhat inspired by Boltzmann and Vontobel.
Universitè Paris Diderot – host of (one half of) the IMJ-PRG.