# The Poincar\'e-Boltzmann Machine: from Statistical Physics to Machine Learning and back

## TL;DR

This paper introduces a topological framework using information cohomology to analyze statistical dependencies in complex systems, bridging statistical physics and machine learning, and providing new insights into multivariate information structures.

## Contribution

It develops a cohomological approach to quantify statistical dependencies and independence without relying on traditional assumptions, linking information theory with topological and physical concepts.

## Key findings

- Cohomology quantifies statistical dependences and independence.
- Topological approach avoids assumptions of i.i.d. variables.
- Introduces free energy landscape for multivariate information.

## Abstract

This paper presents the computational methods of information cohomology applied to genetic expression in and in the companion paper and proposes its interpretations in terms of statistical physics and machine learning. In order to further underline the Hochschild cohomological nature af information functions and chain rules, following, the computation of the cohomology in low degrees is detailed to show more directly that the $k$ multivariate mutual-informations (I_k) are k-coboundaries. The k-cocycles condition corresponds to I_k=0, generalizing statistical independence. Hence the cohomology quantifies the statistical dependences and the obstruction to factorization. The topological approach allows to investigate information in the multivariate case without the assumptions of independent identically distributed variables and without mean field approximations. We develop the computationally tractable subcase of simplicial information cohomology represented by entropy H_k and information I_k landscapes and their respective paths. The I_1 component defines a self-internal energy U_k, and I_k,k>1 components define the contribution to a free energy G_k (the total correlation) of the k-body interactions. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, provides a trivial topological expression of the 2nd law of thermodynamic. The local minima of free-energy, related to conditional information negativity, and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system, and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. I give an interpretation of this complex in terms of frustration in glass and of Van Der Walls k-body interactions for data points.