The Poincaré-Shannon Machine: Statistical Physics and Machine Learning Aspects of Information Cohomology

Previous works established that entropy is characterized uniquely as the first cohomology class in a topos and described some of its applications to the unsupervised classification of gene expression modules or cell types. These studies raised important questions regarding the statistical meaning of the resulting cohomology of information and its interpretation or consequences with respect to usual data analysis and statistical physics. This paper aims to present the computational methods of information cohomology and to propose its interpretations in terms of statistical physics and machine learning. In order to further underline the cohomological nature of information functions and chain rules, the computation of the cohomology in low degrees is detailed to show more directly that the k multivariate mutual information ( I k ) are ( k − 1 ) -coboundaries. The ( k − 1 ) -cocycles condition corresponds to I k = 0 , which generalizes statistical independence to arbitrary degree k. Hence, the cohomology can be interpreted as quantifying the statistical dependences and the obstruction to factorization. I develop the computationally tractable subcase of simp