Topological Information Data Analysis

Pierre Baudot (1) , Mónica Tapia (1) , Daniel Bennequin (2) , Jean-Marc Goaillard (1)

1 UNIS - Inserm U1072 - Unité de Neurobiologie des canaux Ioniques et de la Synapse
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

This paper presents methods that quantify the structure of statistical interactions within a given data set, and were applied in a previous article. It establishes new results on the k-multivariate mutual-information (I k) inspired by the topological formulation of Information introduced in a serie of studies. In particular, we show that the vanishing of all I k for 2 ≤ k ≤ n of n random variables is equivalent to their statistical independence. Pursuing the work of Hu Kuo Ting and Te Sun Han, we show that information functions provide coordinates for binary variables, and that they are analytically independent from the probability simplex for any set of finite variables. The maximal positive I k identifies the variables that co-vary the most in the population, whereas the minimal negative I k identifies synergistic clusters and the variables that differentiate-segregate the most in the population. Finite data size effects and estimation biases severely constrain the effective computation of the information topology on data, and we provide simple statistical tests for the undersampling bias and the k-dependences. We give an example of application of these methods to genetic expression and unsupervised cell-type classification. The methods unravel biologically relevant subtypes, with a sample size of 41 genes and with few errors. It establishes generic basic methods to quantify the epigenetic information storage and a unified epigenetic unsupervised learning formalism. We propose that higher-order statistical interactions and non-identically distributed variables are constitutive characteristics of biological systems that should be estimated in order to unravel their significant statistical structure and diversity. The topological information data analysis presented here allows for precisely estimating this higher-order structure characteristic of biological systems. "When you use the word information, you should rather use the word form"-René Thom

Keywords: information theory; cohomology; information category; topological data analysis; genetic expression; epigenetics; multivariate mutual-information; synergy; statistical independence

Domains: Mathematics [math]; General Topology [math.GN]

Pierre Baudot, Mónica Tapia, Daniel Bennequin, Jean-Marc Goaillard. Topological Information Data Analysis. Entropy, 2019, ⟨10.3390/e21090869⟩. ⟨hal-02457878⟩