Information cohomology of classical vector-valued observables

TL;DR

This paper introduces a new algebraic framework for understanding differential entropy and the dimension of vector-valued random variables using information cohomology, based on their recursive properties.

Contribution

It provides a novel algebraic characterization of differential entropy and space dimension through information cohomology, extending previous concepts to vector-valued observables.

Findings

Complete characterization of 1-cocycles in the cohomology

Differential entropy and dimension as linear combinations of cocycles

Extension of cohomology methods to mixed discrete and continuous observables

Abstract

We provide here a novel algebraic characterization of two information measures associated with a vector-valued random variable, its differential entropy and the dimension of the underlying space, purely based on their recursive properties (the chain rule and the nullity-rank theorem, respectively). More precisely, we compute the information cohomology of Baudot and Bennequin with coefficients in a module of continuous probabilistic functionals over a category that mixes discrete observables and continuous vector-valued observables, characterizing completely the 1-cocycles; evaluated on continuous laws, these cocycles are linear combinations of the differential entropy and the dimension.