Non-commutative Probability Theory for Topological Data Analysis

Abstract

Recent developments [54] have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for describing structure morphisms of homotopy operadic algebras [33].We provide new elementary examples which show a connection between non-commutative probability theory and algebraic topology, based on spectral graph theory. These observations are important for bringing new ideas from non-commutative probability into TDA and stochastic topology, and in the opposite direction.1 To be strict here, we mean multivariate generalizations for the weak versions of these theorems.