Non-commutative Probability Theory for Topological Data Analysis

TL;DR

This paper explores the unexpected links between non-commutative probability theory and algebraic topology, using spectral graph theory to provide elementary examples that bridge these fields and suggest new research directions.

Contribution

It introduces new elementary examples connecting non-commutative probability with algebraic topology through spectral graph theory, advancing interdisciplinary understanding.

Findings

Spectral graph theory examples resemble connections between the fields.

Boolean cumulants are important for morphisms of homotopy operadic algebras.

The work opens new avenues for applying non-commutative probability in TDA and stochastic topology.

Abstract

Recent developments have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for describing morphisms of homotopy operadic algebras. We provide new elementary examples which clearly resemble a connection between algebraic topology and non-commutative probability, 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.