Partial Information Decomposition of Boolean Functions: a Fourier Analysis perspective

TL;DR

This paper explores the relationship between partial information decomposition components and Fourier coefficients in Boolean functions, providing a new perspective on understanding information processing in Boolean gates.

Contribution

It establishes a novel connection between PID components and Fourier analysis for Boolean functions, enhancing interpretability of information decomposition.

Findings

Derived relations between PID components and Fourier coefficients for Boolean gates

Enhanced understanding of information processing in Boolean functions

Provides a foundation for further analysis of multivariate systems

Abstract

Partial information decomposition (PID) partitions the information that a set of sources has about a target variable into synergistic, unique, and redundant contributions. This information-theoretic tool has recently attracted attention due to its potential to characterize the information processing in multivariate systems. However, the PID framework still lacks a solid and intuitive interpretation of its information components. In the aim to improve the understanding of PID components, we focus here on Boolean gates, a much-studied type of source-target mechanisms. Boolean gates have been extensively characterised via Fourier analysis which coefficients have been related to interesting properties of the functions defining the gates. In this paper, we establish for Boolean gates mechanisms a relation between their PID components and Fourier coefficients.