The Fast M\"obius Transform: An algebraic approach to information decomposition

TL;DR

The paper introduces the fast Möbius transform, an algebraic method that significantly reduces computational costs in information decomposition frameworks like PID and $\

Contribution

It presents a novel algebraic formula for efficient estimation of the Möbius function, enabling practical analysis of complex information interactions.

Findings

Efficient decomposition of neural information across frequency bands.

Identification of dynamical interaction properties in baroque music.

Demonstration of the method's feasibility for complex real-world data.

Abstract

The partial information decomposition (PID) and its extension integrated information decomposition ($\Phi$ID) are promising frameworks to investigate information phenomena involving multiple variables. An important limitation of these approaches is the high computational cost involved in their calculation. Here we leverage fundamental algebraic properties of these decompositions to enable a computationally-efficient method to estimate them, which we call the fast M\"obius transform. Our approach is based on a novel formula for estimating the M\"obius function that circumvents important computational bottlenecks. We showcase the capabilities of this approach by presenting two analyses that would be unfeasible without this method: decomposing the information that neural activity at different frequency bands yield about the brain's macroscopic functional organisation, and identifying distinctive dynamical properties of the interactions between multiple voices in baroque music. Overall, our proposed approach illuminates the value of algebraic facets of information decomposition and opens the way to a wide range of future analyses.