The Fast Möbius Transform: An algebraic approach to information decomposition

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.

• The paper introduces a fast Möbius transform method that efficiently computes PID and ΦID, drastically reducing computational complexity in multivariate analysis.
• It applies algebraic techniques with the Möbius inversion theorem to bypass constructing the full redundancy lattice, enabling analysis of systems up to five variables.
• Empirical tests on neural and musical data validate its accuracy and demonstrate potential applications in AI, neuroscience, and genetics.

The Fast Möbius Transform: An Algebraic Approach to Information Decomposition

The paper "The Fast Möbius Transform: An Algebraic Approach to Information Decomposition" presents a novel method for calculating the Partial Information Decomposition (PID) and Integrated Information Decomposition (ΦID), which are frameworks for analyzing the distribution and processing of information across complex systems with multiple variables. The authors focus on addressing the computational inefficiencies associated with these frameworks by introducing the fast Möbius transform, enabling larger systems to be analyzed than previously possible.

Overview

The PID framework decomposes the information a set of predictor variables provides about a target variable into synergistic, redundant, and unique components. ΦID extends this concept to scenarios involving information flow between multiple sets of source and target variables, making it particularly suitable for multivariate time series data. These frameworks have found utility in various fields, including neural and genetic information processing.

The fast Möbius transform fundamentally alters how the decomposition process operates. Traditional methods faced significant computational obstacles, particularly in evaluating the information atoms by subtraction over the redundancy lattice. This method relied on the Möbius function, traditionally complex and computationally expensive to evaluate, especially as the number of variables increases.

The Fast Möbius Transform

The authors introduce an algebraic technique to efficiently compute the Möbius function. The Möbius inversion theorem allows integration over partially ordered sets to be interpreted algebraically. By leveraging these principles, the paper presents explicit formulas (Theorems 1 and 2) to compute the PID and ΦID efficiently without constructing the entire redundancy lattice or solving large systems of equations. This approach thus allows practical implementation of these decompositions for systems with up to five variables, with considerations for potential further extension.

Empirical Evaluations

To validate their method, the authors conducted empirical tests illustrating the method's computational efficiency. They were able to perform a full decomposition on a complete five-variable set using canonical distributions, demonstrating agreement with existing redundancy measures. In practical applications, they applied the transform to neural and musical data, showing its utility in examining complex dependencies between variables — an achievement previously unattainable due to computational constraints.

Implications and Future Directions

The proposed approach opens new pathways in analyzing multivariate dependencies in complex systems, facilitating the exploration of larger systems in fields like artificial intelligence, neuroscience, and genetics. Algebraically-based methods like the fast Möbius transform could extend the reach of information decomposition, making it feasible to analyze datasets traditionally considered too complex.

While this paper resolves significant computational bottlenecks, further research is necessary to address challenges in more extensive systems and refine interpretations of the results in practical scenarios. Future work could explore other algebraic facets of these frameworks and adapt the methods for real-time or large-scale applications in diverse scientific domains.

Overall, "The Fast Möbius Transform" contributes substantially to the computational feasibility of information decomposition, marking a significant advancement in our ability to analyze complex information interactions in multidimensional systems.