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Fast Möbius transform: An algebraic approach to information decomposition

Abel Jansma1,2,3,*, Pedro A. M. Mediano4,5,†, and Fernando E. Rosas6,7,8,9,‡

  • *Contact author: a.a.a.jansma@uva.nl
  • †Contact author: p.mediano@imperial.ac.uk
  • ‡Contact author: f.rosas@sussex.ac.uk
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Phys. Rev. Research 7, 033049 – Published 11 July, 2025

DOI: https://doi.org/10.1103/32sq-z2jf

Abstract

The partial information decomposition (PID) and its extension integrated information decomposition (ΦID) 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 Möbius transform. Our approach is based on a formula for estimating the Möbius function that circumvents important computational bottlenecks and can in some cases offer a double-exponential speedup. 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 yields about the brain's macroscopic functional organization 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.

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Phys. Rev. Research 7, 033049– Published 11 July, 2025
  • Received 5 February 2025
  • Accepted 16 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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