The Fast Möbius Transform: An algebraic approach to information decomposition

TL;DR

This work leverages fundamental algebraic properties of these decompositions to enable a computationally-efficient method to estimate the M\"obius transform, which is based on a novel formula for estimating the M\"obius function that circumvents important computational bottlenecks.

Abstract

The partial information decomposition (PID) and its extension integrated information decomposition ($Φ$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 Möbius transform. Our approach is based on a novel formula for estimating the Möbius 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.

Figures (6)

• Figure 1: The redundancy terms of a powerset of $n$ variables can be partially ordered: if $A$ and $B$ are redundancy terms, then $A\leq B$ if for every $b\in B$ there is an $a \in A$ such that $a\leq b$. Shown here are the transitive reduction (Hasse diagrams) of the redundancy lattices $\mathcal{R}_n$ for $n=2$ (top left), $n=3$ (top right middle) and $n=4$ (bottom, labels not shown) variables. Note that elements of $\mathcal{R}_n$ are the antichains in the powerset of $n$-variables.
• Figure 2: The Möbius function of the redundancy lattice of up to 5 variables. The rows and columns are ordered by the cardinality of the antichains. For $n=4$, only every 20th antichain is indicated along the axes. For $n=5$ none are shown.
• Figure 3: Information about neural functional connectivity provided by the 27 strongest information atoms of the five canonical brain frequency bands. Blue plus signs denote information atoms in the real brain data, and orange lines denote the 99th percentile of a spatial null model (a.k.a. 'spin test').
• Figure 4: Relevance of various categories of $\Phi$ID atoms for the Bach's and Corelli's music scores. Result shows that the dynamics within the music of these two composers differ in the interplay between the whole and the parts, which is captured by the top-down and bottom-up causation categories. Significance was calculated via a t-test between the value of the atoms in each category, and significance is indicated as $p\leq 0.05$ with * and $p\leq \frac{0.05}{6}$ with ** (that is, ** indicates significance after Bonferroni correction). Error bars indicate standard error of the mean.
• Figure 5: The powerset $\mathcal{P}(\{X, Y, Z\})$ of a set of $3$ variables, ordered by set inclusion, forms a poset known as a Boolean algebra. Shown here is the transitive reduction (Hasse diagram) of the Boolean algebra on 3 variables. It is common to keep the directionality implicit by drawing the diagram such that all edges are oriented downwards. Note that $\{X\} \subseteq \{X, Y, Z\}$, but that the Hasse diagram contains no such edge because $\{X\} \subseteq \{X, Y\} \subseteq \{X, Y, Z\}$. For arbitrary $n$, the Hasse diagram of a Boolean algebra describes an $n$-cube.

Theorems & Definitions (14)

• Proposition 1
• proof
• Theorem 1: Fast Möbius transform for PID
• proof : Sketch of proof
• Corollary 1: Fast Möbius transform for $\Phi\mathrm{ID}$
• proof
• Theorem 2
• proof
• proof
• Theorem 3: Birkhoff birkhoff1937rings