Gaussian Information Bottleneck and the Non-Perturbative Renormalization Group

TL;DR

This paper demonstrates a formal equivalence between the information bottleneck method and a class of non-perturbative renormalization group techniques, specifically within Gaussian statistics, revealing new insights into large-scale structure analysis.

Contribution

It establishes a formal mapping between IB and RG, showing their equivalence in Gaussian cases and introducing a new perspective on analyzing large-scale structures in complex systems.

Findings

IB and RG can be mapped onto each other via non-deterministic coarsening maps.

GIB has a semigroup structure with IB-optimal successive transformations.

RG cutoff schemes can be identified within the GIB framework.

Abstract

The renormalization group (RG) is a class of theoretical techniques used to explain the collective physics of interacting, many-body systems. It has been suggested that the RG formalism may be useful in finding and interpreting emergent low-dimensional structure in complex systems outside of the traditional physics context, such as in biology or computer science. In such contexts, one common dimensionality-reduction framework already in use is information bottleneck (IB), in which the goal is to compress an ``input'' signal $X$ while maximizing its mutual information with some stochastic ``relevance'' variable $Y$. IB has been applied in the vertebrate and invertebrate processing systems to characterize optimal encoding of the future motion of the external world. Other recent work has shown that the RG scheme for the dimer model could be ``discovered'' by a neural network attempting to solve an IB-like problem. This manuscript explores whether IB and any existing formulation of RG are formally equivalent. A class of soft-cutoff non-perturbative RG techniques are defined by families of non-deterministic coarsening maps, and hence can be formally mapped onto IB, and vice versa. For concreteness, this discussion is limited entirely to Gaussian statistics (GIB), for which IB has exact, closed-form solutions. Under this constraint, GIB has a semigroup structure, in which successive transformations remain IB-optimal. Further, the RG cutoff scheme associated with GIB can be identified. Our results suggest that IB can be used to impose a notion of ``large scale'' structure, such as biological function, on an RG procedure.