Gaussian information bottleneck and the non-perturbative renormalization group

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.

The Distributed Information Bottleneck reveals the explanatory structure of complex systems

“…Mutual information between partitions of a system allowed the automatic discovery and information-based prescription of relevant features [70][71][72]. The connection between the information bottleneck and the renormalization group [21,22] suggests IB can uncover the most relevant information in a relationship. By constraining the encoder in a standard IB framework to be a linear projection, the authors of Ref.…”

“…Given random variables X and Y serving as an input and an output, the IB defines a spectrum of compressed representations of X that retain only the most relevant information about Y . The potential of IB to analyze relationships was recently strengthened through connections to the renormalization group in statistical physics, one of the field's most powerful tools [21,22]. Although IB serves as a useful framework for examining the process of learning [23,24], it has limited capacity to find useful approximations through optimization, particularly when the relationship between X and Y is deterministic (or nearly so) [25,26].…”

Bayesian renormalization

“…This procedure is related to a variety of different data compression techniques such as the information bottleneck [37] and variational autoencoding [56]. In this way, the Bayesian renormalization scheme makes direct contact with previous insights into the relationship between model building/dimensional reduction for complex systems and the RG such as [19][20][21]36].…”

“…Or worse yet, what if we are interested in renormalizing a model that does not have a physical interpretation at all? Such a situation presents itself in recent work which seeks to import the machinery of renormalization into data science contexts as a tool for performing data compression and improving the interpretability and performance of high dimensional models [19][20][21][22][23][24][25][26][27][28][29].…”

A Neural Programming Language for the Reservoir Computer

“…What makes these neural computers so compelling is that they are exceptional in different ways than modern-day silicon computers: the latter relies on binary representations, rapid sequential processing [6], and segregated memory and CPU [7], while the former utilizes continuum representations [8,9], parallel and distributed processing [10,11], and distributed memory [12]. To harness these distinct computational abilities, prior work has studied a vast array of different network architectures [13,14], learning algorithms [15,16], and information-theoretic frameworks [17][18][19] in both biological and artificial neural networks. Despite these significant advances, the relationship between neural computers and modern-day silicon computers remains an analogy due to our lack of a concrete and low-level programming language, thereby limiting our access to neural computation.…”