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.
Highlights
An overarching theme in the study of complex systems is effective low-dimensionality
We show that Gaussian information bottleneck (GIB) [29] exhibits a semi-group structure in which successive IB coarsenings compose larger IB coarsenings
In Eq (9), we present a soft cutoff scheme which arises from the constraint of GIBoptimality, but it is given in terms of quantities which have no physical context, and so it is hard to say a priori how it relates to existing cutoff schemes structurally
Summary
An overarching theme in the study of complex systems is effective low-dimensionality. We are confident that the laws are insensitive to the particular microscopic configuration of a fluid at any given time These are connected, but different notions of low-dimensionality; the first deals with simplification in model space, while the second refers to the emergence of collective modes, of which relatively few, when compared to the total number of degrees of freedom, will be important. Because Xis defined as a non-deterministic coarsening of X, an exact correspondence between RG and IB demands that the RG scheme uses what is known as a “soft” cutoff This means, for example, that the ubiquitous perturbative momentum shell approach put forth by Wilson cannot be mapped exactly onto IB under the interpretation of Xas some coarsegrained variable. In whichever collective mode basis is chosen, the shape of this IB cutoff scheme is closely related to the Litim regulator which is ubiquitous in NPRG literature [30]
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