Renormalization Group Flow as Optimal Transport

Abstract

We establish that Polchinski's equation for exact renormalization group flow is equivalent to the optimal transport gradient flow of a field-theoretic relative entropy. This provides a compelling information-theoretic formulation of the exact renormalization group, expressed in the language of optimal transport. A striking consequence is that a regularization of the relative entropy is in fact an RG monotone. We compute this monotone in several examples. Our results apply more broadly to other exact renormalization group flow equations, including widely used specializations of Wegner-Morris flow. Moreover, our optimal transport framework for RG allows us to reformulate RG flow as a variational problem. This enables new numerical techniques and establishes a systematic connection between neural network methods and RG flows of conventional field theories.

Bayesian renormalization

“…as has appeared previously in [12,16,17,41,45]. Here C Λ (x, y) is the ERG kernel appearing in the Fokker-Planck equation associated to the ERG, and Σ Λ [ϕ; P Λ ] is called the scheme functional which is determined through the ERG potential V Λ via the equation…”

“…Following the lead of [41], we take the perspective that an ERG flow can be understood as a one parameter family of probability distributions, {P Λ [ϕ]} Λ∈R . Here Λ is a physically meaningful RG scale (typically associated with a momentum cutoff), and ϕ ∈ F corresponds to the field configuration relevant to a given theory.…”

“…where σ 2 is a hyperparameter that sets a scale for the tolerated prediction error. Given a parametric family of probability distributions for Y which can be written in the form (41), we are motivated to rewrite the posterior predictive distribution (39) as an integral over S by forming the pushforward measure via the mapping G 13 .…”

“…In section 2.1 we review exact renormalization in its original physical context. We stress the perspective that a useful subclass of ERG schemes constitute functional diffusion equations, as was originally touted by [41], and recognize the role played by the physical scale in defining such diffusive ERGs. Building on the viewpoint that renormalization and diffusion are equivalent we identify diffusion as a useful device for renormalizing data models even without physical scales in section 2.2.…”

Exact renormalization groups and transportation of measures

“…Further Lipschitz properties of the Langevin transport map can be found in the work of Klartag & Putterman [26] and Neeman [34]. While Cotler and Rezchikov [18] recently made a connection between optimal transport and exact renormalization groups, it was shown by Tanana [37] that, in general, the Langevin transport map is not the same as the optimal transport map. Finally, a connection between renormalization and Ornstein-Uhlenbeck semigroups is discussed by Faris in [21].…”