Renormalization group flow as optimal transport

We show that Polchinski’s equation for exact renormalization group flow is equivalent to the optimal transport gradient flow of a field-theoretic relative entropy.  This gives a surprising information-theoretic formulation of the exact renormalization group, expressed in the language of optimal transport.  We will provide reviews of both the exact renormalization group, as well as the theory of optimal transportation.  Our results allow us to establish a new, non-perturbative RG monotone, and also reformulate RG flow as a variational problem.  The latter enables new numerical techniques and allows us to establish a systematic connection between neural network methods and RG flows of conventional field theories.  Our techniques generalize to other RG flow equations beyond Polchinski’s.

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