Beyond RG: from parameter flow to metric flow

TL;DR

This paper extends the concept of renormalization group (RG) by introducing a metric flow framework that describes how the Fisher Information Metric evolves in complex systems, broadening the scope beyond traditional RG assumptions.

Contribution

It proposes a general framework replacing parameter flow with metric flow on the model manifold, applicable to systems lacking self-similarity symmetry.

Findings

Traditional RG systems have metric flow induced by parameter flow.

In general, the model manifold's geometry deforms under metric flow.

Augmenting the manifold with an extra parameter can reduce metric flow to parameter flow.

Abstract

Complex systems with many degrees of freedom are typically intractable, but some of their behaviors may admit simpler effective descriptions. The question of when such effective descriptions are possible remains open. The paradigmatic approach where such "emergent simplicity" can be understood in detail is the renormalization group (RG). Here, we show that for general systems, without the self-similarity symmetry required by the RG construction, the RG flow of model parameters is replaced by a more general flow of the Fisher Information Metric on the model manifold. We demonstrate that the systems traditionally studied with RG comprise special cases where this metric flow can be induced by a parameter flow, keeping the global geometry of the model-manifold fixed. In general, however, the geometry may deform, and metric flow cannot be reduced to a parameter flow -- though this could be achieved at the cost of augmenting the manifold by one new parameter, as we discuss. We hope that our framework can clarify how ideas from RG may apply in a broader class of complex systems.