Exploring the Landscape of Collective Modes Using IB-RG

The Renormalization Group (RG) is a set of theoretical techniques which interpolate between microscopic models and their macroscopic properties. However, RG typically relies on a definition of the collective modes of a system, organized from least to most relevant. In biological systems, this poses a challenge, since context and function shape collective behaviors in a way that is not known at the outset. In another domain, the information bottleneck approach (IB) describes the best probabilistic compression of an input variable X under a pre-defined “relevance” variable Y. Recent theoretical work has shown that IB can be imposed on RG, allowing one to define the notion of large-scale structure probed by the RG flow [1]. To understand what it means to choose a notion of scale, we explore the behavior of IB-RG flows as the relevance variable statistics are changed. We uncover nontrivial physics in terms of order parameters that vary with with our definition of Y. Our analysis may serve as a model for RG applied to biological systems, in particular neural networks, whose goals are multifaceted and whose collective behavior requires sensitivity to many features in the environment.

[1] Kline and Palmer (2021). arXiv 2107.13700.