Relevance in the Renormalization Group and in Information Theory

The analysis of complex physical systems hinges on ability to sift out the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architecture and training-dependent learned "relevant" features bear to standard objects of physical theory.
Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the information-theoretic notion of relevancy defined in the Information Bottleneck (IB) formalism of compression theory, and the field-theoretic relevance of the Renormalization Group. We show analytically that for statistical physical systems described by a field theory the "relevant" degrees of freedom found using IB compression indeed correspond to primary operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically to high precision. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an avenue to constructively incorporate physical interpretability in applications of machine learning.