Inferring Low-rank Energy-based Models and the Renormalization Group

Complex datasets like neural recordings, high-dimensional images, and protein sequences are often best understood through coarse-grained, low-dimensional representations that isolate the relevant features. The renormalization group works similarly by successively stripping away details and tracking how a system's effective description changes across scales. Building on this perspective, we present a data-driven procedure that infers low-rank energy-based models whose latent variables identify the collective degrees of freedom. By iteratively reducing the number of latent variables and rescaling, we obtain a Wilsonian-like renormalization group flow. In contrast to our method, common PCA-adjacent renormalization approaches integrate out the least susceptible directions of the data rather than the least probable collective degrees of freedom, which do not in general coincide. This procedure works without any additional supervision and does not require spatial or temporal information. With finite data, only a finite number of latent features are identifiable; we demonstrate that the amount of data directly relates to the initial point of the renormalization group flow. Our work provides an interpretable approach to inferring generative models of data that give insight into the underlying structure.