PCA meets RG

TL;DR

This paper draws an analogy between principal components analysis (PCA) and the renormalization group (RG), proposing a new framework for analyzing covariance spectra and applying it to neural and financial data.

Contribution

It introduces a novel analogy between PCA and RG, defining relevant and irrelevant operators based on eigenvalue density, and demonstrates its application to real neural and financial datasets.

Findings

Neural activity shows behavior controlled by a nontrivial fixed point.

Financial data analysis separates noise-dominated modes from meaningful modes.

The approach provides insights into the structure of high-dimensional data.

Abstract

A system with many degrees of freedom can be characterized by a covariance matrix; principal components analysis (PCA) focuses on the eigenvalues of this matrix, hoping to find a lower dimensional description. But when the spectrum is nearly continuous, any distinction between components that we keep and those that we ignore becomes arbitrary; it then is natural to ask what happens as we vary this arbitrary cutoff. We argue that this problem is analogous to the momentum shell renormalization group (RG). Following this analogy, we can define relevant and irrelevant operators, where the role of dimensionality is played by properties of the eigenvalue density. These results also suggest an approach to the analysis of real data. As an example, we study neural activity in the vertebrate retina as it responds to naturalistic movies, and find evidence of behavior controlled by a nontrivial fixed point. Applied to financial data, our analysis separates modes dominated by sampling noise from a smaller but still macroscopic number of modes described by a non--Gaussian distribution.