An exact mapping between the Variational Renormalization Group and Deep Learning

TL;DR

This paper establishes an exact theoretical connection between deep learning architectures, specifically Restricted Boltzmann Machines, and the renormalization group technique from physics, providing insights into why deep learning effectively learns features.

Contribution

It constructs an exact mapping between variational renormalization group methods and deep learning models, revealing a fundamental link between physics and machine learning.

Findings

Deep learning architectures can be viewed as RG-like coarse-graining schemes.

The mapping is demonstrated using the Ising Model in one and two dimensions.

This connection offers a new perspective on feature learning in deep networks.

Abstract

Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. Recently, such techniques have yielded record-breaking results on a diverse set of difficult machine learning tasks in computer vision, speech recognition, and natural language processing. Despite the enormous success of deep learning, relatively little is understood theoretically about why these techniques are so successful at feature learning and compression. Here, we show that deep learning is intimately related to one of the most important and successful techniques in theoretical physics, the renormalization group (RG). RG is an iterative coarse-graining scheme that allows for the extraction of relevant features (i.e. operators) as a physical system is examined at different length scales. We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). We illustrate these ideas using the nearest-neighbor Ising Model in one and two-dimensions. Our results suggests that deep learning algorithms may be employing a generalized RG-like scheme to learn relevant features from data.