Sparse Representations, Inference and Learning

TL;DR

This paper discusses how statistical physics tools, especially the cavity method, can be applied to analyze and solve high-dimensional inference problems like compressed sensing and perceptron learning.

Contribution

It introduces a general framework using statistical physics for studying large inference problems with weak long-range interactions, including theoretical and algorithmic insights.

Findings

Application of replica symmetric analysis to inference problems

Development of cavity method-based algorithms

Insights into fundamental limitations of inference solutions

Abstract

In recent years statistical physics has proven to be a valuable tool to probe into large dimensional inference problems such as the ones occurring in machine learning. Statistical physics provides analytical tools to study fundamental limitations in their solutions and proposes algorithms to solve individual instances. In these notes, based on the lectures by Marc M\'ezard in 2022 at the summer school in Les Houches, we will present a general framework that can be used in a large variety of problems with weak long-range interactions, including the compressed sensing problem, or the problem of learning in a perceptron. We shall see how these problems can be studied at the replica symmetric level, using developments of the cavity methods, both as a theoretical tool and as an algorithm.