Sparse Representations, Inference and Learning
It provides theoretical tools for researchers in machine learning to study fundamental limitations and algorithms in inference problems, but is incremental as it builds on existing statistical physics approaches.
The paper presents a general framework from statistical physics for analyzing large-dimensional inference problems like compressed sensing and perceptron learning, focusing on replica symmetric and cavity methods.
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 Mézard 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.