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.

翻译：近年来，统计物理已被证明是探究大规模推断问题（如机器学习中出现的此类问题）的宝贵工具。统计物理提供了分析工具来研究这些解决方案的基本限制，并提出了求解具体实例的算法。在本讲义中，基于Marc Mézard于2022年在莱苏什暑期学校的授课内容，我们将介绍一个通用框架，可适用于多种具有弱长程相互作用的模型，包括压缩感知问题或感知机中的学习问题。我们将看到如何利用空腔方法的发展，在复制对称层次上研究这些问题——既作为一种理论工具，也作为一种算法。