Author:
Gamarnik David,Moore Cristopher,Zdeborová Lenka
Abstract
Abstract
In this review article we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked tensor model and the generalized linear model. We discuss the random (non-planted) versions of these problems as prototypical optimization problems, as well as the planted versions (with a hidden solution) as prototypical problems in statistical inference and learning. Based on ideas from physics, many of these problems have transitions where they are believed to jump from easy (solvable in polynomial time) to hard (requiring exponential time). We discuss several emerging ideas in theoretical computer science and statistics that provide rigorous evidence for hardness by proving that large classes of algorithms fail in the conjectured hard regime. This includes the overlap gap property, a particular mathematization of clustering or dynamical symmetry-breaking, which can be used to show that many algorithms that are local or robust to changes in their input fail. We also discuss the sum-of-squares hierarchy, which places bounds on proofs or algorithms that use low-degree polynomials such as standard spectral methods and semidefinite relaxations, including the Sherrington–Kirkpatrick model. Throughout the manuscript we present connections to the physics of disordered systems and associated replica symmetry breaking properties.
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Statistical mechanics of the maximum-average submatrix problem;Journal of Statistical Mechanics: Theory and Experiment;2024-01-01
2. Barriers for the performance of graph neural networks (GNN) in discrete random structures;Proceedings of the National Academy of Sciences;2023-11-06
3. Neural-prior stochastic block model;Machine Learning: Science and Technology;2023-08-17
4. Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials;Proceedings of the 55th Annual ACM Symposium on Theory of Computing;2023-06-02