Equivalence of approximate message passing and low-degree polynomials in rank-one matrix estimation

• 成果类型: Article
• 署名作者: Montanari, Andrea; Wein, Alexander S.
• 署名单位: Stanford University; Stanford University; University of California System; University of California Davis
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01322-z
• 发表日期: 2025
• 页码: 181-233
• 关键词: semidefinite relaxations; state evolution; algorithms

摘要

We consider the problem of estimating an unknown parameter vector theta is an element of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\theta }}\in {{\mathbb {R}}}<^>n$$\end{document}, given noisy observations Y=theta theta T/n+Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{Y}}}= {\varvec{\theta }}{\varvec{\theta }}<^>{\textsf{T}}/\sqrt{n}+{{\varvec{Z}}}$$\end{document} of the rank-one matrix theta theta T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\theta }}{\varvec{\theta }}<^>{\textsf{T}}$$\end{document}, where Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{Z}}}$$\end{document} has independent Gaussian entries. When information is available about the distribution of the entries of theta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\theta }}$$\end{document}, spectral methods are known to be strictly sub-optimal. Past work characterized the asymptotics of the accuracy achieved by the optimal estimator. However, no polynomial-time estimator is known that achieves this accuracy. It has been conjectured that this statistical-computation gap is fundamental, and moreover that the optimal accuracy achievable by polynomial-time estimators coincides with the accuracy achieved by certain approximate message passing (AMP) algorithms. We provide evidence towards this conjecture by proving that no estimator in the (broader) class of constant-degree polynomials can surpass AMP.