STATISTICAL THRESHOLDS FOR TENSOR PCA

• 成果类型: Article
• 署名作者: Jagannath, Aukosh; Lopatto, Patrick; Miolane, Leo
• 署名单位: Harvard University; University of Waterloo; University of Waterloo; Inria; Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1547
• 发表日期: 2020
• 页码: 1910-1933
• 关键词: spin-glass model; LARGEST EIGENVALUE; principal components; Mutual information; PHASE-TRANSITION; matrices; deformation; THEOREMS; cavity; LIMITS

摘要

We study the statistical limits of testing and estimation for a rank one deformation of a Gaussian random tensor. We compute the sharp thresholds for hypothesis testing and estimation by maximum likelihood and show that they are the same. Furthermore, we find that the maximum likelihood estimator achieves the maximal correlation with the planted vector among measurable estimators above the estimation threshold. In this setting, the maximum likelihood estimator exhibits a discontinuous BBP-type transition: below the critical threshold the estimator is orthogonal to the planted vector, but above the critical threshold, it achieves positive correlation which is uniformly bounded away from zero.