LONG RANDOM MATRICES AND TENSOR UNFOLDING

• 成果类型: Article
• 署名作者: Ben Arous, Gerard; Huang, Daniel Zhengyu; Huang, Jiaoyang
• 署名单位: New York University; Peking University; University of Pennsylvania
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1958
• 发表日期: 2023
• 页码: 5753-5780
• 关键词: Covariance matrices; LARGEST EIGENVALUE; PHASE-TRANSITION; sparse pca; shrinkage

摘要

In this paper, we consider the singular values and singular vectors of low rank perturbations of large rectangular random matrices, in the regime the matrix is long: we allow the number of rows (columns) to grow polynomi-ally in the number of columns (rows). We prove there exists a critical signal-to-noise ratio (depending on the dimensions of the matrix), and the extreme singular values and singular vectors exhibit a BBP-type phase transition. As a main application, we investigate the tensor unfolding algorithm for the asym-metric rank-one spiked tensor model, and obtain an exact threshold, which is independent of the procedure of tensor unfolding. If the signal-to-noise ratio is above the threshold, tensor unfolding detects the signals; otherwise, it fails to capture the signals.