ALGORITHMIC THRESHOLDS FOR TENSOR PCA

• 成果类型: Article
• 署名作者: Ben Arous, Gerard; Gheissari, Reza; Jagannath, Aukosh
• 署名单位: New York University; University of Waterloo
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1415
• 发表日期: 2020
• 页码: 2052-2087
• 关键词: largest eigenvalue; PHASE-TRANSITION; complexity; matrices

摘要

We study the algorithmic thresholds for principal component analysis of Gaussian k-tensors with a planted rank-one spike, via Langevin dynamics and gradient descent. In order to efficiently recover the spike from natural initializations, the signal-to-noise ratio must diverge in the dimension. Our proof shows that the mechanism for the success/failure of recovery is the strength of the curvature of the spike on the maximum entropy region of the initial data. To demonstrate this, we study the dynamics on a generalized family of high-dimensional landscapes with planted signals, containing the spiked tensor models as specific instances. We identify thresholds of signal-to-noise ratios above which order 1 time recovery succeeds; in the case of the spiked tensor model, these match the thresholds conjectured for algorithms such as approximate message passing. Below these thresholds, where the curvature of the signal on the maximal entropy region is weak, we show that recovery from certain natural initializations takes at least stretched exponential time. Our approach combines global regularity estimates for spin glasses with pointwise estimates to study the recovery problem by a perturbative approach.