Open Access
July 2020 Algorithmic thresholds for tensor PCA
Gérard Ben Arous, Reza Gheissari, Aukosh Jagannath
Ann. Probab. 48(4): 2052-2087 (July 2020). DOI: 10.1214/19-AOP1415


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.


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Gérard Ben Arous. Reza Gheissari. Aukosh Jagannath. "Algorithmic thresholds for tensor PCA." Ann. Probab. 48 (4) 2052 - 2087, July 2020.


Received: 1 August 2018; Revised: 1 November 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224968
MathSciNet: MR4124533
Digital Object Identifier: 10.1214/19-AOP1415

Primary: 62F10 , 62F30 , 62M05
Secondary: 46N30 , 60H30 , 65C05 , 82C44 , 82D30

Keywords: free energy wells , gradient descent , Langevin dynamics , planted signal recovery , Spiked tensor model , Spin glasses , tensor estimation , Tensor PCA

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 4 • July 2020
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