Annals of Probability

Algorithmic thresholds for tensor PCA

Gérard Ben Arous, Reza Gheissari, and Aukosh Jagannath

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Abstract

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.

Article information

Source
Ann. Probab., Volume 48, Number 4 (2020), 2052-2087.

Dates
Received: August 2018
Revised: November 2019
First available in Project Euclid: 20 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1595232103

Digital Object Identifier
doi:10.1214/19-AOP1415

Mathematical Reviews number (MathSciNet)
MR4124533

Zentralblatt MATH identifier
07224968

Subjects
Primary: 62F10: Point estimation 62M05: Markov processes: estimation 62F30: Inference under constraints
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 65C05: Monte Carlo methods 46N30: Applications in probability theory and statistics 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
Tensor PCA tensor estimation spiked tensor model planted signal recovery gradient descent Langevin dynamics spin glasses free energy wells

Citation

Ben Arous, Gérard; Gheissari, Reza; Jagannath, Aukosh. Algorithmic thresholds for tensor PCA. Ann. Probab. 48 (2020), no. 4, 2052--2087. doi:10.1214/19-AOP1415. https://projecteuclid.org/euclid.aop/1595232103


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