Algorithmic thresholds for tensor PCA

TL;DR

This paper investigates the algorithmic thresholds for tensor PCA with a planted spike, revealing how the success of recovery depends on the signal-to-noise ratio and the curvature of the signal in high-dimensional landscapes.

Contribution

It introduces a new framework linking the success of tensor PCA recovery to the curvature of the signal on the maximum entropy region, and identifies thresholds matching conjectured limits.

Findings

Recovery succeeds above certain SNR thresholds

Below thresholds, recovery takes at least stretched exponential time

The approach combines spin glass regularity estimates with perturbative analysis

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 point-wise estimates, to study the recovery problem by a perturbative approach.