Algorithmic thresholds for tensor PCA

Gérard Ben Arous, Reza Gheissari, Aukosh Jagannath

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2052-2087
Number of pages36
JournalAnnals of Probability
Issue number4
StatePublished - Jul 1 2020


  • Free energy wells
  • Gradient descent
  • Langevin dynamics
  • Planted signal recovery
  • Spiked tensor model
  • Spin glasses
  • Tensor PCA
  • Tensor estimation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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