Optimal Spectral Recovery of a Planted Vector in a Subspace

Recovering a planted vector v in an n-dimensional random subspace of R is a generic task related to many problems in machine learning and statistics, such as dictionary learning, subspace recovery, principal component analysis, and non-Gaussian component analysis. In this work, we study computationally efficient estimation and detection of a planted vector v whose l4 norm differs from that of a Gaussian vector with the same l2 norm. For instance, in the special case where v is an Nρ-sparse vector with Bernoulli–Gaussian or Bernoulli–Rademacher entries, our results include the following: (1) We give an improved analysis of a slight variant of the spectral method proposed by Hopkins, Schramm, Shi, and Steurer (2016), showing that it approximately recovers v with high probability in the regime nρ ≪ √ N . This condition subsumes the conditions ρ ≪ 1/ √ n or n √ ρ . √ N required by previous work up to polylogarithmic factors. We achieve l∞ error bounds for the spectral estimator via a leave-one-out analysis, from which it follows that a simple thresholding procedure exactly recovers v with Bernoulli–Rademacher entries, even in the dense case ρ = 1. (2) We study the associated detection problem and show that in the regime nρ ≫ √ N , any spectral method from a large class (and more generally, any low-degree polynomial of the input) fails to detect the planted vector. This matches the condition for recovery and offers evidence that no polynomial-time algorithm can succeed in recovering a Bernoulli–Gaussian vector v when nρ ≫ √ N . Email: cheng.mao@math.gatech.edu. Partially supported by NSF grants DMS-2053333 and DMS-2210734. Email: aswein@ucdavis.edu. Part of this work was done while with the Courant Institute at NYU, partially supported by NSF grant DMS-1712730 and by the Simons Collaboration on Algorithms and Geometry.