Detection Thresholds in Very Sparse Matrix Completion

Abstract

We study the matrix completion problem: an underlying \(m \times n\) matrix P is low rank, with incoherent singular vectors, and a random \(m \times n\) matrix A is equal to P on a (uniformly) random subset of entries of size dn. All other entries of A are equal to zero. The goal is to retrieve information on P from the observation of A. Let \(A_1\) be the random matrix where each entry of A is multiplied by an independent \(\{0,1\}\)-Bernoulli random variable with parameter 1/2. This paper is about when, how and why the non-Hermitian eigen-spectra of the matrices \(A_1 (A - A_1)^*\) and \((A-A_1)^*A_1\) captures more of the relevant information about the principal component structure of A than the eigen-spectra of \(A A^*\) and \(A^* A\). We show that the eigenvalues of the asymmetric matrices \(A_{1} (A - A_{1})^{*}\) and \((A-A_{1})^{*} A_{1}\) with modulus greater than a detection threshold are asymptotically equal to the eigenvalues of \(PP^*\) and \(P^*P\) and that the associated eigenvectors are aligned as well. The central surprise is that by intentionally inducing asymmetry and additional randomness via the \(A_1\) matrix, we can extract more information than if we had worked with the singular value decomposition (SVD) of A. The associated detection threshold is asymptotically exact and is non-universal since it explicitly depends on the element-wise distribution of the underlying matrix P. We show that reliable, statistically optimal but not perfect matrix recovery, via a universal data-driven algorithm, is possible above this detection threshold using the information extracted from the asymmetric eigen-decompositions. Averaging the left and right eigenvectors provably improves estimation accuracy but not the detection threshold. Our results encompass the very sparse regime where d is of order 1 where matrix completion via the SVD of A fails or produces unreliable recovery. We define another variant of this asymmetric principal component analysis procedure that bypasses the randomization step and has a detection threshold that is smaller by a constant factor but with a computational cost that is larger by a polynomial factor of the number of observed entries. Both detection thresholds allow to go beyond the barrier due to the well-known information theoretical limit \(d \asymp \log n\) for exact matrix completion found in the literature.