Subspace Estimation from Unbalanced and Incomplete Data Matrices: $\ell_{2,\infty}$ Statistical Guarantees

TL;DR

This paper develops new statistical guarantees for a spectral method estimating the column space of a low-rank matrix from noisy, incomplete, and unbalanced data, with applications to tensor completion, PCA, and community detection.

Contribution

It provides novel $ ext{ell}_{2, ext{infinity}}$ guarantees for a spectral estimator in unbalanced, incomplete data scenarios, improving upon previous results.

Findings

Establishes matching minimax lower bounds for the estimation problem.

Derives improved guarantees for tensor completion, PCA with missing data, and bipartite community recovery.

Demonstrates the effectiveness of the spectral method under new statistical conditions.

Abstract

This paper is concerned with estimating the column space of an unknown low-rank matrix $\boldsymbol{A}^{\star}\in\mathbb{R}^{d_{1}\times d_{2}}$, given noisy and partial observations of its entries. There is no shortage of scenarios where the observations -- while being too noisy to support faithful recovery of the entire matrix -- still convey sufficient information to enable reliable estimation of the column space of interest. This is particularly evident and crucial for the highly unbalanced case where the column dimension $d_{2}$ far exceeds the row dimension $d_{1}$, which is the focal point of the current paper. We investigate an efficient spectral method, which operates upon the sample Gram matrix with diagonal deletion. While this algorithmic idea has been studied before, we establish new statistical guarantees for this method in terms of both $\ell_{2}$ and $\ell_{2,\infty}$ estimation accuracy, which improve upon prior results if $d_{2}$ is substantially larger than $d_{1}$. To illustrate the effectiveness of our findings, we derive matching minimax lower bounds with respect to the noise levels, and develop consequences of our general theory for three applications of practical importance: (1) tensor completion from noisy data, (2) covariance estimation / principal component analysis with missing data, and (3) community recovery in bipartite graphs. Our theory leads to improved performance guarantees for all three cases.