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

Abstract

This paper is concerned with estimating the column space of an unknown low-rank matrix ${\mathbf{A}}^{\star }\in {\mathbb{R}}^{d_1\times d_2}$, 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.

Accepted to Annals of Statistics

References in corpus (8)

• Provable Tensor Factorization with Missing Data
• Adaptive covariance matrix estimation through block thresholding
• A statistical model for tensor PCA
• Accurate Community Detection in the Stochastic Block Model via Spectral Algorithms
• Unified ${\ell }_{2\to \infty }$ Eigenspace Perturbation Theory for Symmetric Random Matrices
• Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's
• Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality
• Instance-dependent ${\ell }_{\infty }$-bounds for policy evaluation in tabular reinforcement learning

Cited by in corpus (3)

• Biwhitening Reveals the Rank of a Count Matrix
• Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution under Random Designs
• An ${\ell }_p$ theory of PCA and spectral clustering