Subspace Estimation from Unbalanced and Incomplete Data Matrices: $\ell_{2,\infty}$ Statistical Guarantees
Year: 2019
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Princeton University, Tsinghua University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: G1469985387, G1523888516, G1717380892, Collaborative Research: Fine-Grained Statistical Inference in High Dimension: Actionable Information, Bias Reduction, and Optimality, AMPS: Collaborative Research: Analysis of Local Power Grid Properties: From Network Motifs to Tensors, CAREER: Robust Methods for High-Dimensional Signal Processing under Geometric Constraints, RI: Medium: Collaborative Research:Algorithmic High-Dimensional Statistics: Optimality, Computtional Barriers, and High-Dimensional Corrections, G3467583470, NSF-BSF:CIF: Small: A Unified View of Estimation and Information Relationships for Networks and Beyond, G4061742820, G4353487559, G4558904681, G4884717353, CIF: Small: Taming Nonconvexity in High-Dimensional Statistical Estimation, G556410599, G5972442391, G6098521345, G7452299184, CIF: Medium: Collaborative Research: Nonconvex Optimization for High-Dimensional Signal Estimation: Theory and Fast Algorithms, G854152385, G8597684915, G8876996369
OpenAlex ID: W2979328751