Low-rank matrix estimation in multi-response regression with measurement errors: Statistical and computational guarantees

TL;DR

This paper introduces a novel nonconvex estimator for low-rank matrix recovery in multi-response regression with measurement errors, providing statistical guarantees and an efficient computational algorithm with proven convergence.

Contribution

It proposes a new error-corrected nonconvex estimator with theoretical recovery bounds and a proximal gradient method for efficient computation, applicable to various measurement error models.

Findings

Nonasymptotic recovery bounds established for the estimator

Proximal gradient method converges linearly to a near-global solution

Numerical experiments confirm theoretical predictions in high-dimensional settings

Abstract

In this paper, we investigate the matrix estimation problem in the multi-response regression model with measurement errors. A nonconvex error-corrected estimator based on a combination of the amended loss function and the nuclear norm regularizer is proposed to estimate the matrix parameter. Then under the (near) low-rank assumption, we analyse statistical and computational theoretical properties of global solutions of the nonconvex regularized estimator from a general point of view. In the statistical aspect, we establish the nonasymptotic recovery bound for any global solution of the nonconvex estimator, under restricted strong convexity on the loss function. In the computational aspect, we solve the nonconvex optimization problem via the proximal gradient method. The algorithm is proved to converge to a near-global solution and achieve a linear convergence rate. In addition, we also verify sufficient conditions for the general results to be held, in order to obtain probabilistic consequences for specific types of measurement errors, including the additive noise and missing data. Finally, theoretical consequences are demonstrated by several numerical experiments on corrupted errors-in-variables multi-response regression models. Simulation results reveal excellent consistency with our theory under high-dimensional scaling.