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Low-rank matrix estimation via nonconvex spectral regularized methods in errors-in-variables matrix regression

Xin Li, Dongya Wu

TL;DR

The general errors-in-variables matrix regression model is considered and a unified framework for low-rank estimation based on nonconvex spectral regularization is proposed and the performance of the proposed nonconvex estimation method is illustrated by numerical experiments.

Abstract

High-dimensional matrix regression has been studied in various aspects, such as statistical properties, computational efficiency and application to specific instances including multivariate regression, system identification and matrix compressed sensing. Current studies mainly consider the idealized case that the covariate matrix is obtained without noise, while the more realistic scenario that the covariates may always be corrupted with noise or missing data has received little attention. We consider the general errors-in-variables matrix regression model and proposed a unified framework for low-rank estimation based on nonconvex spectral regularization. Then in the statistical aspect, recovery bounds for any stationary points are provided to achieve statistical consistency. In the computational aspect, the proximal gradient method is applied to solve the nonconvex optimization problem and is proved to converge in polynomial time. Consequences for specific matrix compressed sensing models with additive noise and missing data are obtained via verifying corresponding regularity conditions. Finally, the performance of the proposed nonconvex estimation method is illustrated by numerical experiments.

Low-rank matrix estimation via nonconvex spectral regularized methods in errors-in-variables matrix regression

TL;DR

The general errors-in-variables matrix regression model is considered and a unified framework for low-rank estimation based on nonconvex spectral regularization is proposed and the performance of the proposed nonconvex estimation method is illustrated by numerical experiments.

Abstract

High-dimensional matrix regression has been studied in various aspects, such as statistical properties, computational efficiency and application to specific instances including multivariate regression, system identification and matrix compressed sensing. Current studies mainly consider the idealized case that the covariate matrix is obtained without noise, while the more realistic scenario that the covariates may always be corrupted with noise or missing data has received little attention. We consider the general errors-in-variables matrix regression model and proposed a unified framework for low-rank estimation based on nonconvex spectral regularization. Then in the statistical aspect, recovery bounds for any stationary points are provided to achieve statistical consistency. In the computational aspect, the proximal gradient method is applied to solve the nonconvex optimization problem and is proved to converge in polynomial time. Consequences for specific matrix compressed sensing models with additive noise and missing data are obtained via verifying corresponding regularity conditions. Finally, the performance of the proposed nonconvex estimation method is illustrated by numerical experiments.
Paper Structure (10 sections, 10 theorems, 113 equations)

This paper contains 10 sections, 10 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Problem setup
  3. General nonconvex estimator
  4. Regularity conditions
  5. Main results
  6. Statistical recovery bounds
  7. Computational convergence rates
  8. Errors-in-variables matrix regression
  9. Numerical experiments
  10. Conclusion

Key Result

Lemma 1

Let $\Theta,\Theta'\in {\mathbb{R}}^{d_1\times d_2}$ be two given matrices and $d=\min\{d_1,d_2\}$. Let $f:{\mathbb{R}}_+\to {\mathbb{R}}_+$ be a concave increasing function satisfying $f(0)=0$. Then we have that

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 14 more