Nonconvex Low-Rank Matrix Recovery with Arbitrary Outliers via Median-Truncated Gradient Descent

TL;DR

This paper introduces a robust truncated gradient descent algorithm for low-rank matrix recovery that effectively handles arbitrary outliers, achieving linear convergence with fewer measurements and demonstrating superior empirical performance.

Contribution

The paper proposes a novel median-truncated gradient descent method that enhances robustness against outliers in low-rank matrix recovery, with improved initialization and convergence guarantees.

Findings

Converges linearly to the ground truth with a near-optimal number of measurements.

Effective in the presence of a constant fraction of arbitrarily corrupted measurements.

Numerical experiments validate the method's superior robustness and accuracy.

Abstract

Recent work has demonstrated the effectiveness of gradient descent for directly recovering the factors of low-rank matrices from random linear measurements in a globally convergent manner when initialized properly. However, the performance of existing algorithms is highly sensitive in the presence of outliers that may take arbitrary values. In this paper, we propose a truncated gradient descent algorithm to improve the robustness against outliers, where the truncation is performed to rule out the contributions of samples that deviate significantly from the {\em sample median} of measurement residuals adaptively in each iteration. We demonstrate that, when initialized in a basin of attraction close to the ground truth, the proposed algorithm converges to the ground truth at a linear rate for the Gaussian measurement model with a near-optimal number of measurements, even when a constant fraction of the measurements are arbitrarily corrupted. In addition, we propose a new truncated spectral method that ensures an initialization in the basin of attraction at slightly higher requirements. We finally provide numerical experiments to validate the superior performance of the proposed approach.