Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

TL;DR

This paper introduces HSGD, a fast and efficient non-convex gradient descent algorithm tailored for robust low-rank Hankel matrix completion, effectively handling sparse corruptions and outliers.

Contribution

It proposes a novel structured gradient descent method leveraging Hankel structure, with proven linear convergence and superior efficiency over existing methods.

Findings

HSGD achieves high computational efficiency and sample efficiency.

Theoretical recovery guarantees with linear convergence are established.

Empirical tests show HSGD outperforms state-of-the-art methods on synthetic and real data.

Abstract

We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from the partial observation. In this paper, we explore the convenient Hankel structure and propose a novel non-convex algorithm, coined Hankel Structured Gradient Descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing- and sample-efficient compared to the state-of-the-arts. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.