Leave-One-Out Analysis for Nonconvex Robust Matrix Completion with General Thresholding Functions

TL;DR

This paper introduces a novel leave-one-out analysis for a simple nonconvex robust matrix completion method, demonstrating linear convergence and improved sampling complexity for general thresholding functions.

Contribution

It provides the first leave-one-out analysis for a nonconvex RMC method, showing linear convergence and enhanced sampling efficiency for various thresholding functions.

Findings

Achieves linear convergence for general thresholding functions.

First leave-one-out analysis on a nonconvex RMC method.

Improves sampling complexity over existing methods.

Abstract

We study the problem of robust matrix completion (RMC), where the partially observed entries of an underlying low-rank matrix is corrupted by sparse noise. Existing analysis of the non-convex methods for this problem either requires the explicit but empirically redundant regularization in the algorithm or requires sample splitting in the analysis. In this paper, we consider a simple yet efficient nonconvex method which alternates between a projected gradient step for the low-rank part and a thresholding step for the sparse noise part. Inspired by leave-one out analysis for low rank matrix completion, it is established that the method can achieve linear convergence for a general class of thresholding functions, including for example soft-thresholding and SCAD. To the best of our knowledge, this is the first leave-one-out analysis on a nonconvex method for RMC. Additionally, when applying our result to low rank matrix completion, it improves the sampling complexity of existing result for the singular value projection method.