Robust Matrix Completion with Deterministic Sampling via Convex Optimization

TL;DR

This paper introduces a convex optimization approach for robust matrix completion using deterministic sampling patterns, which are more hardware-friendly than random sampling, and proves conditions for exact recovery.

Contribution

It proposes the restricted approximate infinity-isometry property and demonstrates exact recovery guarantees under deterministic sampling, low-rank, and incoherence conditions.

Findings

Exact recovery is possible with high probability under specified conditions.

Deterministic sampling can replace random sampling in robust matrix completion.

The method is robust to a small fraction of outliers.

Abstract

This paper deals with the problem of robust matrix completion -- retrieving a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition. Though seemingly not an unresolved issue, we point out that the compressed matrix in our case is sampled in a deterministic pattern instead of those random ones on which existing studies depend. In fact, deterministic sampling is much more hardware-friendly than random ones. The limited resources on many platforms leave deterministic sampling the only choice to sense a matrix, resulting in the significance of investigating robust matrix completion with deterministic pattern. In such spirit, this paper proposes \textit{restricted approximate $\infty$-isometry property} and proves that, if a \textit{low-rank} and \textit{incoherent} square matrix and certain deterministic sampling pattern satisfy such property and two existing conditions called \textit{isomerism} and \textit{relative well-conditionedness}, the exact recovery from its sampled counterpart grossly corrupted by a small fraction of outliers via convex optimization happens with very high probability.