Optimal Exact Matrix Completion Under new Parametrization

TL;DR

This paper introduces new matrix completion algorithms that exactly recover low-rank matrices with less observation data, especially for highly coherent matrices, by leveraging a novel sparsity-number relation.

Contribution

The paper proposes algorithms that improve exact matrix completion efficiency and require less prior information, especially for highly coherent matrices, using a new sparsity-number concept.

Findings

Achieves exact recovery with fewer observations than previous methods.

Effectively recovers highly coherent matrices with minimal data.

Experimental results demonstrate the algorithms' superior performance.

Abstract

We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call \textit{sparsity-number} here). Using this relation, we propose matrix completion algorithms that exactly recovers the target matrix. These algorithms are superior to previous works in two important ways. First, our algorithms exactly recovers $\mu_0$-coherent column space matrices by probability at least $1 - \epsilon$ using much smaller observations complexity than $\mathcal{O}(\mu_0 rn \mathrm{log}\frac{r}{\epsilon})$ the state of art. Specifically, many of the previous adaptive sampling methods require to observe the entire matrix when the column space is highly coherent. However, we show that our method is still able to recover this type of matrices by observing a small fraction of entries under many scenarios. Second, we propose an exact completion algorithm, which requires minimal pre-information as either row or column space is not being highly coherent. At the end of the paper, we provide experimental results that illustrate the strength of the algorithms proposed here.