Lifelong Matrix Completion with Sparsity-Number

TL;DR

This paper introduces a single-phase matrix completion algorithm based on sparsity-number that matches the efficiency of multi-phase methods and outperforms previous algorithms by making more informed decisions.

Contribution

It proposes a novel single-phase matrix completion algorithm leveraging sparsity-number, extending to a two-phase method, and demonstrating comparable efficiency to multi-phase algorithms.

Findings

The proposed algorithm achieves theoretical lower bounds in matrix recovery.

Experimental results show improved efficiency over previous methods.

The single-phase approach simplifies the process while maintaining performance.

Abstract

Matrix completion problem has been previously studied under various adaptive and passive settings. Previously, researchers have proposed passive, two-phase and single-phase algorithms using coherence parameter, and multi phase algorithm using sparsity-number. It has been shown that the method using sparsity-number reaching to theoretical lower bounds in many conditions. However, the aforementioned method is running in many phases through the matrix completion process, therefore it makes much more informative decision at each stage. Hence, it is natural that the method outperforms previous algorithms. In this paper, we are using the idea of sparsity-number and propose and single-phase column space recovery algorithm which can be extended to two-phase exact matrix completion algorithm. Moreover, we show that these methods are as efficient as multi-phase matrix recovery algorithm. We provide experimental evidence to illustrate the performance of our algorithm.