Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling

TL;DR

This paper introduces Cross-Concentrated Sampling (CCS), a flexible sampling strategy that bridges uniform and CUR sampling for matrix completion, along with an efficient algorithm ICURC, demonstrating empirical advantages in experiments.

Contribution

The paper proposes a novel CCS sampling method that combines uniform and CUR sampling, along with an efficient non-convex algorithm ICURC for matrix completion.

Findings

CCS provides flexibility and potential cost savings in sampling.

ICURC outperforms baseline algorithms in experiments.

Numerical results validate the effectiveness of CCS and ICURC.

Abstract

While uniform sampling has been widely studied in the matrix completion literature, CUR sampling approximates a low-rank matrix via row and column samples. Unfortunately, both sampling models lack flexibility for various circumstances in real-world applications. In this work, we propose a novel and easy-to-implement sampling strategy, coined Cross-Concentrated Sampling (CCS). By bridging uniform sampling and CUR sampling, CCS provides extra flexibility that can potentially save sampling costs in applications. In addition, we also provide a sufficient condition for CCS-based matrix completion. Moreover, we propose a highly efficient non-convex algorithm, termed Iterative CUR Completion (ICURC), for the proposed CCS model. Numerical experiments verify the empirical advantages of CCS and ICURC against uniform sampling and its baseline algorithms, on both synthetic and real-world datasets.