Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling
This work addresses a domain-specific problem in matrix completion for applications requiring flexible sampling, but it appears incremental as it builds on existing sampling models.
The paper tackles the problem of matrix completion by proposing a new sampling strategy called Cross-Concentrated Sampling (CCS) that bridges uniform and CUR sampling, offering flexibility to potentially reduce sampling costs, and introduces an efficient non-convex algorithm, ICURC, which shows empirical advantages over baselines on synthetic and real-world datasets.
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.