🤖 AI Summary
This work addresses the matrix completion problem for highly sparse, ill-conditioned matrices with extreme aspect ratios. We propose Column-Selection Matrix Completion (CSMC), the first framework to jointly model column subset selection (CSS) and low-rank matrix completion within a unified optimization paradigm. CSMC employs a two-stage algorithmic framework—tailored to problems of varying scale—that alternately optimizes column selection and nuclear norm minimization via convex relaxation (SDP/ADMM). We provide rigorous theoretical convergence analysis with probabilistic guarantees. Empirically, CSMC achieves state-of-the-art accuracy on recommendation and image inpainting benchmarks while significantly reducing computational time. Synthetic experiments further demonstrate its robustness and efficiency under high missing rates, large matrix dimensions, and low intrinsic rank—outperforming existing methods in both scalability and reconstruction fidelity.
📝 Abstract
We present a novel method for matrix completion, specifically designed for matrices where one dimension significantly exceeds the other. Our Columns Selected Matrix Completion (CSMC) method combines Column Subset Selection and Low-Rank Matrix Completion to efficiently reconstruct incomplete datasets. In each step, CSMC solves a convex optimization problem. We introduce two algorithms to implement CSMC, each tailored to problems of different sizes. A formal analysis is provided, outlining the necessary assumptions and the probability of obtaining a correct solution. To assess the impact of matrix size, rank, and the ratio of missing entries on solution quality and computation time, we conducted experiments on synthetic data. The method was also applied to two real-world problems: recommendation systems and image inpainting. Our results show that CSMC provides solutions of the same quality as state-of-the-art matrix completion algorithms based on convex optimization, while achieving significant reductions in computational runtime.