Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting

We present a novel method for matrix completion, specifically designed for matrices where one dimension is significantly larger than the other. Our Columns Selected Matrix Completion (CSMC) method combines Column Subset Selection with Low-Rank Matrix Completion to efficiently reconstruct incomplete datasets. CSMC substantially reduces computational cost while preserving the solution quality of state-of-the-art convex matrix completion techniques. Each stage of CSMC involves solving a convex optimization problem. We introduce two algorithmic implementations of CSMC, each tailored to problems of different scales. A formal analysis is provided, outlining the necessary assumptions and the probability of exact recovery. To evaluate the impact of matrix size, rank, and the ratio of missing entries on solution quality and runtime, we conducted experiments on synthetic data. The method was also applied to two real-world tasks: recommendation systems and image inpainting. Our results show that CSMC achieves substantial runtime improvements while maintaining competitive accuracy compared to leading convex optimization-based methods.