We introduce a two-step method for the matrix recovery problem. Our approach combines the theoretical foundations of the Column Subset Selection and Low-rank Matrix Completion problems. The proposed method, in each step, solves a convex optimization task. We present two algorithms that implement our Columns Selected Matrix Completion (CSMC) method, each dedicated to a different size problem. We performed a formal analysis of the presented method, in which we formulated the necessary assumptions and the probability of finding a correct solution. In the second part of the paper, we present the results of the experimental work. Numerical experiments verified the correctness and performance of the algorithms. To study the influence of the matrix size, rank, and the proportion of missing elements on the quality of the solution and the computation time, we performed experiments on synthetic data. The presented method was applied to two real-life problems problems: prediction of movie rates in a recommendation system and image inpainting. Our thorough analysis shows that CSMC provides solutions of comparable quality to matrix completion algorithms, which are based on convex optimization. However, CSMC offers notable savings in terms of runtime.

翻译：我们提出了一种用于矩阵恢复问题的两步法。该方法结合了列子集选择与低秩矩阵补全问题的理论基础，在每一步中求解一个凸优化任务。我们提出了两种实现列子集选择矩阵补全（CSMC）方法的算法，分别适用于不同规模的问题。我们对所提方法进行了形式化分析，制定了必要的假设条件，并推导了找到正确解的概率。在论文的第二部分，我们展示了实验工作结果。数值实验验证了算法的正确性与性能。为研究矩阵规模、秩以及缺失元素比例对解质量和计算时间的影响，我们在合成数据上进行了实验。该方法还应用于两个实际问题：推荐系统中的电影评分预测和图像修复。我们的全面分析表明，CSMC能够提供与基于凸优化的矩阵补全算法质量相当的解，但在运行时间上具有显著优势。