Abstract: Aiming at the problems that the traditional low-rank matrix factorization methods commonly suffered from low computational efficiency, excessive memory consumption, and insufficient decomposition accuracy when handling large-scale data, based on the core theory of CUR decomposition, combined with random sampling and optimization strategies, we proposed two new random low-rank matrix factorization methods: column random factorization and column-row random factorization. By optimizing the algorithmic randomness design, we introduced dedicated iterative optimization strategies, effectively improving the operational deficiencies of traditional algorithms. The comparative experimental results show that, compared with traditional decomposition methods, the two proposed methods have superior computational efficiency, faster convergence speed, and significantly enhanced noise robustness. They effectively refine the technical framework of low-rank decomposition for large-scale data, provide efficient and stable new solution for massive data processing, as well as reliable technical support for engineering applications in related fields such as machine learning and data mining.

Key words: low-rank matrix factorization, random low-rank factorization, CUR decomposition