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The low-rank matrix factorization as a L1 norm minimization problem hasrecently attracted much attention due to its intrinsic robustness to thepresence of outliers and missing data. In this paper, we propose a new method,called the divide-and-conquer method, for solving this problem. The main ideais to break the original problem into a series of smallest possiblesub-problems, each involving only unique scalar parameter. Each of thesesubproblems is proved to be convex and has closed-form solution. By recursivelyoptimizing these small problems in an analytical way, efficient algorithm,entirely avoiding the time-consuming numerical optimization as an inner loop,for solving the original problem can naturally be constructed. Thecomputational complexity of the proposed algorithm is approximately linear inboth data size and dimensionality, making it possible to handle large-scale L1norm matrix factorization problems. The algorithm is also theoretically provedto be convergent. Based on a series of experiment results, it is substantiatedthat our method always achieves better results than the currentstate-of-the-art methods on $L1$ matrix factorization calculation in bothcomputational time and accuracy, especially on large-scale applications such asface recognition and structure from motion.

Divide-and-Conquer Method for L1 Norm Matrix Factorization in the Presence of Outliers and Missing Data