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

TL;DR

This paper introduces a divide-and-conquer algorithm for L1 norm matrix factorization that efficiently handles outliers and missing data, outperforming existing methods in speed and accuracy, especially on large-scale problems.

Contribution

The paper presents a novel divide-and-conquer approach that breaks the L1 matrix factorization problem into convex subproblems with closed-form solutions, enabling efficient large-scale computation.

Findings

The proposed algorithm has approximately linear complexity in data size and dimensionality.

It is theoretically proven to converge.

Experimental results show it outperforms state-of-the-art methods in speed and accuracy.

Abstract

The low-rank matrix factorization as a L1 norm minimization problem has recently attracted much attention due to its intrinsic robustness to the presence 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 idea is to break the original problem into a series of smallest possible sub-problems, each involving only unique scalar parameter. Each of these subproblems is proved to be convex and has closed-form solution. By recursively optimizing 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. The computational complexity of the proposed algorithm is approximately linear in both data size and dimensionality, making it possible to handle large-scale L1 norm matrix factorization problems. The algorithm is also theoretically proved to be convergent. Based on a series of experiment results, it is substantiated that our method always achieves better results than the current state-of-the-art methods on $L1$ matrix factorization calculation in both computational time and accuracy, especially on large-scale applications such as face recognition and structure from motion.