On the L1-Norm Approximation of a Matrix by Another of Lower Rank

In the past decade, there has been a growing documented effort to approximate a matrix by another of lower rank minimizing the L1-norm of the residual matrix. In this paper, we first show that the problem is NP-hard. Then, we introduce a theorem on the sparsity of the residual matrix. The theorem sets the foundation for a novel algorithm that outperforms all existing counterparts in the L1-norm error minimization metric and exhibits high outlier resistance in comparison to usual L2-norm error minimization in machine learning applications.