$L_1$-norm regularized $L_1$-norm best-fit line problem

Abstract

This work develops a sparse and outlier-insensitive method to fit a one-dimensional subspace that can be used as a replacement for eigenvector methods such as principal component analysis (PCA). The method is insensitive to outlier observations by formulating procedures as optimization problems that seek the best-fit line according to the $\ell^1$ norm. It is also capable of producing sparse principal components by leveraging an additional penalty term induce sparsity. The algorithm has a worst-case time complexity of $O{(m^2n \log n)}$ and, under certain conditions, produces a globally optimal solution. An implementation of this algorithm in the parallel and heterogeneous environment NVIDIA CUDA is tested on synthetic and real world datasets and compared to existing methods. The results demonstrate the scalability and efficiency of the proposed approach.