Optimal Algorithms for $L_1$-subspace Signal Processing

Abstract

We describe ways to define and calculate L 1 -norm signal subspaces which are less sensitive to outlying data than L 2 -calculated subspaces. We start with the computation of the L 1 maximum-projection principal component of a data matrix containing N signal samples of dimension D. We show that while the general problem is formally NP-hard in asymptotically large N , D, the case of engineering interest of fixed dimension D and asymptotically large sample size N is not. In particular, for the case where the sample size is less than the fixed dimension (N < D), we present in explicit form an optimal algorithm of computational cost 2 N . For the case N ≥ D, we present an optimal algorithm of complexity O(N D ). We generalize to multiple L 1 -max-projection components and present an explicit optimal L 1 subspace calculation algorithm of complexity O(N DK−K+1 ) where K is the desired number of L 1 principal components (subspace rank). We conclude with illustrations of L 1 -subspace signal processing in the fields of data dimensionality reduction, direction-of-arrival estimation, and image conditioning/restoration.