Some Options for L1-subspace Signal Processing

We describe ways to define and calculate L1-norm signal subspaces which are less sensitive to outlying data than L2-calculated subspaces. We focus on the computation of the L1 maximum-projection principal component of a data matrix containing N signal samples of dimension D and conclude that the general problem is formally NP-hard in asymptotically large N, D. We prove, however, that the case of engineering interest of fixed dimension D and asymptotically large sample support N is not and we present an optimal algorithm of complexity O(N(exp D)). We generalize to multiple L1-max-projection components and present an explicit optimal L1 subspace calculation algorithm in the form of matrix nuclear-norm evaluations. We conclude with illustrations of L1-subspace signal processing in the fields of data dimensionality reduction and direction-of-arrival estimation.