# Some Options for L1-Subspace Signal Processing

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 focus on the computation of the L_1 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^D). We generalize to multiple L_1-max-projection components and present an explicit optimal L_1 subspace calculation algorithm in the form of matrix nuclear-norm evaluations. We conclude with illustrations of L_1-subspace signal processing in the fields of data dimensionality reduction and direction-of-arrival estimation.

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