  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.
