Optimal Algorithms for $L_{1}$-subspace Signal Processing

We describe ways to define and calculate L
₁
-norm signal subspaces that 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 , 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.