Efficient L1-Norm Principal-Component Analysis via Bit Flipping

It was shown recently that the K L1-norm principal components (L1-PCs) of a real-valued data matrix X ∈ R
^D×N
(N data samples of D dimensions) can be exactly calculated with cost O(2
^NK
) or, when advantageous, O(N
^{dK - K + 1}
) where d=rank (X), K<;d. In applications where X is large (e.g., “big” data of large N and/or “heavy” data of large d), these costs are prohibitive. In this paper, we present a novel suboptimal algorithm for the calculation of the K <; d L1-PCs of X of cost O (ND min {N, D} + N
²
K
²
(K
²
+ d)), which is comparable to that of standard L2-norm PC analysis. Our theoretical and experimental studies show that the proposed algorithm calculates the exact optimal L1-PCs with high frequency and achieves higher value in the L1-PC optimization metric than any known alternative algorithm of comparable computational cost. The superiority of the calculated L1-PCs over standard L2-PCs (singular vectors) in characterizing potentially faulty data/measurements is demonstrated with experiments in data dimensionality reduction and disease diagnosis from genomic data.