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 is an element of R-DxN (N data samples of D dimensions) can be exactly calculated with cost O(2(NK)) or, when advantageous, O(NdK-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(NDmin{N, D} + (NK2)-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.