L1-norm principal-component analysis in L2-norm-reduced-rank data subspaces

Standard Principal-Component Analysis (PCA) is known to be very sensitive to outliers among the processed data.¹ On the other hand, it has been recently shown that L1-norm-based PCA (L1-PCA) exhibits sturdy resistance against outliers, while it performs similar to standard PCA when applied to nominal or smoothly corrupted data.^{2, 3} Exact calculation of the K L1-norm Principal Components (L1-PCs) of a rank-r data matrix X∈ R^D×N costs O(2^NK), in the general case, and O(N^(r-1)K+1) when r is fixed with respect to N.^{2, 3} In this work, we examine approximating the K L1-PCs of X by the K L1-PCs of its L2-norm-based rank-d approximation (K≤d≤r), calculable exactly with reduced complexity O(N^(d-1)K+1). Reduced-rank L1-PCA aims at leveraging both the low computational cost of standard PCA and the outlier-resistance of L1-PCA. Our novel approximation guarantees and experiments on dimensionality reduction show that, for appropriately chosen d, reduced-rank L1-PCA performs almost identical to L1-PCA.