Reduced-Rank L1-Norm Principal-Component Analysis with Performance Guarantees

Standard Principal-Component Analysis (PCA) is known to be sensitive to outliers among the processed data. On the other hand, 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 [1]. Exact calculation of the K L1-norm Principal Components (L1-PCs) of a rank-r data matrix mathbf X in mathbb {R}-{D times N} costs mathcal {O}(N-{(r -1)K + 1}) [1], [2]. In this work, we present reduced-rank L1-PCA (RR L1-PCA): a hybrid approach that approximates the K L1-PCs of mathbf X by the L1-PCs of its L2-norm-based rank-d approximation (d leq r), calculable exactly with reduced complexity mathcal {O}(N-{(d -1)K + 1}). The proposed method combines the denoising capabilities and low computation cost of standard PCA with the outlier-resistance of L1-PCA. RR L1-PCA is accompanied by formal performance guarantees as well as thorough numerical studies that corroborate its computational and corruption resistance merits.