ℓ¹ Major Component Detection and Analysis (ℓ¹ MCDA): Foundations in Two Dimensions

Abstract: Principal Component Analysis (PCA) is widely used for identifying the major components of statistically distributed point clouds. Robust versions of PCA, often based in part on the ℓ¹ norm (rather than the ℓ2 norm), are increasingly used, especially for point clouds with many outliers. Neither standard PCA nor robust PCAs can provide, without additional assumptions, reliable information for outlier-rich point clouds and for distributions with several main directions (spokes). We carry out a fundamental and complete reformulation of the PCA approach in a framework based exclusively on the ℓ¹ norm and heavy-tailed distributions. The ℓ¹ Major Component Detection and Analysis (ℓ¹ MCDA) that we propose can determine the main directions and the radial extent of 2D data from single or multiple superimposed Gaussian or heavy-tailed distributions without and with patterned artificial outliers (clutter). In nearly all cases in the computational results, 2D ℓ¹ MCDA has accuracy superior to that of standard PCA and of two robust PCAs, namely, the projection-pursuit method of Croux and Ruiz-Gazen and the ℓ¹ factorization method of Ke and Kanade. (Standard PCA is, of course, superior to ℓ¹ MCDA for Gaussian-distributed point clouds.) The computing time of ℓ¹ MCDA is competitive with the computing times of the two robust PCAs.