ℓ1 Major Component Detection and Analysis (ℓ1 MCDA) in Three and Higher Dimensional Spaces

Abstract

Based on the recent development of two dimensional ℓ1 major component detection and analysis (ℓ1 MCDA), we develop a scalable ℓ1 MCDA in the n-dimensional space to identify the major directions of star-shaped heavy-tailed statistical distributions with irregularly positioned “spokes” and “clutters”. In order to achieve robustness and efficiency, the proposed ℓ1 MCDA in n-dimensional space adopts a two-level median fit process in a local neighbor of a given direction in each iteration. Computational results indicate that in terms of accuracy ℓ1 MCDA is competitive with two well-known PCAs when there is only one major direction in the data, and ℓ1 MCDA can further determine multiple major directions of the n-dimensional data from superimposed Gaussians or heavy-tailed distributions without and with patterned artificial outliers. With the ability to recover complex spoke structures with heavy-tailed noise and clutter in the data, ℓ1 MCDA has potential to generate better semantics than other methods.

Keywords

• multidimensional heavy-tailed distribution
• ℓ1-norm
• major component
• n-dimensional
• outlier
• pattern recognition
• robust principal component analysis