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

• Main Authors: Zhibin Deng, John E. Lavery, Shu-Cherng Fang, Jian Luo
• Format: Article
• Language: English
• Published: MDPI AG 2014-08-01
• Series: Algorithms
• Subjects: multidimensional heavy-tailed distribution; ℓ1-norm; major component; n-dimensional; outlier; pattern recognition; robust principal component analysis
• Online Access: http://www.mdpi.com/1999-4893/7/3/429

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.