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Algorithms 2014, 7(3), 429-443; doi:10.3390/a7030429

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

1
School of Management, Chinese Academy of Sciences, Beijing 100190, China
2
Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA
3
Mathematical Sciences Division and Computing Sciences Division, Army Research Office, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC 27709-2211, USA
*
Author to whom correspondence should be addressed.
Received: 21 May 2014 / Revised: 21 July 2014 / Accepted: 23 July 2014 / Published: 19 August 2014
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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 multidimensional heavy-tailed distribution; 1-norm; major component; n-dimensional; outlier; pattern recognition; robust principal component analysis
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Deng, Z.; Lavery, J.E.; Fang, S.-C.; Luo, J. ℓ1 Major Component Detection and Analysis (ℓ1 MCDA) in Three and Higher Dimensional Spaces. Algorithms 2014, 7, 429-443.

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