L1-norm principal-component analysis in L2-norm-reduced-rank data subspaces

Panos P. Markopoulos; Dimitris A. Pados; George N. Karystinos; Michael Langberg

doi:10.1117/12.2263733

5 May 2017 L1-norm principal-component analysis in L2-norm-reduced-rank data subspaces

Panos P. Markopoulos, Dimitris A. Pados, George N. Karystinos, Michael Langberg

Proceedings Volume 10211, Compressive Sensing VI: From Diverse Modalities to Big Data Analytics; 1021104 (2017) https://doi.org/10.1117/12.2263733
Event: SPIE Commercial + Scientific Sensing and Imaging, 2017, Anaheim, CA, United States

Abstract

Standard Principal-Component Analysis (PCA) is known to be very sensitive to outliers among the processed data.¹ On the other hand, it has been recently shown that 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.^{2, 3} Exact calculation of the K L1-norm Principal Components (L1-PCs) of a rank-r data matrix X∈ R^D×N costs O(2^NK), in the general case, and O(N^(r-1)K+1) when r is fixed with respect to N.^{2, 3} In this work, we examine approximating the K L1-PCs of X by the K L1-PCs of its L2-norm-based rank-d approximation (K≤d≤r), calculable exactly with reduced complexity O(N^(d-1)K+1). Reduced-rank L1-PCA aims at leveraging both the low computational cost of standard PCA and the outlier-resistance of L1-PCA. Our novel approximation guarantees and experiments on dimensionality reduction show that, for appropriately chosen d, reduced-rank L1-PCA performs almost identical to L1-PCA.

Conference Presentation

Citation Download Citation

Panos P. Markopoulos, Dimitris A. Pados, George N. Karystinos, and Michael Langberg "L1-norm principal-component analysis in L2-norm-reduced-rank data subspaces", Proc. SPIE 10211, Compressive Sensing VI: From Diverse Modalities to Big Data Analytics, 1021104 (5 May 2017); https://doi.org/10.1117/12.2263733

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