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

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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∈ RD×N costs O(2NK), 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.

Original languageEnglish
Title of host publicationCompressive Sensing VI
Subtitle of host publicationFrom Diverse Modalities to Big Data Analytics
EditorsFauzia Ahmad
ISBN (Electronic)9781510609235
StatePublished - 2017
Externally publishedYes
EventCompressive Sensing VI: From Diverse Modalities to Big Data Analytics 2017 - Anaheim, United States
Duration: 12 Apr 201713 Apr 2017

Publication series

NameProceedings of SPIE - The International Society for Optical Engineering
ISSN (Print)0277-786X
ISSN (Electronic)1996-756X


ConferenceCompressive Sensing VI: From Diverse Modalities to Big Data Analytics 2017
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© 2017 SPIE.


  • Dimensionality reduction
  • L1-norm
  • eigen-decomposition
  • faulty measurements
  • outlier resistance
  • subspace signal processing


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