L1-norm Principal-Component Analysis of Complex Data

TL;DR

This paper explores the theoretical foundations and algorithms for L1-norm Principal-Component Analysis of complex data, revealing its NP-hardness and proposing suboptimal solutions that demonstrate robustness against outliers.

Contribution

It establishes the NP-hardness of complex L1-PCA and introduces the first suboptimal algorithms for its solution, advancing the understanding of complex data analysis.

Findings

Complex L1-PCA is NP-hard.

Proposed suboptimal algorithms for complex L1-PCA.

Robustness of complex L1-PCA against outliers.

Abstract

L1-norm Principal-Component Analysis (L1-PCA) of real-valued data has attracted significant research interest over the past decade. However, L1-PCA of complex-valued data remains to date unexplored despite the many possible applications (e.g., in communication systems). In this work, we establish theoretical and algorithmic foundations of L1-PCA of complex-valued data matrices. Specifically, we first show that, in contrast to the real-valued case for which an optimal polynomial-cost algorithm was recently reported by Markopoulos et al., complex L1-PCA is formally NP-hard in the number of data points. Then, casting complex L1-PCA as a unimodular optimization problem, we present the first two suboptimal algorithms in the literature for its solution. Our experimental studies illustrate the sturdy resistance of complex L1-PCA against faulty measurements/outliers in the processed data.