L1-Norm Principal-Component Analysis of Complex Data

L1-norm Principal-Component Analysis (L1-PCA) of real-valued data has attracted significant research interest over the past decade. L1-PCA of complex-valued data remains to date unexplored despite the many possible applications (in communication systems, for example). In this paper, 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, Karystinos, and Pados, complex L1-PCA is formally NP-hard. Then, casting complex L1-PCA as a unimodular optimization problem, we present the first two suboptimal algorithms in the literature for its solution. Extensive experimental studies included in this paper illustrate the sturdy resistance of complex L1-PCA against faulty measurements/outliers in the processed data.