Complex-Valued L1-Norm Principal Component Analysis for Signal Processing and Machine Learning
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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 three algorithms for its solution. One suboptimal algorithm for the for the general K ≥ 1 case and two for the special K = 1 case from which one is the fastest known suboptimal and the other is the first optimal in the literature. Our experimental studies illustrate the sturdy resistance of complex L1-PCA against faulty measurements/outliers in the processed data.