  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.
