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. 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 NPhard 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. Index Terms — Data analytics, dimensionality reduction, erroneous data, faulty measurements, L1-norm, machine learning, principal-component analysis, outlier resistance. ∗Corresponding author. Some preliminary studies related to this paper were presented at the IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, Sweden, in June 2015 [1]. This work was supported in part by the National Science Foundation under Grant ECCS-1462341 and the Office of the Vice President for Research of the Rochester Institute of Technology.