Principal Component Analysis With Complex Kernel: The Widely Linear Model

Nonlinear complex representations, via the use of complex kernels, can be applied to model and capture the nonlinearities of complex data. Even though the theoretical tools of complex reproducing kernel Hilbert spaces (CRKHS) have been recently successfully applied to the design of digital filters and regression and classification frameworks, there is a limited research on component analysis and dimensionality reduction in CRKHS. The aim of this brief is to properly formulate the most popular component analysis methodology, i.e., Principal Component Analysis (PCA), in CRKHS. In particular, we define a general widely linear complex kernel PCA framework. Furthermore, we show how to efficiently perform widely linear PCA in small sample sized problems. Finally, we show the usefulness of the proposed framework in robust reconstruction using Euler data representation.