Complex Gaussian Processes for Regression and Their Connection to WLMMSE

The Gaussian process (GP) is a well-established Bayesian nonparametric tool for inference in nonlinear estimation problems. When GPs are used for regression, the goal is to estimate a target signal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${y}$</tex-math></inline-formula> from an input vector <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {x}$</tex-math></inline-formula> without assuming that they are linearly related, but with a probabilistic model <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p({y}|\mathbf {x})$</tex-math></inline-formula> that is Gaussian distributed. Therefore, GPs can be understood as a natural nonlinear extension to MMSE estimation. For real-valued GPs, this has been analyzed in the existing literature, and it is concluded that they are the natural nonlinear Bayesian extension to the linear minimum mean-squared error (LMMSE) estimation. In this letter, we show that, consequently, complex-valued GP regression (GPR) models are the natural nonlinear Bayesian extension of the widely linear minimum mean squared-error (WLMMSE) estimation. As in the real-valued case, complex-valued GPs are able to better model many regression problems by making use of the information that the complementary kernel or pseudo-kernel provides.