Complex-Valued Gaussian Processes for Regression

In this paper we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on results in complex-valued linear theory and Gaussian random processes we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution approach and for several scenarios. Just in the particular case where the outputs are proper, the pseudo-kernel cancels. Also, the hyperparameters of the covariance {can be learnt} maximizing the marginal likelihood using Wirtinger's calculus and patterned complex-valued matrix derivatives. In the experiments included, we show how CGPR successfully solve systems where real and imaginary parts are correlated. Besides, we successfully solve the nonlinear channel equalization problem by developing a recursive solution with basis removal. We report remarkable improvements compared to previous solutions: a 2-4 dB reduction of the MSE with {just a quarter} of the training samples used by previous approaches.