Complex Gaussian Processes for Regression.

Area of Science:

• Machine Learning
• Statistical Modeling
• Complex Analysis

Background:

• Traditional kernel methods for nonlinear regression face limitations in complex fields.
• Existing complexification approaches often fail to capture the nuances of real and imaginary parts in data.
• A gap exists in robust Bayesian regression techniques for complex-valued, nonlinear systems.

Purpose of the Study:

• To propose a novel Bayesian solution for nonlinear regression in complex fields.
• To develop a complex-valued Gaussian process for regression (CGPR) incorporating a pseudo-kernel.
• To address limitations of existing complexification methods in kernel-based regression.

Main Methods:

• Developed a complex-valued formulation for Gaussian processes for regression (CGPR).
• Incorporated a pseudo-kernel, derived from complex-valued linear theory and Gaussian random processes.
• Utilized a convolution approach for designing covariance and pseudo-covariance functions.
• Employed Wirtinger's calculus and patterned complex-valued matrix derivatives for hyperparameter optimization.

Main Results:

• CGPR successfully models systems with correlated real and imaginary parts.
• Achieved effective nonlinear channel equalization using a recursive solution with basis removal.
• Demonstrated significant improvements over previous methods, including a 2-4 dB reduction in mean squared error.
• Required only a quarter of the training samples compared to prior approaches.

Conclusions:

• The proposed CGPR with a pseudo-kernel offers a powerful Bayesian solution for complex nonlinear regression.
• This method significantly enhances performance and efficiency in handling complex-valued data.
• CGPR provides a robust framework for applications like nonlinear channel equalization.