Complex-Valued Kernel Methods for Regression

TL;DR

This paper introduces the widely RKHS (WRKHS), a new complex-valued kernel method that incorporates a pseudo-kernel to improve regression performance, especially when real and imaginary parts are correlated or have different relations.

Contribution

It develops the WRKHS framework, including the design of complex-valued kernels and pseudo-kernels, addressing limitations of previous complex RKHS approaches.

Findings

Remarkable improvements in regression scenarios with correlated real and imaginary parts.

Enhanced non-linear channel equalization using WRKHS.

The pseudo-kernel enables learning of more complex functions in complex-valued fields.

Abstract

Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudo kernel, is needed to learn any function in complex-valued fields. Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS). When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of previous approaches. We address the kernel and pseudo-kernel design, paying attention to the kernel and the pseudo-kernel being complex-valued. In the experiments included we report remarkable improvements in simple scenarios where real a imaginary parts have different similitude relations for given inputs or cases where real and imaginary parts are correlated. In the context of these novel results we revisit the problem of non-linear channel equalization, to show that the WRKHS helps to design more efficient solutions.