Complex-Valued Kernel Methods for Regression
This work addresses regression problems in complex-valued fields, such as non-linear channel equalization, by providing a more efficient solution, though it appears incremental as it builds on existing RKHS frameworks.
The paper tackles the limitation of existing complex-valued RKHS for regression by introducing a pseudo-kernel and a novel widely RKHS (WRKHS), which shows remarkable improvements in scenarios with different similitude relations or correlations between real and imaginary parts.
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.