Widely Linear Complex-Valued Kernel Methods for Regression

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 pseudokernel, 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.

The Generalized Complex Kernel Least-Mean-Square Algorithm

“…Notice that this kernel and pseudo-kernel follow the structure introduced in [11]. The first entry ofŷ(i) in (13) yields the proposed generalized complex KLMS (gCKLMS):…”

“…Therefore, these algorithms use a real-valued kernel k(x, x ) = 2k rr (x, x ). The condition k rj (x(i), x(l)) = 0 implies that the real and the imaginary parts are not related and that one of them does not provide information to learn the other [11].…”

“…We end this discussion about the kernels by bringing here a suitable real-valued function for k rr , k jj , k rj and k jr proposed in [11]. This is the adaptation to complex-valued inputs of the real-valued Gaussian kernel:…”

“…where γ is the kernel parameter. This real function provides a measure of similarity between the complex-valued inputs that is simple but effective for complex-valued signals: inputs closer to other input in the complex field are considered more similar than inputs that are further away [11]. We will use it in our experiments.…”

“…In this paper, we propose a novel generalized formulation for the adaptive complex KLMS algorithm. In light of the findings in [11], we herein develop the generalized complex KLMS algorithm, that we call gCKLMS, which includes a kernel and a pseudo-kernel term. We show that the kernel and pseudo-kernel in the gCKLSM have the same structures found in [11], and we can use the analysis in that work to design these two functions.…”