Complex correntropy function: Properties, and application to a channel equalization problem

Abstract

The use of correntropy as a similarity measure has been increasing in different scenarios due to the well-known ability to extract high-order statistic information from data. Recently, a new similarity measure between complex random variables was defined and called complex correntropy. Based on a Gaussian kernel, it extends the benefits of correntropy to complex-valued data. However, its properties have not yet been formalized. This paper studies the properties of this new similarity measure and extends this definition to positive-definite kernels. Complex correntropy is applied to a channel equalization problem as good results are achieved when compared with other algorithms such as the complex least mean square (CLMS), complex recursive least squares (CRLS), and least absolute deviation (LAD).

Performance evaluation of the maximum complex correntropy criterion with adaptive kernel width update

“…E[ • ] is the expected value operator. The work presented in [8] demonstrated that the complex correntropy generalizes the regular correntropy concept to complex-valued data while keeping important properties such as symmetry, bounded, high-order statistical measure, and a probabilistic meaning, specially when the complex Gaussian kernel, defined in (2) is used.…”

“…Figure 1 summarizes a system identification problem in which the MCCC was successful applied in [7]. The MCCC has been also applied to a channel equalization problem [8], but always employing a fixed-point solution algorithm. This approach depends on a matrix inversion, and this operation is sometimes unavailable or the increase in the computational cost is not ideal [13].…”

“…4 depends on complex-valued parameter (d, y), although it is always a real-valued function when the complex Gaussian kernel from Eq. 2is applied [15]. This violates the Cauchy-Riemann conditions, thus making the gradient function not analytical in the complex domain [16].…”

“…Thus, the kernel size, also called kernel width, is a free parameter that is inherent to the kernel used to estimate the complex correntropy. It works as a scale parameter that controls the steady-state performance, convergence rate, and impulsive noise rejection [15]. Since it is a free parameter, the kernel width must be chosen by the user, whose value changes according to data and application nature.…”

“…This is why it has been widely used as a cost function in optimization problems such as adaptive filtering in an approach called maximum correntropy criterion (MCC), thus providing better performance than second-order methods in non-Gaussian noise environments [2][3][4][5][6]. Recently, the correntropy concept has been extended to complex-valued random variables using the maximum complex correntropy criterion (MCCC) [7,8]. Both MCC and MCCC employ a free parameter called kernel width or kernel size.…”