Complex Correntropy: Probabilistic Interpretation and Application to Complex-Valued Data

Abstract

Recent studies have demonstrated that correntropy is an efficient tool for analyzing higher-order statistical moments in nonGaussian noise environments.Although correntropy has been used with complex data, no theoretical study was pursued to elucidate its properties, nor how to best use it for optimization . This paper presents a probabilistic interpretation for correntropy using complex-valued data called complex correntropy. A recursive solution for the maximum complex correntropy criterion (MCCC) is introduced based on a fixedpoint solution. This technique is applied to a simple system identification case study, and the results demonstrate prominent advantages when compared to the complex recursive least squares (RLS) algorithm. By using such probabilistic interpretation, correntropy can be applied to solve several problems involving complex data in a more straightforward way.

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

“…Recently, the correntropy function was extended to the case of complex-valued data. This approach is called complex correntropy and is defined as [7]:…”

“…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.…”

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

“…One can notice that, for all tested kernel sizes and GSNR values, MCCC has the smallest standard deviation among all tested algorithms thus demonstrating its robustness. It is also worth to mention that this method has fast convergence rates since it is a fixed-point method (Guimãraes et al, 2017). According to Figure 3-a, the MCCC performance regarding the channel equalization problem is strictly related to the selection of a proper kernel size, which is a free parameter and must be wisely chosen by the user.…”

“…So, due to the limitation to deal with real-valued data, it is difficult to use correntropy in a straightforward way as applied to problems involving complex-valued data. Inspired by the probabilistic interpretation demonstrated in (Liu et al, 2006), our research group has extended the concept of classic correntropy to include processing from complex-valued data (Guimãraes et al, 2017), as the method has been defined as complex correntropy. This similarity measure is based on the probability density function (PDF) applied to multidimensional spaces using the Parzen estimator with a Gaussian Kernel.…”

“…This similarity measure is based on the probability density function (PDF) applied to multidimensional spaces using the Parzen estimator with a Gaussian Kernel. Additionally, a new cost function called Maximum Complex Correntropy Criterion (MCCC) is defined, whose performance has proven to be superior to that regarding classical algorithms for processing complex-valued data (Guimãraes et al, 2017). However, the properties of complex correntropy have yet not been properly formalized.…”