Wirtinger演算在再生核Hilbert空间和复核LMS上的推广
Extension of Wirtinger's Calculus to Reproducing Kernel Hilbert Spaces and the Complex Kernel LMS
摘要：
在过去的十年中，非线性处理的核方法已成功地用于机器学习社区。在这些方法中采用的主要数学工具是再生核希尔伯特空间的概念。然而，到目前为止，重点一直放在批处理技术上。直到最近，在线技术才被认为是自适应信号处理任务的背景下。此外，这些努力只集中在实值数据序列。据我们所知，没有自适应的基于内核的战略已经开发，到目前为止，复值信号。此外，虽然真实的再生内核被用于越来越多的机器学习问题，复杂的内核还没有被使用，尽管他们在处理复杂信号的应用程序中的潜在利益，与通信是一个典型的例子。在本文中，我们提出了一个通用的框架来攻击复杂信号的自适应滤波的问题，使用真实的再生内核，利用一种技术称为\textit{complexification}的真实RKHS，或复杂的再生内核，突出使用复杂的高斯内核。为了得到梯度的运营商，需要定义相关的复杂RKHS，我们采用了强大的工具Wirtinger的微积分，这最近引起了人们的注意，在信号处理界。为此，在本文中，Wirtinger的演算的概念进行了扩展，第一次，包括复杂的RKHS和使用它来获得几个实现的复杂的内核最小均方（CKLMS）算法。实验证明，CKLMS提供了显着的性能改善几个线性和非线性算法，当处理非线性。
英文摘要：
Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. The primary mathematical tool employed in these methods is the notion of the Reproducing Kernel Hilbert Space. However, so far, the emphasis has been on batch techniques. It is only recently, that online techniques have been considered in the context of adaptive signal processing tasks. Moreover, these efforts have only been focussed on real valued data sequences. To the best of our knowledge, no adaptive kernel-based strategy has been developed, so far, for complex valued signals. Furthermore, although the real reproducing kernels are used in an increasing number of machine learning problems, complex kernels have not, yet, been used, in spite of their potential interest in applications that deal with complex signals, with Communications being a typical example. In this paper, we present a general framework to attack the problem of adaptive filtering of complex signals, using either real reproducing kernels, taking advantage of a technique called \textit{complexification} of real RKHSs, or complex reproducing kernels, highlighting the use of the complex gaussian kernel. In order to derive gradients of operators that need to be defined on the associated complex RKHSs, we employ the powerful tool of Wirtinger's Calculus, which has recently attracted attention in the signal processing community. To this end, in this paper, the notion of Wirtinger's calculus is extended, for the first time, to include complex RKHSs and use it to derive several realizations of the Complex Kernel Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS offers significant performance improvements over several linear and nonlinear algorithms, when dealing with nonlinearities.
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