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首页> 外文期刊>Neural Networks and Learning Systems, IEEE Transactions on >Adaptive Learning in Complex Reproducing Kernel Hilbert Spaces Employing Wirtinger's Subgradients
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Adaptive Learning in Complex Reproducing Kernel Hilbert Spaces Employing Wirtinger's Subgradients

机译：使用维特林格次梯度的复杂再现核希尔伯特空间中的自适应学习

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摘要

This paper presents a wide framework for non-linear online supervised learning tasks in the context of complex valued signal processing. The (complex) input data are mapped into a complex reproducing kernel Hilbert space (RKHS), where the learning phase is taking place. Both pure complex kernels and real kernels (via the complexification trick) can be employed. Moreover, any convex, continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response. The only requirement is the subgradient of the adopted loss function to be available in an analytic form. In order to derive analytically the subgradients, the principles of the (recently developed) Wirtinger's calculus in complex RKHS are exploited. Furthermore, both linear and widely linear (in RKHS) estimation filters are considered. To cope with the problem of increasing memory requirements, which is present in almost all online schemes in RKHS, the sparsification scheme, based on projection onto closed balls, has been adopted. We demonstrate the effectiveness of the proposed framework in a non-linear channel identification task, a non-linear channel equalization problem and a quadrature phase shift keying equalization scheme, using both circular and non circular synthetic signal sources.
机译：本文为复杂值信号处理环境中的非线性在线监督学习任务提供了一个广泛的框架。 （复杂的）输入数据被映射到一个复杂的再现内核希尔伯特空间（RKHS）中，在此发生学习阶段。可以使用纯复杂内核和实际内核（通过复杂化技巧）。此外，任何凸的，连续的和不一定可微的函数都可以用来测量特定系统的输出与所需响应之间的损耗。唯一的要求是所采用的损失函数的次梯度以解析形式可用。为了分析地得出次梯度，利用了（最近开发的）复杂RKHS中的Wirtinger演算的原理。此外，考虑了线性和广义线性（在RKHS中）估计滤波器。为了解决在RKHS中几乎所有在线方案中都存在的内存需求增加的问题，已采用了基于投影到封闭球上的稀疏方案。我们使用循环和非循环合成信号源，证明了所提出框架在非线性信道识别任务，非线性信道均衡问题和正交相移键控均衡方案中的有效性。

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