This paper develops adaptive graph filters that operate in reproducing kernel Hilbert spaces. We consider both centralized and fully distributed implementations. We first define nonlinear graph filters that operate on graph-shifted versions of the input signal. We then propose a centralized graph kernel least mean squares (GKLMS) algorithm to identify nonlinear graph filters' model parameters. To reduce the dictionary size of the centralized GKLMS, we apply the principles of coherence check and random Fourier features (RFF). The resulting algorithms have performance close to that of the GKLMS algorithm. Additionally, we leverage the graph structure to derive the distributed graph diffusion KLMS (GDKLMS) algorithms. We show that, unlike the coherence check-based approach, the GDKLMS based on RFF avoids the use of a pre-trained dictionary through its data-independent fixed structure. We conduct a detailed performance study of the proposed RFF-based GDKLMS, and the conditions for its convergence both in mean and mean-squared senses are derived. Extensive numerical simulations show that GKLMS and GDKLMS can successfully identify nonlinear graph filters and adapt to model changes. Furthermore, RFF-based strategies show faster convergence for model identification and exhibit better tracking performance in model-changing scenarios.

中文翻译：

---

再现核希尔伯特空间中的自适应图滤波器：设计和性能分析


本文开发了在再现内核希尔伯特空间中运行的自适应图滤波器。我们考虑集中式和完全分布式的实现。我们首先定义对输入信号的图移位版本进行操作的非线性图滤波器。然后，我们提出了一种集中式图核最小均方（GKLMS）算法来识别非线性图滤波器的模型参数。为了减少集中式 GKLMS 的字典大小，我们应用了一致性检查和随机傅里叶特征（RFF）的原理。所得算法的性能接近 GKLMS 算法。此外，我们利用图结构来推导分布式图扩散 KLMS（GDKLMS）算法。我们表明，与基于一致性检查的方法不同，基于 RFF 的 GDKLMS 通过其与数据无关的固定结构避免了使用预训练字典。我们对所提出的基于 RFF 的 GDKLMS 进行了详细的性能研究，并导出了其在均值和均方意义上收敛的条件。大量的数值模拟表明，GKLMS 和 GDKLMS 可以成功识别非线性图滤波器并适应模型变化。此外，基于 RFF 的策略在模型识别方面表现出更快的收敛性，并在模型更改场景中表现出更好的跟踪性能。