Abstract
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.
Highlights
G RAPH signal processing (GSP) has recently received increased attention due to its wide applicability to model, process, and analyze signals and large data sets, ranging from daily-life social networks to sensor networks for industrial and Manuscript received July 3, 2020; revised October 21, 2020 and December 11, 2020; accepted December 12, 2020
This paper introduced nonlinear adaptive graph filters for model estimation in the reproducing kernel Hilbert space
To overcome the growing dimension problem encountered in the centralized graph kernel least mean squares (GKLMS) algorithm, coherence check based dictionary-sparsification and random Fourier features (RFF) were proposed
Summary
G RAPH signal processing (GSP) has recently received increased attention due to its wide applicability to model, process, and analyze signals and large data sets, ranging from daily-life social networks to sensor networks for industrial and Manuscript received July 3, 2020; revised October 21, 2020 and December 11, 2020; accepted December 12, 2020. Some of the prior works account for the input signals’ network-related characteristics, such as smoothness across the graph, existing RKHS-based approaches do not consider graph-shifted signals. This paper introduces nonlinear graph filters and presents two adaptive methods for function estimation over graphs, namely. ELIAS et al.: ADAPTIVE GRAPH FILTERS IN REPRODUCING KERNEL HILBERT SPACES: DESIGN AND PERFORMANCE ANALYSIS the centralized graph kernel least mean squares (GKLMS) and the graph diffusion kernel least mean squares (GDKLMS). Preliminary results on this topic have been presented in [58].
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