Graph Filter Design for Distributed Network Processing: A Comparison between Adaptive Algorithms

Abstract

Graph filters (GFs) have attracted great interest since they can be directly implemented in a diffused way. Thus it is interesting to investigate using GFs to implement signal processing operations in a distributed manner. However, in most GF models, the input signals are assumed to be time-invariant, static, or change at a very low rate. In addition to that, the GF coefficients are usually set to be node-invariant, i.e. the same for all the nodes. Yet, in general, the input signals may evolve with time and the underlying GF may have parameters dependant on the nodes. Therefore, in this paper, we consider dynamic input signals with two sets of GF coefficients, nodevariant, i.e. vary on different nodes, and node-invariant. Then, we apply LMS and RLS algorithms for GF design, along with two others called adapt-then-combine (ATC) and combined RLS (CRLS) to estimate the GF coefficients. We study and compare the performance of the algorithms and show that in the case of node-invariant GF coefficients, CRLS gives the best performance with lowest mean-square-displacement (MSD), whereas, for nodevariant case, RLS represents the best results. The effect of bias in the input signal has also been examined.

Graph Filters for Signal Processing and Machine Learning on Graphs

“…To alleviate this problem, [71] presents a modified adaptation step (25a) based on Newton's method where Hessian information is considered but at an increased (per iteration) computational cost. Finally, the work in [72] considers recursive least squares (RLS)-based adaptive estimators, observing a better performance compared to LMS.…”

“…A direct comparison of Propositions 1 and 4 reveals the added expressivity of node varying graph filters since the stringent requirement of simultaneous diagonalization in Proposition 1 is replaced by a milder condition here. Furthermore, in terms of data-driven design, modifications to the adaptive methodologies presented in Section V-B have also been extended to node varying filters in [71], [72].…”

“…Reference [72] compares different adaptive solutions for this task and shows that versions of the recursive least squares algorithm converge faster. The work in [268] extends these techniques to nonlinear filters, where kernels are used to model nonliear relations between the nodes while still ensuring the objective function is linear in the parameters.…”