Minimum-Degree Distributed Graph Filter Design

Abstract

Here we consider the problem of designing finite-impulse-response (FIR) graph filter (GF) in a fully distributed way. For a directed graph with $\mathsf {N}$ nodes, each node designs filter coefficients in a distributed manner, when the knowledge of the graph structure, recognized as global information, is unavailable to each node. By modeling graph signal shifting with observations at a node as a linear dynamical system, we establish fundamental connections between local response of shifting at a node, concerned in the graph signal processing (GSP) field, and the observability of the system, investigated in control theory. The observability, as a measure of how well internal states of a dynamical system can be learned from node's observations, is reflected by the minimal polynomial of a matrix pair related to the system. Specifically, by introducing a notion of observable graph frequencies to a node, we show that the output signals (observations) at a node only contain the spectral components of its so-called observable graph frequencies. Furthermore, we unveil that the observability of a node to the spectral components of a GS is related to the rank of its observability matrix from the perspective of control theory. Our work reveals that partial signal outputs at a node are sufficient to design the FIR GF locally in terms of node-variant (NV) GFs. These findings further enable us to characterize the minimum-degree NV GF, where a minimum number of its shifted GSs are involved in the filter's output, and devise a distributed GF design algorithm for it.