Autoregressive Moving Average Graph Filtering

TL;DR

This paper introduces ARMA graph filters capable of approximating any frequency response, suitable for static and dynamic graph signals, with stable and efficient distributed implementations for denoising and interpolation.

Contribution

The work develops a novel ARMA-based graph filtering method that is independent of the graph structure and applicable to time-varying signals, extending classical graph filtering techniques.

Findings

ARMA graph filters can approximate any desired frequency response.

The proposed filters are effective for static and time-varying graph signals.

The filters are stable under certain conditions and suitable for distributed implementation.

Abstract

One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogues of classical filters, but intended for signals defined on graphs. This work brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which (i) are able to approximate any desired graph frequency response, and (ii) give exact solutions for tasks such as graph signal denoising and interpolation. The design philosophy, which allows us to design the ARMA coefficients independently from the underlying graph, renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph are changing over time. We show that in case of a time-varying graph signal our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domains. We also derive sufficient conditions for filter stability when the graph and signal are time-varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.