Autoregressive Moving Average Graph Filtering

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 specific graph signal denoising and interpolation problems. The philosophy, 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 topology 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 domain. We also derive the graph filter behavior, as well as 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. Keywords-- distributed graph filtering, signal processing on graphs, infinite impulse response graph filters, autoregressive moving average graph filters, time-varying graph signals, time-varying graphs. Due to their ability to capture the complex relationships present in many high-dimensional datasets, graphs have emerged as a favorite tool for data analysis.