Autoregressive Moving Average Graph Filtering (1602.04436v2)

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.

• The paper introduces Autoregressive Moving Average (ARMA) graph filters, generalizing distributed filtering for static and time-varying graph signals and topologies.
• ARMA graph filters can approximate any desired graph frequency response independently of topology, offering universality for time-varying settings.
• The method has a formal framework, stability conditions, and is validated by simulations showing robustness over FIR methods in dynamic scenarios.

Overview of Autoregressive Moving Average Graph Filtering

The paper "Autoregressive Moving Average Graph Filtering" by Elvin Isufi, Andreas Loukas, Andrea Simonetto, and Geert Leus presents a significant advancement in the area of graph signal processing. It introduces Autoregressive Moving Average (ARMA) graph filters as a generalized approach for distributed graph filtering that can be effectively applied to both static and time-varying graph signals and topologies.

ARMA Graph Filters and Their Design

Graph signal processing extends classical signal processing techniques to signals defined over graph structures. This is crucial for analyzing data with complex relationships encapsulated in high-dimensional graphs. Graph filters, analogous to classical filters, operate on the graph Fourier coefficients, offering a way to manipulate these graph signals for tasks such as denoising, interpolation, and classification. The paper highlights the limitations of previous graph filtering methods which relied heavily on FIR graph filters, often failing under time-varying conditions due to their limited adaptability.

The presented ARMA graph filters are developed to overcome these limitations. The authors propose two types of ARMA implementations: parallel and periodic. Importantly, these filters can approximate any desired graph frequency response independently of the underlying graph topology. This universality empowers their use in time-varying settings, a departure from previous designs that were sensitive to changes in graph topology or signal variations.

Formal Mathematical Framework

The paper delineates the mathematical underpinnings necessary for ARMA filter design, providing sufficient conditions for filter stability and offering a characterization of their performance across varying graph realizations. The ARMA filters possess a dual-operational capacity, functioning across both the graph frequency domain and the temporal frequency domain for time-variant signals. This dual-domain operation is critical for effectively filtering signals that evolve over time and varied graph structures.

The design methodology employs a variant of Shanks' method to ensure the ARMA filters' stability while preserving desired selectivity. Analytical results enable the authors to offer explicit constructions for ARMA filters that execute specific graph signal processing functions like Tikhonov and Wiener-based denoising and interpolation with precise outcomes.

Numerical and Simulation Results

The theoretical propositions are substantiated through numerical simulations that demonstrate the ARMA filters' robustness and practicality in dynamic scenarios. In particular, the simulations showcase their effectiveness over conventional FIR methods in managing both static and time-varying inputs, further validating the universal applicability and precise performance of the designed filters.

Implications and Future Prospects

The implications of ARMA graph filters extend to a variety of applications where data is inherently structured over graphs, including sensor networks, social networking data analysis, and biological network studies. The flexibility and universality of these filters suggest potential further enhancement and scalability in increasingly dynamic data environments.

Looking ahead, the research lays groundwork for future inquiries into more complex multi-dimensional ARMA filters and their possible disjoint design between graph and temporal domains. Analytical design tools to secure stability and performance in even broader settings would enhance the utility of such filtering techniques.

In conclusion, this paper contributes a robust framework for distributed graph signal processing, expanding capabilities notably in scenarios wherein graphs and their signals are subject to dynamic changes. This work paves the way for more advanced signal processing techniques on graphs, aligning with the growing complexity and dynamism of real-world datasets.