# Autoregressive Moving Average Graph Filtering

arXiv: https://arxiv.org/abs/1602.04436 · HTML: https://mlsanity.com/paper/1602.04436/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2016.

## Built on by (most cited first, 30 of 35)

- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Bridging the Gap between Spatial and Spectral Domains: A Unified Framework for Graph Neural Networks](https://mlsanity.com/paper/2107.10234/) (2021)
- [Beyond Low-Pass Filters: Adaptive Feature Propagation on Graphs](https://mlsanity.com/paper/2103.14187/) (2021)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Joint Detection and Localization of Stealth False Data Injection Attacks in Smart Grids using Graph Neural Networks](https://mlsanity.com/paper/2104.11846/) (2021)
- [How Expressive are Transformers in Spectral Domain for Graphs?](https://mlsanity.com/paper/2201.09332/) (2022)
- [GaitFi: Robust Device-Free Human Identification via WiFi and Vision Multimodal Learning](https://mlsanity.com/paper/2208.14326/) (2022)
- [Stability to Deformations of Manifold Filters and Manifold Neural Networks](https://mlsanity.com/paper/2106.03725/) (2021)
- [Graph-Time Convolutional Neural Networks](https://mlsanity.com/paper/2103.01730/) (2021)
- [Graph Denoising with Framelet Regularizer](https://mlsanity.com/paper/2111.03264/) (2021)
- [Fairness-Aware Graph Filter Design](https://mlsanity.com/paper/2303.11459/) (2023)
- [Graph-Aware Diffusion for Signal Generation](https://mlsanity.com/paper/2510.05036/) (2025)
- [Node-Variant Graph Filters in Graph Neural Networks](https://mlsanity.com/paper/2106.00089/) (2021)
- [MGC: A Complex-Valued Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/2110.07570/) (2021)
- [Geometric Multimodal Deep Learning with Multi-Scaled Graph Wavelet Convolutional Network](https://mlsanity.com/paper/2111.13361/) (2021)
- [Deep representation learning: Fundamentals, Perspectives, Applications, and Open Challenges](https://mlsanity.com/paper/2211.14732/) (2022)
- [BiGCN: A Bi-directional Low-Pass Filtering Graph Neural Network](https://mlsanity.com/paper/2101.05519/) (2021)
- [Fairness-aware Optimal Graph Filter Design](https://mlsanity.com/paper/2310.14432/) (2023)
- [UnSeGArmaNet: Unsupervised Image Segmentation using Graph Neural Networks with Convolutional ARMA Filters](https://mlsanity.com/paper/2410.06114/) (2024)
- [Learning Optimal Graph Filters for Clustering of Attributed Graphs](https://mlsanity.com/paper/2211.04634/) (2022)
- [Adaptive Least Mean Squares Graph Neural Networks and Online Graph Signal Estimation](https://mlsanity.com/paper/2401.15304/) (2024)
- [Spatiotemporal Forecasting Meets Efficiency: Causal Graph Process Neural Networks](https://mlsanity.com/paper/2405.18879/) (2024)
- [Infinite Impulse Response Graph Neural Networks for Cyberattack Localization in Smart Grids](https://mlsanity.com/paper/2206.12527/) (2022)
- [Graph Filter Transfer via Probability Density Ratio Weighting](https://mlsanity.com/paper/2210.14633/) (2022)
- [Adaptive Least Mean pth Power Graph Neural Networks](https://mlsanity.com/paper/2405.04111/) (2024)
- [Fast Wrong-way Cycling Detection in CCTV Videos: Sparse Sampling is All You Need](https://mlsanity.com/paper/2405.07293/) (2024)
- [TRGR: Transmissive RIS-aided Gait Recognition Through Walls](https://mlsanity.com/paper/2407.21566/) (2024)
- [From ChebNet to ChebGibbsNet](https://mlsanity.com/paper/2412.01789/) (2024)
- [On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective](https://mlsanity.com/paper/2506.01213/) (2025)

## Abstract (arXiv)

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.