# A Novel Unified Extended Matrix for Graph Signal Processing: Theory and Application

arXiv: https://arxiv.org/abs/2508.16633 · HTML: https://mlsanity.com/paper/2508.16633/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 60 references that are in the map.
- First posted on arXiv 16 Aug 2025.

## Builds on (most cited first, 18 of 18)

- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Discrete Signal Processing on Graphs: Frequency Analysis](https://mlsanity.com/paper/1307.0468/) (2013)
- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Signal Processing on Graphs: Causal Modeling of Unstructured Data](https://mlsanity.com/paper/1503.00173/) (2015)
- [Sampling of graph signals with successive local aggregations](https://mlsanity.com/paper/1504.04687/) (2015)
- [Reconstruction of Time-varying Graph Signals via Sobolev Smoothness](https://mlsanity.com/paper/2207.06439/) (2022)
- [Joint Network Topology Inference in the Presence of Hidden Nodes](https://mlsanity.com/paper/2306.17364/) (2023)
- [Graph Neural Networks With Lifting-based Adaptive Graph Wavelets](https://mlsanity.com/paper/2108.01660/) (2021)
- [Graph Laplacian for Image Anomaly Detection](https://mlsanity.com/paper/1802.09843/) (2018)
- [Graph Fourier Transform: A Stable Approximation](https://mlsanity.com/paper/2001.05042/) (2020)
- [Sampling and Uniqueness Sets in Graphon Signal Processing](https://mlsanity.com/paper/2401.06279/) (2024)
- [Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows](https://mlsanity.com/paper/1711.09598/) (2017)
- [Multi-dimensional graph fractional Fourier transform and its application](https://mlsanity.com/paper/2109.04358/) (2021)
- [A Generalized Nyquist-Shannon Sampling Theorem Using the Koopman Operator](https://mlsanity.com/paper/2303.01927/) (2023)
- [Graph Fourier Transform based Audio Zero-watermarking](https://mlsanity.com/paper/2109.08007/) (2021)
- [Learning Optimal Graph Filters for Clustering of Attributed Graphs](https://mlsanity.com/paper/2211.04634/) (2022)
- [Online Signed Sampling of Bandlimited Graph Signals](https://mlsanity.com/paper/2402.10493/) (2024)

## Abstract (arXiv)

Graph signal processing has become an essential tool for analyzing data structured on irregular domains. While conventional graph shift operators (GSOs) are effective for certain tasks, they inherently lack flexibility in modeling dependencies between non-adjacent nodes, limiting their ability to represent complex graph structures. To address this limitation, this paper proposes the unified extended matrix (UEM) framework, which integrates the extended-adjacency matrix and the unified graph representation matrix through parametric design, so as to be able to flexibly adapt to different graph structures and reveal more graph signal information. Theoretical analysis of the UEM is conducted, demonstrating positive semi-definiteness and eigenvalue monotonicity under specific conditions. Then, we propose graph Fourier transform based on UEM (UEM-GFT), which can adaptively tune spectral properties to enhance signal processing performance. Experimental results on synthetic and real-world datasets demonstrate that the UEM-GFT outperforms existing GSO-based methods in anomaly detection tasks, achieving superior performance across varying network topologies.