Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Locally Stationary Graph Processes

Abdullah Canbolat, Elif Vural

TL;DR

This work introduces Locally Stationary Graph Processes (LSGP) to model nonstationary graph signals whose statistics vary across the graph. The core idea is to express the signal as a sum of component WSSc graph processes with node-wise memberships that vary smoothly, enabling a vertex-frequency perspective via a spectrum matrix M. A convex, SDP-based learning framework estimates the memberships and spectral kernels from realizations (including missing data), and the model supports extension/restriction to subgraphs and local WSS approximations under spectral separation. The authors also develop a partition-based approach to scale LSGP to large graphs by locally learning simpler WSSt processes on subgraphs, with theoretical bounds on cross-subgraph covariance. Experimental results on synthetic and real data for graph signal interpolation show competitive performance against state-of-the-art methods, highlighting the method’s flexibility, locality, and potential scalability.

Abstract

Stationary graph process models are commonly used in the analysis and inference of data sets collected on irregular network topologies. While most of the existing methods represent graph signals with a single stationary process model that is globally valid on the entire graph, in many practical problems, the characteristics of the process may be subject to local variations in different regions of the graph. In this work, we propose a locally stationary graph process (LSGP) model that aims to extend the classical concept of local stationarity to irregular graph domains. We characterize local stationarity by expressing the overall process as the combination of a set of component processes such that the extent to which the process adheres to each component varies smoothly over the graph. We propose an algorithm for computing LSGP models from realizations of the process, and also study the approximation of LSGPs locally with WSS processes. Experiments on signal interpolation problems show that the proposed process model provides accurate signal representations competitive with the state of the art.

Locally Stationary Graph Processes

TL;DR

This work introduces Locally Stationary Graph Processes (LSGP) to model nonstationary graph signals whose statistics vary across the graph. The core idea is to express the signal as a sum of component WSSc graph processes with node-wise memberships that vary smoothly, enabling a vertex-frequency perspective via a spectrum matrix M. A convex, SDP-based learning framework estimates the memberships and spectral kernels from realizations (including missing data), and the model supports extension/restriction to subgraphs and local WSS approximations under spectral separation. The authors also develop a partition-based approach to scale LSGP to large graphs by locally learning simpler WSSt processes on subgraphs, with theoretical bounds on cross-subgraph covariance. Experimental results on synthetic and real data for graph signal interpolation show competitive performance against state-of-the-art methods, highlighting the method’s flexibility, locality, and potential scalability.

Abstract

Stationary graph process models are commonly used in the analysis and inference of data sets collected on irregular network topologies. While most of the existing methods represent graph signals with a single stationary process model that is globally valid on the entire graph, in many practical problems, the characteristics of the process may be subject to local variations in different regions of the graph. In this work, we propose a locally stationary graph process (LSGP) model that aims to extend the classical concept of local stationarity to irregular graph domains. We characterize local stationarity by expressing the overall process as the combination of a set of component processes such that the extent to which the process adheres to each component varies smoothly over the graph. We propose an algorithm for computing LSGP models from realizations of the process, and also study the approximation of LSGPs locally with WSS processes. Experiments on signal interpolation problems show that the proposed process model provides accurate signal representations competitive with the state of the art.
Paper Structure (28 sections, 8 theorems, 56 equations, 8 figures, 4 tables, 2 algorithms)

This paper contains 28 sections, 8 theorems, 56 equations, 8 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Graph Signal Processing
  5. Wide Sense Stationary Processes on Graphs
  6. Time-Varying Moving Average Processes
  7. Locally Stationary Graph Processes
  8. Proposed Process Model
  9. Vertex-Frequency Spectrum
  10. Extension and Restriction of LSGPs
  11. Learning LSGP Models
  12. Locally Approximating LSGPs with WSS Processes
  13. Covariance Analysis of LSGPs
  14. Proposed Algorithm for Locally Approximating LSGPs
  15. Experimental Results
  16. ...and 13 more sections

Key Result

Theorem 1

A locally stationary graph process with variation rate $C$ can be expressed as $\mathbf{x}=\mathbf{H} \mathbf{w}$, where the filter $\mathbf{H} = \sum_{k=1}^K \mathbf{G}_k \mathbf{U}_\mathcal{G} h_k(\mathbf{\Lambda}_ \mathcal{G}) \mathbf{U}_\mathcal{G} ^T$ can be written in the form with $\mathbf{M}= \sum_{k=1}^K \mathbf{g}_k \mathbf{h}_k^T$. Here the matrix $\mathbf{M} \in \mathbb{R}^{N\times

Figures (8)

  • Figure 1: (a) Variation of the covariance discrepancy with respect to the number of realizations. (b) Variation of the estimation errors with the SNR.
  • Figure 2: NME of compared algorithms on COVID-19 and Molène data sets
  • Figure 3: Results obtained on the NOAA and the USA COVID-19 data sets
  • Figure 4: Variation of the covariance discrepancy with the model complexity
  • Figure 5: MAE of compared algorithms on COVID-19 and Molène data sets
  • ...and 3 more figures

Theorems & Definitions (20)

  • Definition 1: Wide Sense Stationary Process
  • Definition 2: Locally Stationary Graph Process
  • Theorem 1: Vertex-Frequency Spectrum
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 10 more