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Gegenbauer Graph Neural Networks for Time-varying Signal Reconstruction

Jhon A. Castro-Correa, Jhony H. Giraldo, Mohsen Badiey, Fragkiskos D. Malliaros

TL;DR

The paper introduces GegenGNN, a time-varying graph signal reconstruction framework that leverages Gegenbauer polynomial-based graph convolutions (GegenConv) to generalize Chebyshev filters. It uses an encoder-decoder GNN architecture that processes time-difference signals with a cascade of Gegenbauer convolutions and a linear fusion layer, trained with a loss combining mean squared error and a Sobolev-based regularization to capture temporal dynamics. Extensive experiments on four real-world datasets show GegenGNN outperforms state-of-the-art GSP and GNN baselines, with ablations demonstrating the benefits of the Gegenbauer parameter $\alpha$ and the temporal loss. The approach relaxes strict smoothness assumptions, offers competitive efficiency, and has potential applications in sensor networks, weather forecasting, and related time-varying graph signal tasks.

Abstract

Reconstructing time-varying graph signals (or graph time-series imputation) is a critical problem in machine learning and signal processing with broad applications, ranging from missing data imputation in sensor networks to time-series forecasting. Accurately capturing the spatio-temporal information inherent in these signals is crucial for effectively addressing these tasks. However, existing approaches relying on smoothness assumptions of temporal differences and simple convex optimization techniques have inherent limitations. To address these challenges, we propose a novel approach that incorporates a learning module to enhance the accuracy of the downstream task. To this end, we introduce the Gegenbauer-based graph convolutional (GegenConv) operator, which is a generalization of the conventional Chebyshev graph convolution by leveraging the theory of Gegenbauer polynomials. By deviating from traditional convex problems, we expand the complexity of the model and offer a more accurate solution for recovering time-varying graph signals. Building upon GegenConv, we design the Gegenbauer-based time Graph Neural Network (GegenGNN) architecture, which adopts an encoder-decoder structure. Likewise, our approach also utilizes a dedicated loss function that incorporates a mean squared error component alongside Sobolev smoothness regularization. This combination enables GegenGNN to capture both the fidelity to ground truth and the underlying smoothness properties of the signals, enhancing the reconstruction performance. We conduct extensive experiments on real datasets to evaluate the effectiveness of our proposed approach. The experimental results demonstrate that GegenGNN outperforms state-of-the-art methods, showcasing its superior capability in recovering time-varying graph signals.

Gegenbauer Graph Neural Networks for Time-varying Signal Reconstruction

TL;DR

The paper introduces GegenGNN, a time-varying graph signal reconstruction framework that leverages Gegenbauer polynomial-based graph convolutions (GegenConv) to generalize Chebyshev filters. It uses an encoder-decoder GNN architecture that processes time-difference signals with a cascade of Gegenbauer convolutions and a linear fusion layer, trained with a loss combining mean squared error and a Sobolev-based regularization to capture temporal dynamics. Extensive experiments on four real-world datasets show GegenGNN outperforms state-of-the-art GSP and GNN baselines, with ablations demonstrating the benefits of the Gegenbauer parameter and the temporal loss. The approach relaxes strict smoothness assumptions, offers competitive efficiency, and has potential applications in sensor networks, weather forecasting, and related time-varying graph signal tasks.

Abstract

Reconstructing time-varying graph signals (or graph time-series imputation) is a critical problem in machine learning and signal processing with broad applications, ranging from missing data imputation in sensor networks to time-series forecasting. Accurately capturing the spatio-temporal information inherent in these signals is crucial for effectively addressing these tasks. However, existing approaches relying on smoothness assumptions of temporal differences and simple convex optimization techniques have inherent limitations. To address these challenges, we propose a novel approach that incorporates a learning module to enhance the accuracy of the downstream task. To this end, we introduce the Gegenbauer-based graph convolutional (GegenConv) operator, which is a generalization of the conventional Chebyshev graph convolution by leveraging the theory of Gegenbauer polynomials. By deviating from traditional convex problems, we expand the complexity of the model and offer a more accurate solution for recovering time-varying graph signals. Building upon GegenConv, we design the Gegenbauer-based time Graph Neural Network (GegenGNN) architecture, which adopts an encoder-decoder structure. Likewise, our approach also utilizes a dedicated loss function that incorporates a mean squared error component alongside Sobolev smoothness regularization. This combination enables GegenGNN to capture both the fidelity to ground truth and the underlying smoothness properties of the signals, enhancing the reconstruction performance. We conduct extensive experiments on real datasets to evaluate the effectiveness of our proposed approach. The experimental results demonstrate that GegenGNN outperforms state-of-the-art methods, showcasing its superior capability in recovering time-varying graph signals.
Paper Structure (23 sections, 18 equations, 4 figures, 4 tables)

This paper contains 23 sections, 18 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notation
  5. Graph Signals
  6. Reconstruction of Smooth Time-varying Graph Signals
  7. Learning Graphs from Data
  8. Gegenbauer Graph Neural Network
  9. Spectral Graph Convolution
  10. Gegenbauer Polynomials
  11. Computation of the Gegenbauer Basis
  12. Graph Neural Network Architecture
  13. Experimental Evaluation
  14. Implementation Details
  15. Datasets
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Pipeline of our Gegenbauer-based Graph Neural Network (GegenGNN) for recovering time-varying graph signals. The graph construction is performed using either $k$-nearest neighbors or learned from time-varying data. GegenGNN is an encoder-decoder architecture, with the graph Laplacian matrix $\mathbf{L}$ and the time difference signal $\mathbf{X}\mathbf{D}_h$ serving as inputs. Each layer of GegenGNN is powered by a cascade of Gegenbauer-based convolutions, which are then merged by a linear combination layer. Our method incorporates the Sobolev smoothness term to account for time dependence in the graph signal.
  • Figure 2: Convolutional layer of GegenGNN. Each layer of GegenGNN consists of a cascade of Gegenbauer-based convolutions of increasing order $\rho=0,\dots, \zeta-1$. The outputs from all $\zeta$ convolutions are then merged using a linear combination layer, which includes learnable parameters $\mu_\rho$.
  • Figure 3: Location of the SW06 experiment. The sensors were deployed in a three-dimensional fashion designed to study the wavefront of gravitational nonlinear internal waves (IWs).
  • Figure 4: Comparison of GegenGNN with several methods from the literature was performed on four real-world datasets. The evaluation was based on the root mean square error (RMSE) metric, considering different sampling densities $m$.