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Adaptive Least Mean Squares Graph Neural Networks and Online Graph Signal Estimation

Yi Yan, Changran Peng, Ercan Engin Kuruoglu

TL;DR

The paper addresses online estimation of time-varying graph signals corrupted by noise and missing data. It introduces LMS-GNN, a hybrid architecture that combines adaptive graph filters with Graph Neural Networks to learn time-varying, bandlimited filters from data and update them online via residual-driven backpropagation. The method outputs $ oldsymbol{x}_L[t] ightarrow oldsymbol{x}[t+1] $ by leveraging multiple denoising layers and a final time-varying update, demonstrated on real temperature graphs where it outperforms adaptive GSP methods and offline GNNs. This yields a simple, interpretable, and computationally efficient approach for real-time graph signal prediction in noisy, incomplete environments.

Abstract

The online prediction of multivariate signals, existing simultaneously in space and time, from noisy partial observations is a fundamental task in numerous applications. We propose an efficient Neural Network architecture for the online estimation of time-varying graph signals named the Adaptive Least Mean Squares Graph Neural Networks (LMS-GNN). LMS-GNN aims to capture the time variation and bridge the cross-space-time interactions under the condition that signals are corrupted by noise and missing values. The LMS-GNN is a combination of adaptive graph filters and Graph Neural Networks (GNN). At each time step, the forward propagation of LMS-GNN is similar to adaptive graph filters where the output is based on the error between the observation and the prediction similar to GNN. The filter coefficients are updated via backpropagation as in GNN. Experimenting on real-world temperature data reveals that our LMS-GNN achieves more accurate online predictions compared to graph-based methods like adaptive graph filters and graph convolutional neural networks.

Adaptive Least Mean Squares Graph Neural Networks and Online Graph Signal Estimation

TL;DR

The paper addresses online estimation of time-varying graph signals corrupted by noise and missing data. It introduces LMS-GNN, a hybrid architecture that combines adaptive graph filters with Graph Neural Networks to learn time-varying, bandlimited filters from data and update them online via residual-driven backpropagation. The method outputs by leveraging multiple denoising layers and a final time-varying update, demonstrated on real temperature graphs where it outperforms adaptive GSP methods and offline GNNs. This yields a simple, interpretable, and computationally efficient approach for real-time graph signal prediction in noisy, incomplete environments.

Abstract

The online prediction of multivariate signals, existing simultaneously in space and time, from noisy partial observations is a fundamental task in numerous applications. We propose an efficient Neural Network architecture for the online estimation of time-varying graph signals named the Adaptive Least Mean Squares Graph Neural Networks (LMS-GNN). LMS-GNN aims to capture the time variation and bridge the cross-space-time interactions under the condition that signals are corrupted by noise and missing values. The LMS-GNN is a combination of adaptive graph filters and Graph Neural Networks (GNN). At each time step, the forward propagation of LMS-GNN is similar to adaptive graph filters where the output is based on the error between the observation and the prediction similar to GNN. The filter coefficients are updated via backpropagation as in GNN. Experimenting on real-world temperature data reveals that our LMS-GNN achieves more accurate online predictions compared to graph-based methods like adaptive graph filters and graph convolutional neural networks.
Paper Structure (10 sections, 7 equations, 2 figures, 1 table)

This paper contains 10 sections, 7 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Algorithm Derivation
  4. Spectral Graph Neural Networks
  5. Adaptive Filtering
  6. Least Mean Squares Graph Neural Networks
  7. Experiments
  8. Experiment Setup
  9. Results and Discussion
  10. Conclusion

Figures (2)

  • Figure 1: The MSE for the predictions from $t = 1$ to $95$, with the VAR = 0.1 for the zero-mean Gaussian noise. (t = [0, 24] is the training set, and t = [25, 95] is the testing set.)
  • Figure 2: The MAE for the predictions from $t = 1$ to $95$, with the VAR = 0.1 for the zero-mean Gaussian noise. (t = [0, 24] is the training set, and t = [25, 95] is the testing set.)