  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.
