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.

基于 Gegenbauer 多项式的图卷积操作符与 GegenGNN 结构的时间图神经网络通过增强学习模块以提高下游任务的准确性和复杂度，展示出优于现有方法的能力，用于恢复时间变化的图信号。

时间变化信号重建的 Gegenbauer 图神经网络

Gegenbauer Graph Neural Networks for Time-varying Signal Reconstruction

Generative models for 3D geometric data arise in many important applications
in 3D computer vision and graphics. In this paper, we focus on 3D deformable
shapes that share a common topological structure, such as human faces and
bodies. Morphable Models and their variants, despite their linear formulation,
have been widely used for shape representation, while most of the recently
proposed nonlinear approaches resort to intermediate representations, such as
3D voxel grids or 2D views. In this work, we introduce a novel graph
convolutional operator, acting directly on the 3D mesh, that explicitly models
the inductive bias of the fixed underlying graph. This is achieved by enforcing
consistent local orderings of the vertices of the graph, through the spiral
operator, thus breaking the permutation invariance property that is adopted by
all the prior work on Graph Neural Networks. Our operator comes by construction
with desirable properties (anisotropic, topology-aware, lightweight,
easy-to-optimise), and by using it as a building block for traditional deep
generative architectures, we demonstrate state-of-the-art results on a variety
of 3D shape datasets compared to the linear Morphable Model and other graph
convolutional operators.

本文提出了一种新颖的图卷积算子，通过对网格的本地顺序强制执行连续性，实现对固定底层图形的归纳偏差建模，并将其作为传统深度生成建筑的构件，以多种 3D 形状数据集为例，与线性 Morphable Model 和其他图卷积算子相比，展示出最先进的结果。