Dynamic systems of graph signals are encountered in various applications, including social networks, power grids, and transportation. While such systems can often be described as state space (SS) models, tracking graph signals via conventional tools based on the Kalman filter (KF) and its variants is typically challenging. This is due to the nonlinearity, high dimensionality, irregularity of the domain, and complex modeling associated with real-world dynamic systems of graph signals. In this work, we study the tracking of graph signals using a hybrid model-based/data-driven approach. We develop the GSP-KalmanNet , which tracks the hidden graphical states from the graphical measurements by jointly leveraging graph signal processing (GSP) tools and deep learning (DL) techniques. The derivations of the GSP-KalmanNet are based on extending the KF to exploit the inherent graph structure via designing a graph frequency domain filtering and replacing the Kalman gain (KG) with a graph filter that minimizes the prediction error. Thus, it considerably simplifies the computational complexity entailed in processing high-dimensional signals and increases the robustness to small topology changes. Then, we use data to learn the KG, namely, the graph filter, following the recently proposed KalmanNet framework, which copes with partial and approximated modeling, without forcing a specific model over the noise statistics. Restricting the KG to a graph filter in the proposed GSP-KalmanNet reduces learned parameters, thereby enhancing stability. Our empirical results demonstrate that the GSP-KalmanNet achieves enhanced accuracy and run time performance, and improved robustness to model misspecifications compared with both model-based and data-driven benchmarks.

中文翻译：

---

GSP-KalmanNet：通过神经辅助卡尔曼滤波跟踪图形信号


图信号的动态系统在各种应用中都会遇到，包括社交网络、电网和交通。虽然此类系统通常可以描述为状态空间 (SS) 模型，但通过基于卡尔曼滤波器 (KF) 及其变体的传统工具跟踪图形信号通常具有挑战性。这是由于非线性、高维、域的不规则性以及与现实世界的图信号动态系统相关的复杂建模。在这项工作中，我们使用基于模型/数据驱动的混合方法研究图形信号的跟踪。我们开发了 GSP-KalmanNet，它通过联合利用图形信号处理 (GSP) 工具和深度学习 (DL) 技术来跟踪图形测量中隐藏的图形状态。 GSP-KalmanNet 的推导基于扩展 KF，通过设计图频域滤波并用图滤波器替换卡尔曼增益 (KG) 来利用固有的图结构，从而最小化预测误差。因此，它大大简化了处理高维信号所需的计算复杂性，并提高了对小拓扑变化的鲁棒性。然后，我们使用数据来学习 KG，即图过滤器，遵循最近提出的 KalmanNet 框架，该框架可以处理部分和近似建模，而不需要在噪声统计上强制使用特定模型。在所提出的 GSP-KalmanNet 中将 KG 限制为图过滤器可以减少学习参数，从而增强稳定性。我们的实证结果表明，与基于模型和数据驱动的基准相比，GSP-KalmanNet 提高了准确性和运行时性能，并提高了对模型错误指定的鲁棒性。