The Gaussian process state-space models (GPSSMs) represent a versatile class of data-driven nonlinear dynamical system models. However, the presence of numerous latent variables in GPSSM incurs unresolved issues for existing variational inference approaches, particularly under the more realistic non-mean-field (NMF) assumption, including extensive training effort, compromised inference accuracy, and infeasibility for online applications, among others. In this paper, we tackle these challenges by incorporating the ensemble Kalman filter (EnKF), a well-established model-based filtering technique, into the NMF variational inference framework to approximate the posterior distribution of the latent states. This novel marriage between EnKF and GPSSM not only eliminates the need for extensive parameterization in learning variational distributions, but also enables an interpretable, closed-form approximation of the evidence lower bound (ELBO). Moreover, owing to the streamlined parameterization via the EnKF, the new GPSSM model can be easily accommodated in online learning applications. We demonstrate that the resulting EnKF-aided online algorithm embodies a principled objective function by ensuring data-fitting accuracy while incorporating model regularizations to mitigate overfitting. We also provide detailed analysis and fresh insights for the proposed algorithms. Comprehensive evaluation across diverse real and synthetic datasets corroborates the superior learning and inference performance of our EnKF-aided variational inference algorithms compared to existing methods.

中文翻译：

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集成卡尔曼滤波与高斯过程 SSM 相遇，用于非均值场和在线推理


高斯过程状态空间模型 （GPSSM） 代表了一类多功能的数据驱动非线性动力系统模型。然而，GPSSM 中存在许多潜在变量，这给现有的变分推理方法带来了未解决的问题，特别是在更现实的非均值场 （NMF） 假设下，包括大量的训练工作、受损的推理准确性以及在线应用程序的不可行性等。在本文中，我们通过将集成卡尔曼滤波 （EnKF）（一种成熟的基于模型的过滤技术）整合到 NMF 变分推理框架中来应对这些挑战，以近似潜在状态的后验分布。EnKF 和 GPSSM 之间的这种新颖结合不仅消除了学习变分分布中广泛参数化的需要，而且还实现了证据下限 （ELBO） 的可解释、封闭式近似。此外，由于通过 EnKF 简化了参数设置，新的 GPSSM 模型可以轻松适应在线学习应用程序。我们证明，由此产生的 EnKF 辅助在线算法通过确保数据拟合的准确性，同时结合模型正则化来减轻过拟合，从而体现了一个原则性的目标函数。我们还为所提出的算法提供了详细的分析和新的见解。对各种真实和合成数据集的全面评估证实了与现有方法相比，我们的 EnKF 辅助变分推理算法具有卓越的学习和推理性能。