Gaussian process state-space models (GPSSMs) are a flexible and principled approach for modeling dynamical systems. However, existing variational learning and inference methods for GPSSMs often necessitate optimizing a substantial number of variational distribution parameters, leading to inadequate performance and efficiency. To overcome this issue, we propose incorporating the ensemble Kalman filter (EnKF), a well-established model-based filtering technique, into the variational inference framework to approximate the posterior distribution of latent states. This utilization of EnKF can effectively exploit the dependencies between latent states and GP dynamics, while eliminating the need for parameterizing the variational distribution, thereby significantly reducing the number of variational parameters. Moreover, we show that our proposed algorithm allows straightforward evaluation of an approximated evidence lower bound (ELBO) in variational inference via simply summating multiple terms with readily available closed-form solutions. Leveraging automatic differentiation tools, we hence can maximize the ELBO and train the GPSSM efficiently. We also extend the proposed algorithm to an online setting and provide detailed algorithmic analyses and insights. Extensive evaluation on diverse real and synthetic datasets demonstrates the superiority of our EnKF-aided variational inference algorithms in terms of learning and inference performance compared to existing methods.

翻译：高斯过程状态空间模型（GPSSM）是建模动态系统的一种灵活且原理性的方法。然而，现有针对GPSSM的变分学习与推断方法通常需要优化大量变分分布参数，导致性能与效率不足。为解决该问题，本文提出将成熟的基于模型的滤波技术——集成卡尔曼滤波（EnKF）——融入变分推断框架，用于近似隐状态的后验分布。这种EnKF的利用能够有效挖掘隐状态与高斯过程动力学之间的依赖关系，同时消除对变分分布参数化的需求，从而显著减少变分参数数量。此外，我们证明所提出的算法可通过简单求和多个具有闭式解形式的项，直接评估变分推断中的近似证据下界（ELBO）。借助自动微分工具，我们因此能够最大化ELBO并高效训练GPSSM。我们还将所提算法扩展至在线场景，并提供详细的算法分析与见解。在多样化的真实与合成数据集上进行的广泛评估表明，与现有方法相比，本文提出的EnKF辅助变分推断算法在学习与推断性能方面具有优越性。