Gaussian process state-space models (GPSSMs) are a versatile and principled family of nonlinear dynamical system models. However, existing variational learning and inference methods for GPSSMs often necessitate optimizing a substantial number of variational 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 accommodate 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（GPSSMs）是一类通用且原理性的非线性动态系统模型。然而，现有针对GPSSMs的变分学习与推断方法通常需要优化大量变分参数，导致性能与效率不足。为克服此问题，我们提出将集合卡尔曼滤波（EnKF）——一种成熟的基于模型的滤波技术——纳入变分推断框架，以近似潜在状态的后验分布。这种对EnKF的利用能有效挖掘潜在状态与GP动态之间的依赖关系，同时消除对变分分布进行参数化的需求，从而显著减少变分参数的数量。此外，我们表明，所提算法可通过简单累加多个具有现成闭式解的项，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，我们因此能最大化ELBO并高效训练GPSSM。我们还将所提算法拓展至在线设置，并提供详细的算法分析与洞见。在多样化的真实与合成数据集上的大量评估表明，与现有方法相比，我们基于EnKF辅助的变分推断算法在学习与推断性能方面均具优越性。