Recursive Inference for Heterogeneous Multi-Output GP State-Space Models with Arbitrary Moment Matching

TL;DR

This work extends online Gaussian Process State-Space Models (GPSSMs) by introducing a heterogeneous multi-output kernel and a per-dimension inducing-point management scheme, enabling independent learning across outputs. It develops a unified moment-matching inference framework that does not rely on first-order linearization, supporting EKF, UKF, and ADF in online settings, with a numerically stable Cholesky-based implementation. The proposed RGPSSM-H demonstrates higher learning flexibility and accuracy under strong nonlinearity and noise, while offering substantial computational efficiency gains (e.g., online runtime reductions of about 70% relative to state-of-the-art online GPSSMs and offline accuracy parity with 1/100 of the runtime). Through synthetic kink learning, time-varying parameter identification, hypersonic vehicle dynamics, and real-world quadrotor data, the method proves effective for rapidly adapting online models in complex dynamical environments. The practical impact lies in enabling robust, fast, online learning of heterogeneous, high-order nonlinear dynamics for control and decision-making tasks.

Abstract

Accurate learning of system dynamics is becoming increasingly crucial for advanced control and decision-making in engineering. However, real-world systems often exhibit multiple channels and highly nonlinear transition dynamics, challenging traditional modeling methods. To enable online learning for these systems, this paper formulates the system as Gaussian process state-space models (GPSSMs) and develops a recursive learning method. The main contributions are threefold. First, a heterogeneous multi-output kernel is designed, allowing each output dimension to adopt distinct kernel types, hyperparameters, and input variables, improving expressiveness in multi-dimensional dynamics learning. Second, an inducing-point management algorithm enhances computational efficiency through independent selection and pruning for each output dimension. Third, a unified recursive inference framework for GPSSMs is derived, supporting general moment matching approaches, including the extended Kalman filter (EKF), unscented Kalman filter (UKF), and assumed density filtering (ADF), enabling accurate learning under strong nonlinearity and significant noise. Experiments on synthetic and real-world datasets show that the proposed method matches the accuracy of SOTA offline GPSSMs with only 1/100 of the runtime, and surpasses SOTA online GPSSMs by around 70% in accuracy under heavy noise while using only 1/20 of the runtime.

Figures (11)

• Figure 1: Algorithmic workflow of the RGPSSM zheng2024recursive, where $\bm m_{x_t} = \mathbb{E}_{q(\bm{u}, \bm{x}_t)}[\bm{x}_t]$.
• Figure 2: Best learning performance of the kink transition function over five random experiments using various methods across different levels of measurement noise ($\sigma^2_m \in \{0.008,\, 0.08,\, 0.8\}$, from left to right).
• Figure 3: Filtering and smoothing estimation of time-varying parameters. Filtering estimation refers to $p(\bm\theta_t|\bm y_{1:t})$ and smoothing estimation refers to $p(\bm\theta_t|\bm y_{1:T})$, where $T$ is the total number of measurements. The shaded area indicates the 95% confidence interval, and the '+' marker indicates the locations of the inducing points.
• Figure 4: Evolution curves of the length scales of the kernel.
• Figure 5: First row: filtering estimation of the angular rate $\bm\omega$. Second row: prediction of the moment coefficient model bias $\Delta \bm C(\cdot)$ at each prediction step. The shaded area indicates the 95% confidence interval.