Sparse Adaptive Kernel Kalman Filter for Nonlinear Non-Gaussian State Estimation

Abstract

The adaptive kernel Kalman filter (AKKF) provides a state estimation framework for nonlinear and non-Gaussian systems by synergizing data-space particle propagation with kernel-space Kalman updates. However, expanding the particle set to improve tracking accuracy inevitably induces severe computational burden and numerical ill-conditioning due to dense Gram matrix operations. To tackle that, this paper proposes a sparse adaptive kernel Kalman filter (SAKKF). The filter employs a Nyström-based low-rank approximation to structurally alleviate the computational complexity of kernel matrix evaluations. Crucially, an error feedback-driven dynamic mechanism is designed to adaptively modulate the cardinality and spatial distribution of the inducing points. This gradient-free, dynamic trigger strategy ensures real-time adaptability without the heavy overhead of conventional continuous optimization. Both numerical simulations and real-world vehicle positioning experiments demonstrate that the SAKKF significantly mitigates computational complexity and enhances numerical stability, while maintaining competitive estimation accuracy. Note to Practitioners—Real-time state estimation in complex systems, such as vehicle navigation and target tracking, frequently encounters highly nonlinear dynamics and non-Gaussian noises. While advanced kernel-based filtering methods offer superior accuracy in such environments, their practical deployment on resource-constrained embedded platforms is often hindered by the high computational complexity and potential numerical instability associated with large-scale matrix operations. This paper introduces an SAKKF. Instead of processing all sampled particles, the SAKKF utilizes a dynamically adjusted subset of inducing points to compress the computational workload. By automatically adapting the number and locations of these points based on real-time estimation errors, the algorithm avoids the need for manual tuning and heavy gradient calculations. Experimental results confirm that this approach effectively reduces processing time, delivering a reliable balance between estimation accuracy and computational efficiency. This framework can be directly applied to vehicle navigation systems, with future potential for extension into multi-sensor fusion and distributed tracking architectures.