Computation-Aware Kalman Filtering and Smoothing

2024-05-14

arXiv.org

AI Summary

To address the computational and memory bottlenecks of Kalman filtering/smoothing in high-dimensional spatiotemporal regression under the Gauss–Markov model—caused by state dimension growth—this paper introduces the first *computation-aware probabilistic numerical method*. The approach builds upon low-rank covariance approximations but explicitly models the numerical error induced by truncation, enabling a tunable trade-off between computational cost and predictive uncertainty quantification. Integrating matrix-free iterative solvers, GPU acceleration, and rigorous error propagation modeling, the method achieves *linear time and space complexity* on large-scale climate datasets. Empirical evaluation demonstrates significantly improved uncertainty calibration compared to conventional approximations, which often yield overconfident (overly optimistic) predictions. This advancement bridges probabilistic numerics and scalable inference for high-dimensional dynamical systems.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. Since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.

Problem

• Addresses high computational complexity in Kalman filtering and smoothing.
• Mitigates scaling issues in high-dimensional Gauss-Markov models.
• Enables tunable trade-off between computational cost and predictive uncertainty.

Innovation

• Matrix-free iterative algorithm for high-dimensional inference
• GPU acceleration enhances computational efficiency
• Tunable trade-off between cost and predictive uncertainty