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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2409.20004 |
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| _version_ | 1866909464136777728 |
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| author | Krämer, Nicholas |
| author_facet | Krämer, Nicholas |
| contents | Practical implementations of Gaussian smoothing algorithms have received a great deal of attention in the last 60 years. However, almost all work focuses on estimating complete time series (''fixed-interval smoothing'', $\mathcal{O}(K)$ memory) through variations of the Rauch--Tung--Striebel smoother, rarely on estimating the initial states (''fixed-point smoothing'', $\mathcal{O}(1)$ memory). Since fixed-point smoothing is a crucial component of algorithms for dynamical systems with unknown initial conditions, we close this gap by introducing a new formulation of a Gaussian fixed-point smoother. In contrast to prior approaches, our perspective admits a numerically robust Cholesky-based form (without downdates) and avoids state augmentation, which would needlessly inflate the state-space model and reduce the numerical practicality of any fixed-point smoother code. The experiments demonstrate how a JAX implementation of our algorithm matches the runtime of the fastest methods and the robustness of the most robust techniques while existing implementations must always sacrifice one for the other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20004 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerically Robust Fixed-Point Smoothing Without State Augmentation Krämer, Nicholas Numerical Analysis Machine Learning Systems and Control Computation Practical implementations of Gaussian smoothing algorithms have received a great deal of attention in the last 60 years. However, almost all work focuses on estimating complete time series (''fixed-interval smoothing'', $\mathcal{O}(K)$ memory) through variations of the Rauch--Tung--Striebel smoother, rarely on estimating the initial states (''fixed-point smoothing'', $\mathcal{O}(1)$ memory). Since fixed-point smoothing is a crucial component of algorithms for dynamical systems with unknown initial conditions, we close this gap by introducing a new formulation of a Gaussian fixed-point smoother. In contrast to prior approaches, our perspective admits a numerically robust Cholesky-based form (without downdates) and avoids state augmentation, which would needlessly inflate the state-space model and reduce the numerical practicality of any fixed-point smoother code. The experiments demonstrate how a JAX implementation of our algorithm matches the runtime of the fastest methods and the robustness of the most robust techniques while existing implementations must always sacrifice one for the other. |
| title | Numerically Robust Fixed-Point Smoothing Without State Augmentation |
| topic | Numerical Analysis Machine Learning Systems and Control Computation |
| url | https://arxiv.org/abs/2409.20004 |