The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929196330123264 |
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| author | Schmidt, Jonathan Hennig, Philipp Nick, Jörg Tronarp, Filip |
| author_facet | Schmidt, Jonathan Hennig, Philipp Nick, Jörg Tronarp, Filip |
| contents | Inference and simulation in the context of high-dimensional dynamical systems remain computationally challenging problems. Some form of dimensionality reduction is required to make the problem tractable in general. In this paper, we propose a novel approximate Gaussian filtering and smoothing method which propagates low-rank approximations of the covariance matrices. This is accomplished by projecting the Lyapunov equations associated with the prediction step to a manifold of low-rank matrices, which are then solved by a recently developed, numerically stable, dynamical low-rank integrator. Meanwhile, the update steps are made tractable by noting that the covariance update only transforms the column space of the covariance matrix, which is low-rank by construction. The algorithm differentiates itself from existing ensemble-based approaches in that the low-rank approximations of the covariance matrices are deterministic, rather than stochastic. Crucially, this enables the method to reproduce the exact Kalman filter as the low-rank dimension approaches the true dimensionality of the problem. Our method reduces computational complexity from cubic (for the Kalman filter) to \emph{quadratic} in the state-space size in the worst-case, and can achieve \emph{linear} complexity if the state-space model satisfies certain criteria. Through a set of experiments in classical data-assimilation and spatio-temporal regression, we show that the proposed method consistently outperforms the ensemble-based methods in terms of error in the mean and covariance with respect to the exact Kalman filter. This comes at no additional cost in terms of asymptotic computational complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07774 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions Schmidt, Jonathan Hennig, Philipp Nick, Jörg Tronarp, Filip Machine Learning Inference and simulation in the context of high-dimensional dynamical systems remain computationally challenging problems. Some form of dimensionality reduction is required to make the problem tractable in general. In this paper, we propose a novel approximate Gaussian filtering and smoothing method which propagates low-rank approximations of the covariance matrices. This is accomplished by projecting the Lyapunov equations associated with the prediction step to a manifold of low-rank matrices, which are then solved by a recently developed, numerically stable, dynamical low-rank integrator. Meanwhile, the update steps are made tractable by noting that the covariance update only transforms the column space of the covariance matrix, which is low-rank by construction. The algorithm differentiates itself from existing ensemble-based approaches in that the low-rank approximations of the covariance matrices are deterministic, rather than stochastic. Crucially, this enables the method to reproduce the exact Kalman filter as the low-rank dimension approaches the true dimensionality of the problem. Our method reduces computational complexity from cubic (for the Kalman filter) to \emph{quadratic} in the state-space size in the worst-case, and can achieve \emph{linear} complexity if the state-space model satisfies certain criteria. Through a set of experiments in classical data-assimilation and spatio-temporal regression, we show that the proposed method consistently outperforms the ensemble-based methods in terms of error in the mean and covariance with respect to the exact Kalman filter. This comes at no additional cost in terms of asymptotic computational complexity. |
| title | The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2306.07774 |