The Rank-Reduced Kalman Filter : Approximate Dynamical-Low-Rank Filtering In High Dimensions
(2023) 37th Conference on Neural Information Processing Systems, NeurIPS 2023 36.- Abstract
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... (More)
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 quadratic in the state-space size in the worst-case, and can achieve 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.
(Less)
- author
- Schmidt, Jonathan ; Nick, Jörg ; Hennig, Philipp and Tronarp, Filip LU
- organization
- publishing date
- 2023
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- host publication
- Advances in Neural Information Processing Systems
- volume
- 36
- publisher
- Morgan Kaufmann Publishers
- conference name
- 37th Conference on Neural Information Processing Systems, NeurIPS 2023
- conference location
- New Orleans, United States
- conference dates
- 2023-12-10 - 2023-12-16
- external identifiers
-
- scopus:85191166081
- language
- English
- LU publication?
- yes
- id
- 3a2171a5-862f-4d75-b40c-4f59af123b12
- date added to LUP
- 2024-05-08 10:14:50
- date last changed
- 2024-05-08 10:16:15
@inproceedings{3a2171a5-862f-4d75-b40c-4f59af123b12,
abstract = {{<p>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 quadratic in the state-space size in the worst-case, and can achieve 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.</p>}},
author = {{Schmidt, Jonathan and Nick, Jörg and Hennig, Philipp and Tronarp, Filip}},
booktitle = {{Advances in Neural Information Processing Systems}},
language = {{eng}},
publisher = {{Morgan Kaufmann Publishers}},
title = {{The Rank-Reduced Kalman Filter : Approximate Dynamical-Low-Rank Filtering In High Dimensions}},
volume = {{36}},
year = {{2023}},
}