# On dimension reduction in Gaussian filters

arXiv: https://arxiv.org/abs/1508.06452 · HTML: https://mlsanity.com/paper/1508.06452/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Aug 2015.

## Built on by (most cited first, 4 of 4)

- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Computation-Aware Kalman Filtering and Smoothing](https://mlsanity.com/paper/2405.08971/) (2024)
- [Ensemble transport smoothing. Part II: Nonlinear updates](https://mlsanity.com/paper/2210.17435/) (2022)
- [RMFGP: Rotated Multi-fidelity Gaussian process with Dimension Reduction for High-dimensional Uncertainty Quantification](https://mlsanity.com/paper/2204.04819/) (2022)

## Abstract (arXiv)

A priori dimension reduction is a widely adopted technique for reducing the computational complexity of stationary inverse problems. In this setting, the solution of an inverse problem is parameterized by a low-dimensional basis that is often obtained from the truncated Karhunen-Loeve expansion of the prior distribution. For high-dimensional inverse problems equipped with smoothing priors, this technique can lead to drastic reductions in parameter dimension and significant computational savings. In this paper, we extend the concept of a priori dimension reduction to non-stationary inverse problems, in which the goal is to sequentially infer the state of a dynamical system. Our approach proceeds in an offline-online fashion. We first identify a low-dimensional subspace in the state space before solving the inverse problem (the offline phase), using either the method of "snapshots" or regularized covariance estimation. Then this subspace is used to reduce the computational complexity of various filtering algorithms - including the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within a novel subspace-constrained Bayesian prediction-and-update procedure (the online phase). We demonstrate the performance of our new dimension reduction approach on various numerical examples. In some test cases, our approach reduces the dimensionality of the original problem by orders of magnitude and yields up to two orders of magnitude in computational savings.