Wasserstein bounds for non-linear Gaussian filters
Toni Karvonen, Simo Särkkä
TL;DR
This work addresses the problem of quantifying the error of non-linear Gaussian filters by bounding the Wasserstein distance between the true joint distribution $\mathbf{Z}=(\mathbf{X},\mathbf{Y})$ and its Gaussian filter approximation $\widetilde{\mathbf{Z}}=(\widetilde{\mathbf{X}},\widetilde{\mathbf{Y}})$. It combines Stein's method with second-order Poincaré inequalities to derive a computable bound that depends on the first- and second-derivatives of the system and measurement maps $\mathbf{f}$ and $\mathbf{h}$, as well as discrepancies in the joint means and covariances; crucially, the bound vanishes when the models are affine, confirming exact Gaussianity. The main result, Theorem mainTheorem, provides an explicit, albeit conservative, bound that captures how Gaussian moment-matching and non-linear transformations propagate error through one-step filtering. A numerical example with univariate growth models demonstrates the bound's conservativeness and practical relevance for assessing non-linear Gaussian filters in safety-critical settings. Overall, the paper offers a principled metric-based stability tool to evaluate and compare Gaussian filtering approximations and suggests directions toward formal stability analysis in non-linear state-space models.
Abstract
Most Kalman filters for non-linear systems, such as the unscented Kalman filter, are based on Gaussian approximations. We use Poincaré inequalities to bound the Wasserstein distance between the true joint distribution of the prediction and measurement and its Gaussian approximation. The bounds can be used to assess the performance of non-linear Gaussian filters and determine those filtering approximations that are most likely to induce error.