# Closed-form Filtering for Non-linear Systems

arXiv: https://arxiv.org/abs/2402.09796 · HTML: https://mlsanity.com/paper/2402.09796/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 22 references that are in the map.
- First posted on arXiv 15 Feb 2024.

## Builds on (most cited first, 6 of 6)

- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [Sampling from Arbitrary Functions via PSD Models](https://mlsanity.com/paper/2110.10527/) (2021)
- [Duality for nonlinear filtering](https://mlsanity.com/paper/2207.07709/) (2022)

## Abstract (arXiv)

Sequential Bayesian Filtering aims to estimate the current state distribution of a Hidden Markov Model, given the past observations. The problem is well-known to be intractable for most application domains, except in notable cases such as the tabular setting or for linear dynamical systems with gaussian noise. In this work, we propose a new class of filters based on Gaussian PSD Models, which offer several advantages in terms of density approximation and computational efficiency. We show that filtering can be efficiently performed in closed form when transitions and observations are Gaussian PSD Models. When the transition and observations are approximated by Gaussian PSD Models, we show that our proposed estimator enjoys strong theoretical guarantees, with estimation error that depends on the quality of the approximation and is adaptive to the regularity of the transition probabilities. In particular, we identify regimes in which our proposed filter attains a TV ε-error with memory and computational complexity of O(ε-1) and O(ε-3/2) respectively, including the offline learning step, in contrast to the O(ε-2) complexity of sampling methods such as particle filtering.