Closed-form Filtering for Non-linear Systems
This work addresses the computational bottleneck in state estimation for non-linear systems, offering a more efficient alternative to sampling-based methods like particle filtering.
The paper tackles the intractable problem of sequential Bayesian filtering for non-linear systems by proposing a new class of filters based on Gaussian PSD Models, achieving a TV ε-error with memory and computational complexities of O(ε^{-1}) and O(ε^{-3/2}) respectively, compared to O(ε^{-2}) for particle filtering.
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.