Transport Techniques for Optimal Filtering

点击翻译按钮获取中文摘要

Optimal filtering, also known as data assimilation, is a real-time estimation problem for the current state of a dynamical system of interest. The aim is to combine predictions from a theoretical model of the system with real-time observations of the system in an optimal way. This problem statement is very broad and has applications as diverse as weather prediction, financial forecasting, target tracking and robot localisation. These application areas require filtering techniques which can be performed in real-time (also known as online) and give good performance in high dimensional state spaces, preferably with theoretical guarantees. It is these requirements which typically present the greatest challenges to current state-of-the-art filtering techniques.A common perspective on filtering is that of Bayesian inference, which gives a probabilistic approach where the assimilation step is an application of Bayes' theorem. However, Bayesian inference comes with its own challenges since the object of interest, the posterior distribution, is intractable in all but the simplest of cases. A significant portion of work in the filtering community relates to developing computational techniques for sequential Bayesian inference which perform well in real-time, high dimensional settings. Notable approaches include the Kalman filter (and its extensions), particle filters and variational approaches. Unfortunately, all these methods suffer from some drawbacks: Kalman filters only provide simple Gaussian approximations to the filtering distribution, particle filter estimates have variance which scales poorly with the state dimension and variational techniques do not typically provide consistent estimates. A further approach instead frames the Bayesian update step as a Schrödinger bridge problem. Solving this Schrödinger bridge problem results in a transition density which directly transforms a collection of particles approximating the previous filtering distribution into an approximation of the current filtering distribution without the need for any importance sampling weights (which are required in a particle filter). This approach has been acknowledged for a number of years, but it is only recently that the computational tools which make this approach practicable have emerged. These computational tools include methods for approximating diffusion processes under time reversals and conditioning modifications. The aim of this project is to use these recent computational tools for solving the Schrödinger bridge problem to develop novel filtering methodology, and thus answer the question of 'how to efficiently perform filtering by solving the Schrödinger bridge problem?'The specific objectives of this project begin by refining existing theoretical frameworks for filtering as a Schrödinger bridge problem, including exploring theoretical guarantees and properties of the approach, particularly behaviour in high dimensional state spaces. Various modern computational tools will then be applied to solve the Schrödinger bridge problem and implement the approach in practice. The implementation will be evaluated in comparison to existing approaches such as particle filters. A key objective of the approach is to achieve better performance in high dimensional state spaces than particle filters, so this will be emphasised during the comparisons. Success in these objectives would increase the scope of filtering methods to high dimensional non-linear filtering problems, therefore impacting various application and research areas.This project falls within the EPSRC 'control engineering' and EPSRC 'statistics and applied probability' research areas.