Transport Techniques for Optimal Filtering

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.

The primary CDT impact will be training 75 PhD graduates as the next generation of leaders in statistics and statistical machine learning. These graduates will lead in industry, government, health care, and academic research. They will bridge the gap between academia and industry, resulting in significant knowledge transfer to both established and start-up companies. Because this cohort will also learn to mentor other researchers, the CDT will ultimately address a UK-wide skills gap. The students will also be crucial in keeping the UK at the forefront of methodological research in statistics and machine learning.
After graduating, students will act as multipliers, educating others in advanced methodology throughout their career. There are a range of further impacts:
- The CDT has a large number of high calibre external partners in government, health care, industry and science. These partnerships will catalyse immediate knowledge transfer, bringing cutting edge methodology to a large number of areas. Knowledge transfer will also be achieved through internships/placements of our students with users of statistics and machine learning.
- Our Women in Mathematics and Statistics summer programme is aimed at students who could go on to apply for a PhD. This programme will inspire the next generation of statisticians and also provide excellent leadership training for the CDT students.
- The students will develop new methodology and theory in the domains of statistics and statistical machine learning. It will be relevant research, addressing the key questions behind real world problems. The research will be published in the best possible statistics journals and machine learning conferences and will be made available online. To maximize reproducibility and replicability, source code and replication files will be made available as open source software or, when relevant to an industrial collaboration, held as a patent or software copyright.