Transport Techniques for Optimal Filtering

最优滤波，也被称为数据同化，是一个实时估计问题的当前状态的动力系统的利益。目的是以最佳方式将来自系统的理论模型的联合收割机预测与系统的实时观测相结合。这个问题陈述非常广泛，应用范围广泛，如天气预测，金融预测，目标跟踪和机器人定位。这些应用领域需要滤波技术，其可以实时（也称为在线）执行并且在高维状态空间中给出良好的性能，优选地具有理论保证。正是这些要求通常对当前最先进的过滤技术提出了最大的挑战。关于过滤的一个常见观点是贝叶斯推理，它给出了一种概率方法，其中同化步骤是贝叶斯定理的应用。然而，贝叶斯推理也有自己的挑战，因为感兴趣的对象，即后验分布，除了最简单的情况外，在所有情况下都是难以处理的。在过滤社区的一个重要部分的工作涉及到开发计算技术的顺序贝叶斯推理，以及在实时，高维设置。值得注意的方法包括卡尔曼滤波器（及其扩展），粒子滤波器和变分方法。不幸的是，所有这些方法都有一些缺点：卡尔曼滤波器只提供简单的高斯近似的过滤分布，粒子滤波器估计的方差与状态维度和变分技术的规模差，通常不提供一致的估计。另一种方法将贝叶斯更新步骤框定为薛定谔桥问题。解决这个薛定谔桥问题的结果是一个过渡密度，它直接将近似于前一个滤波分布的粒子集合转换为当前滤波分布的近似值，而不需要任何重要性采样权重（粒子滤波器中需要）。这种方法已经被承认了很多年，但直到最近才出现了使这种方法切实可行的计算工具。这些计算工具包括近似扩散过程下的时间反转和条件修改的方法。该项目的目的是使用这些最新的计算工具来解决薛定谔桥问题，以开发新的滤波方法，从而回答“如何通过解决薛定谔桥问题有效地执行滤波？“这个项目的具体目标开始通过完善现有的理论框架过滤作为一个薛定谔桥问题，包括探索理论保证和属性的方法，特别是在高维状态空间的行为。各种现代计算工具将被应用于解决薛定谔桥问题，并在实践中实施的方法。将评估的实施相比，现有的方法，如粒子滤波器。该方法的一个关键目标是在高维状态空间中实现比粒子滤波器更好的性能，因此在比较过程中将强调这一点。这些目标的成功将增加高维非线性滤波问题的滤波方法的范围，从而影响各种应用和研究领域。本项目福尔斯EPSRC“控制工程”和EPSRC“统计和应用概率”研究领域。

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.