Transport Techniques for Optimal Filtering

最优滤波，也称为数据同化，是对感兴趣的动力系统当前状态的实时估计问题。其目的是将系统理论模型的预测与系统的实时观测以最佳方式结合起来。这一问题陈述非常广泛，应用广泛，包括天气预报、金融预报、目标跟踪和机器人定位。这些应用领域需要可以实时(也称为在线)执行的过滤技术，并在高维状态空间中提供良好的性能，最好有理论上的保证。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，贝叶斯推断有其自身的挑战，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。过滤领域的大量工作与开发用于顺序贝叶斯推理的计算技术有关，该技术在实时、高维环境中表现良好。值得注意的方法包括卡尔曼过滤器(及其扩展)、粒子过滤器和变分方法。不幸的是，所有这些方法都有一些缺点：卡尔曼滤波只提供对滤波分布的简单高斯近似，粒子滤波估计具有随状态维度变化很小的方差，而变分技术通常不能提供一致的估计。相反，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会产生一个过渡密度，它直接将近似先前过滤分布的粒子集合转换为当前过滤分布的近似，而不需要任何重要的采样权重(粒子过滤器中需要的)。这种方法已经被承认多年了，但直到最近才出现了使这种方法可行的计算工具。这些计算工具包括在时间反转和条件化修改下近似扩散过程的方法。这个项目的目的是利用这些解决薛定谔桥问题的最新计算工具来开发新的滤波方法，从而回答“如何通过解决薛定谔桥问题来有效地执行滤波？”这个项目的具体目标首先将现有的滤波理论框架提炼为薛定谔桥问题，包括探索该方法的理论保证和性质，特别是在高维状态空间中的行为。然后将应用各种现代计算工具来解决薛定谔桥问题，并在实践中实施该方法。将与粒子过滤器等现有方法进行比较，对实现进行评估。该方法的一个关键目标是在高维状态空间中获得比粒子过滤器更好的性能，因此在比较过程中将强调这一点。这些目标的成功将扩大高维非线性滤波问题的滤波方法范围，从而影响到各种应用和研究领域。本项目属于EPSRC控制工程和EPSRC统计与应用概率研究领域。

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.