Computing and inverting the signatures of rough paths

路径模拟变量在特定状态空间中的演化。状态空间可以表示物理量，如气体粒子的位置，或数据，如未来的海平面。这些例子的一个共同特征是它们是随机过程。因为在任何时候，一条随机路径都可以向任何方向移动，所以它的轨迹通常是不稳定的，也不是平滑的。微积分的卓越理论已经被发展出来，用来描述这些振荡路径是如何相互影响的。第一个重大成功是伊藤的理论适用于系统驱动的布朗运动，一个典型的数学模型的随机粒子运动。另一个突破发生在20世纪90年代末，粗糙路径理论的出现。与伊藤的构造不同，粗糙路径理论能够处理比布朗运动更不规则的方向运动的路径。它还导致了表面生长建模的突破，这一成就在2014年被授予菲尔兹奖给Martin Hairer。与此同时，粗糙路径理论的许多成功的应用已经建立，从新的数值和统计方法，国际获奖的中文手写识别算法。这些应用程序中的大多数都使用一种称为签名的工具来分析不规则路径。签名是专门构建的，用于描述在例如正方形中随机移动的路径，它们可以填充整个正方形。签名的第一项捕获路径的一维方面，例如位移。第二项代表二维的方面，例如面积等。在签名中的连续项将告诉我们关于路径的越来越高维的信息。签名具有复杂的结构，这意味着许多基本问题尚未解决。例如：问题1：如何计算随机路径的签名的平均值？问题2：签名与路径的其他关键特征有什么关系？随着基于粗糙路径的方法显示出其最初的前景，这些问题已经成为阻碍进一步发展的主要挑战。这种情况是我们当前提案的主要动机。我们将首先研究签名上的函数的性质，而不是直接研究签名。至关重要的是，签名的最新进展使用了这些函数的定性性质。它们的数量方面仍然没有得到充分利用，可能是由于它们的结构复杂。我们将开发新的方法来理解这些结构，利用新的方法来使用其他数学领域的重要工具，包括李代数，双曲几何和随机分析。对问题1和2的研究有望揭示签名与数学中其他重要思想之间的深层关系，例如长度的概念。这是一个值得追求的目标，因为许多数学上的突破都是从两个迄今为止不相关的思想联系起来产生的，费马大定理的证明就是一个著名的例子。该项目的一个关键要素是将我们在粗糙路径理论方面的新成果传播到我们通常的概率论受众之外，因为最大的收益将来自于那些还没有意识到粗糙路径理论及其与他们工作的潜在相关性的人。科学家们已经观察到，许多现实世界的随机过程，如河流流量和股票价格，具有粗糙的路径行为。如果我们能够解决问题1，它将扩展现有的应用程序的签名，这些现实世界的进程。对于问题2，任何进展都将为基于签名的方法为何有效提供重要的见解，并可能导致对使用签名的识别方法等的效率的切实改进。

A path models the evolution of a variable in a certain state space. The state space could represent physical quantities, such as the position of a gas particle, or data such as future sea levels. A common feature in these examples is that they are random processes. Since at each time a random path could move in any direction, its trajectory would be erratic and not smooth in general. Remarkable theories of calculus have been developed to describe how these oscillatory paths affect each other. A first major success was Itô's theory which applies to systems driven by Brownian motion, a canonical mathematical model for random particle motion. Another breakthrough occurred in the late 1990s with the advent of rough path theory. Unlike Itô's construction, rough path theory is able to handle paths that move in much more irregular directions than Brownian motion. It has also led to breakthroughs on the modelling of surface growth, an achievement recognized by the award of the Fields Medal to Martin Hairer in 2014. Meanwhile, many successful applications of rough path theory have been established, ranging from new numerical and statistical methods to an international award-winning algorithm for Chinese handwriting recognition. Most of these applications use a tool, known as the signature, to analyze irregular paths. The signature is purpose-built to describe paths that move so randomly in for example, a square, that they can fill the entire square. The first term of the signature captures the one dimensional aspects of the path, such as the displacement. The second term represents two dimensional aspects such as the area, and so on. Successive terms in the signature will tell us higher and higher dimensional information about the path. The signature has a complex structure and this means that many fundamental problems have remained unresolved. For example: Problem 1: How do we calculate the average values of signatures of random paths?Problem 2: How is the signature related to the other key features of paths? As rough path-based methods demonstrate their initial promise, these problems have emerged as the main challenges hindering further development. This state of affairs is the main motivation for our current proposal.Instead of studying the signature directly, we will first examine the properties of functions on signatures. Crucially, most recent advances on signatures have used the qualitative properties of these functions. Their quantitative aspects have remained underused, possibly due to their complex structure. We will develop new methods for understanding these structures, making novel use of important tools from other areas of mathematics, including Lie algebra, hyperbolic geometry and stochastic analysis.The study of Problems 1 and 2 is expected to reveal the deep relationship between the signature and other important ideas in mathematics, such as the notion of length. This is a worthwhile pursuit because many mathematical breakthroughs were born out of linking two hitherto unrelated ideas, with the proof of Fermat's Last Theorem being a famous example. A key element of this project is to disseminate our new results in rough path theory beyond our usual audience in probability theory, as the biggest gains will come from reaching those who have not been aware of rough path theory and its potential relevance to their work.There will also be impact beyond academia. Scientists have observed that many real-world random processes, such as river flow and stock prices, have rough path behaviour. If we can resolve Problem 1, it will extend the existing applications of signatures to these real-world processes. For Problem 2, any progress will provide crucial insights into why signature-based methods work and could lead to tangible improvements to the efficiency of, for instance, recognition methods that use the signature.