Signature transformation of paths from rough analysis

对研究背景的简要描述，包括潜在的影响路径的数学概念抓住了连续时间有序的值序列的概念。这些对象及其泛化在纯数学和应用数学中广泛存在。例如，对随机过程的样本路径的分析构成了随机分析的重要部分，而时间序列分析是现代统计学中的一个既定工具。抽象路径本质上是无限维的对象，希望寻找能够捕获一些感兴趣的特征的低维摘要。近年来，实现这一目标的一种数学原则性方法变得突出起来。这种方法涉及使用(路径)特征变换，与传统的基于采样的方法不同，它的根源在于通过了解路径对任何光滑的非线性受控微分系统的影响来捕获路径。例如，签名变换捕获了底层路径的所有非线性，因为来自$p$变化路径的每个连续实值函数都可以被应用于路径签名的线性函数任意很好地逼近：路径函数的学习变成了对签名的线性回归。此外，由于$n$阶签名项的范数按因子衰减，高阶项往往非常小，可以自然地省略并且签名被截断，使得截断签名变换成为路径的自然和易处理的有限维表示。目的和对象本研究的目的是利用签名变换在与时间序列分析和数据科学相关的几个领域中的性质，例如(1)最优传输，(2)序列聚类，(3)自然语言处理(NLP)，和(4)强化学习(RL)。研究方法的新颖性这些方法都是新颖的，因为它们是在上述领域中最早探索签名变换的方法之一。原因有两个：首先，签名变换可能是从业者难以理解的复杂数学工具，因为它支撑着与受控微分方程组、粗路径分析和统计理论相关的复杂纯数学。其次，签名方法在数据科学中的利用最近才开始(不到10年前)，并仍在扩展。与EPSRC的战略和研究领域保持一致本项目属于EPSRC随机系统数学(EP/S023925/1)研究领域，其中统计学、应用概率和数学分析是(https://epsrc.ukri.org/research/ourportfolio/researchareas/).Any公司或合作者参与的一些主题或研究领域我的主管：Thomas Cass和Dan Crisan。《签名变换与最优传输》和《签名变换与序列聚类》是与托马斯·卡斯联合开展的项目。《Signature Transform and NLP》是与克瑞斯·萨尔维的合作项目。《Signature Transform and RL》是杨凌毅和克丽丝·萨尔维的合作项目。

Brief description of the context of the research including potential impactThe mathematical notion of a path captures the concept of a continuously time-ordered sequence of values. These objects and their generalisations, occur widely throughout both pure and applied mathematics. For example, the analysis of the sample paths of a stochastic process forms a significant part of stochastic analysis, while time series analysis is an established tool in modern statistics. Abstract paths are inherently infinite-dimensional objects, and it is desirable to seek low-dimensional summaries which capture some features of interest. A mathematically-principled approach to effecting this has gained prominence in recent years. This approach involves using the (path) signature transform which, in distinction to traditional methods based on sampling, is rooted in capturing the path by understanding its effects on any smooth non-linear controlled differential system.Representing paths in terms of signatures also offer several computational advantages. For example, the signature transform captures all the non-linearity of the underlying path in the sense that every continuous real-valued function from $p$-variation paths can be arbitrarily well approximated by a linear function applied on the path signature: the learning of functions of path become linear regressions on signatures. Also, because the norm of the $n$-order signature terms decay factorially, higher order terms tend to be very small and can be naturally left out and the signature truncated, making the truncated signature transform a natural and tractable finite dimensional representation of paths.Aims and objectivesThe goal of this research is to leverage the properties of the signature transform in several areas related to time-series analysis and data science, such as (1) optimal transport, (2) sequence clustering, (3) natural language processing (NLP), and (4) reinforcement learning (RL).Novelty of the research methodologyAll the methodologies are novel as they are among the first ones to exploit the signature transform in the above mentioned fields. The reasons are twofold: first, the signature transform can be a complex mathematical tool to understand for practitioners as it underpins complex pure mathematics related to the theory of controlled differential equations, rough path analysis and statistics. Second, the leveraging of the signature method in data science has only recently started (less than 10 years ago) and is still spreading out.Alignment to EPSRC's strategies and research areasThis project falls within the EPSRC Mathematics of Random Systems (EP/S023925/1) research area' where Statistics and applied probability and Mathematical analysis are some of the themes or research areas (https://epsrc.ukri.org/research/ourportfolio/researchareas/).Any companies or collaborators involvedMy supervisors: Thomas Cass and Dan Crisan. "Signature transform and optimal transport" and "Signature transform and sequence clustering" is a joint project with Thomas Cass. "Signature transform and NLP" is a joint project with Cris Salvi. "Signature transform and RL" is a joint project with Lingyi Yang and Cris Salvi.