Signature transformation of paths from rough analysis

简要描述研究的背景，包括潜在的影响。路径的数学概念抓住了连续时间排序的值序列的概念。这些对象和它们的推广，在纯数学和应用数学中广泛存在。例如，随机过程的样本路径分析是随机分析的重要组成部分，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，并且期望寻求捕获感兴趣的一些特征的低维摘要。近年来，一种实现这一目标的原则性方法越来越突出。这种方法涉及到使用（路径）签名变换，与传统的基于采样的方法不同，其根源在于通过理解路径对任何光滑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，在这个意义上，来自$p$-变化路径的每个连续实值函数可以被应用于路径签名的线性函数任意地很好地近似：路径函数的学习成为签名上的线性回归。此外，由于$n$阶签名项的范数因子地衰减，高阶项往往非常小，并且可以自然地被省略并且签名被截断，使得截断的签名变换成为路径的自然且易于处理的有限维表示。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）。研究方法的新奇所有方法都是新颖的，因为它们是在上述领域中利用签名变换的第一种方法。原因有二：首先，签名变换可以是从业者理解的复杂数学工具，因为它支持与控制微分方程、粗略路径分析和统计学的理论相关的复杂纯数学。其次，签名方法在数据科学中的应用最近才开始与EPSRC的战略和研究领域保持一致本项目福尔斯属于EPSRC随机系统数学（EP/S 023925/1）研究领域，其中统计和应用概率以及数学分析是一些主题或研究领域（https：//epsrc.ukri.org/research/ourportfolio/researchareas/）。任何公司或合作者参与我的主管：托马斯卡斯和丹克里桑。“签名变换和最佳传输”和“签名变换和序列聚类”是与托马斯卡斯的联合项目。“Signature transform and NLP”是与Cris Salvi的合作项目。“Signature transform and RL”是与Lingyi Yang和Cris Salvi的联合项目。

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.