Signature transformation of paths from rough analysis

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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.