Signature asymptotics, empirical processes, and optimal transport

TL;DR

This paper connects signature asymptotics with empirical processes and Wasserstein distances, extending Hambly-Lyons limit to broader classes of paths and providing a way to compute the limit via differential equations.

Contribution

It establishes a theoretical link between signature asymptotics, empirical process theory, and Wasserstein distances, broadening the Hambly-Lyons limit to less smooth paths.

Findings

Reinterpreted Hambly-Lyons limit through Wasserstein distances.

Extended convergence results from $C^3$ to $C^2$ paths.

Provided an explicit method to compute the limit using differential equations.

Abstract

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning , deep learning and more recently in kernel methods. In this article, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path. The general question of rates of convergence for these objects has been studied in depth in the recent monograph of Bobkov and Ledoux. By using these results, we generalise the original result of Hambly and Lyons from $C^3$ curves to a broad class of $C^2$ ones. We conclude by providing an explicit way to compute the limit in terms of a second-order differential equation.