BibTex format
@unpublished{Cass:2021,
author = {Cass, T and Turner, WF and Messadene, R},
title = {Signature asymptotics, empirical processes, and optimal transport},
url = {http://arxiv.org/abs/2107.11203v4},
year = {2021}
}
Publications
Citation
RIS format (EndNote, RefMan)
TY - UNPB
AB - Rough path theory provides one with the notion of signature, a graded familyof tensors which characterise, up to a negligible equivalence class, andordered stream of vector-valued data. In the last few years, use of thesignature has gained traction in time-series analysis, machine learning , deeplearning and more recently in kernel methods. In this article, we lay down thetheoretical foundations for a connection between signature asymptotics, thetheory of empirical processes, and Wasserstein distances, opening up thelandscape and toolkit of the second and third in the study of the first. Ourmain contribution is to show that the Hambly-Lyons limit can be reinterpretedas a statement about the asymptotic behaviour of Wasserstein distances betweentwo independent empirical measures of samples from the same underlyingdistribution. In the setting studied here, these measures are derived fromsamples from a probability distribution which is determined by geometricalproperties of the underlying path. The general question of rates of convergencefor these objects has been studied in depth in the recent monograph of Bobkovand Ledoux. By using these results, we generalise the original result of Hamblyand Lyons from $C^3$ curves to a broad class of $C^2$ ones. We conclude byproviding an explicit way to compute the limit in terms of a second-orderdifferential equation.
AU - Cass,T
AU - Turner,WF
AU - Messadene,R
PY - 2021///
TI - Signature asymptotics, empirical processes, and optimal transport
UR - http://arxiv.org/abs/2107.11203v4
ER -