Topologies on unparameterised rough path space

TL;DR

This paper extends the study of topologies on unparameterised path space from paths of bounded variation to weakly geometric rough paths, analyzing various topological properties and their implications for signature-based methods.

Contribution

It generalizes previous results to rough paths, examining three classes of topologies and their properties, including separability, Lusin property, and conditions for being Polish.

Findings

Metrisable topologies are separable and Lusin but not locally compact.

The quotient topology is Hausdorff but not metrisable.

The explicit metric topology is not complete and not locally compact; it is Polish when p=1.

Abstract

The signature of a $p$-weakly geometric rough path summarises a path up to a generalised notion of reparameterisation. The quotient space of equivalence classes on which the signature is constant yields unparameterised path space. The study of topologies on unparameterised path space, initiated in [CT24b] for paths of bounded variation, has practical bearing on the use of signature based methods in a variety applications. This note extends the majority of results from [CT24b] to unparameterised weakly geometric rough path space. We study three classes of topologies: metrisable topologies for which the quotient map is continuous; the quotient topology derived from the underlying path space; and an explicit metric between the tree-reduced representatives of each equivalence class. We prove that topologies of the first type (under an additional assumption) are separable and Lusin, but not locally compact or completely metrisable. The quotient topology is Hausdorff but not metrisable, while the metric generating the third topology is not complete and its topology is not locally compact. We also show that the third topology is Polish when $p=1$.