Reconstruction for the Signature of a Rough Path

TL;DR

This paper develops an explicit, universal method to reconstruct a rough path from its signature, addressing the gap in understanding how to recover the original path from its signature.

Contribution

It provides a constructive approach to invert the signature map, enabling the recovery of rough paths from their signatures in a general setting.

Findings

Provides an explicit reconstruction method for rough paths from signatures

Establishes universality of the reconstruction approach

Advances understanding of the inverse signature problem

Abstract

Recently it was proved that the group of rough paths modulo tree-like equivalence is isomorphic to the corresponding signature group through the signature map S (a generalized notion of taking iterated path integrals). However, the proof of this uniqueness result does not contain any information on how to "see" the trajectory of a (tree-reduced) rough path from its signature, and a constructive understanding on the uniqueness result (in particular on the inverse of S) has become an interesting and important question. The aim of the present paper is to reconstruct a rough path from its signature in an explicit and universal way.