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2014

Inverting the signature of a path

Lyons, Terry, Xu, Weijun

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The aim of this article is to develop an explicit procedure that enables one to reconstruct any $C^1$ path (at natural parametrization) from its signature.

We also explicitly quantify the distance between the reconstructed path and the original path in terms of the number of terms in the signature that are used for the construction and the modulus of continuity of the derivative of the path.
A key ingredient in the construction is the use of a procedure of symmetrization that separates the behavior of the path at small and large scales.

Inverting the signature of a path · Around

Built on

Iterated integrals and exponential homomorphisms
K.-T. Chen · 1954
Earlier work this paper cites.
Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula
K.-T. Chen · 1957
Earlier work this paper cites.
Integration of paths - a faithful representation of paths by noncommutative formal power series
K.-T. Chen · 1958
Earlier work this paper cites.
Differential equations driven by rough signals
T. Lyons · 1998
Earlier work this paper cites.

Similar

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The uniqueness of signature problem in the non-Markov setting
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The signature of a rough path: uniqueness
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Then

On an inversion theorem for Stratonovich’s signatures of multidimensional diffusion paths
X. Geng · 2016
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Signature inversion for monotone paths
J. Chang · 2017
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Reconstruction for the signature of a rough path
X. Geng · 2017
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Hyperbolic development and inversion of the signature
T. Lyons · 2017
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