[Paper Review] Hyperbolic development and inversion of signature

Terry Lyons, Weijun Xu|arXiv (Cornell University)|Jul 1, 2015

Quantum chaos and dynamical systems7 references4 citations

TL;DR

This paper presents a novel method to reconstruct any piecewise linear path from its signature using hyperbolic development onto hyperbolic space. By analyzing the asymptotic behavior of the path's development as a parameter λ→∞, the method recovers the path's direction and length of each linear segment exponentially fast, enabling exact inversion of the signature map for piecewise linear paths.

ABSTRACT

We develop a simple procedure that allows one to explicitly reconstruct any piecewise linear path from its signature. The construction is based on the development of the path onto the hyperbolic space.

Motivation & Objective

• To develop an explicit inversion procedure that reconstructs a piecewise linear path from its signature, a fundamental problem in signature theory.
• To provide a geometric and analytic framework based on hyperbolic space development to achieve path reconstruction, independent of prior symmetrization-based methods.
• To establish that the direction and length of each linear segment can be recovered from the signature via asymptotic analysis of the hyperbolic development.
• To prove that the reconstruction error decays exponentially with respect to a scaling parameter λ, ensuring high-precision recovery.

Proposed method

• The method uses the hyperbolic development of a path onto the hyperbolic space H^d, where the path is reparametrized and scaled by a parameter λ.
• It applies a sequence of Lorentz transformations corresponding to each linear segment of the path, updating the development endpoint recursively using hyperbolic trigonometric identities.
• The key step involves analyzing the asymptotic behavior of the development as λ→∞, showing that the direction vector η_λ,k converges to the true segment direction θ_k exponentially fast.
• The method relies on induction over segments, using bounds on hyperbolic norms and sinh/cosh identities to control the convergence rate.
• It exploits the fact that the signature is invariant under reparametrization, allowing the use of natural parametrization on [0,1] for simplicity.
• The reconstruction of segment length is achieved via the logarithmic decay rate of |η_λ,n − θ_n|, which yields lim_{λ→∞} (1/λ) log|η_λ,n − θ_n| = −l_n.

Experimental results

Research questions

• RQ1 Can a piecewise linear path be reconstructed exactly from its signature using geometric methods?
• RQ2 What is the asymptotic behavior of the hyperbolic development of a path as the scaling parameter λ→∞?
• RQ3 How fast can the direction of each linear segment be recovered from the signature?
• RQ4 Is the length of each segment recoverable from the signature using this geometric inversion procedure?

Key findings

• The direction of each linear segment is recovered exponentially fast: |η_λ,k − θ_k| = O(e^{−λ l_k}) as λ→∞.
• The logarithmic decay rate of the reconstruction error gives the exact length of the last segment: lim_{λ→∞} (1/λ) log|η_λ,n − θ_n| = −l_n.
• The method successfully reconstructs the entire piecewise linear path from its signature using only the hyperbolic development process and asymptotic analysis.
• The convergence is uniform and independent of the path’s initial configuration, provided the segments are not collinear.
• The reconstruction is exact in the limit λ→∞, with no dependence on the path’s initial position or orientation.
• The method does not rely on Chen’s identity or symmetrization, offering a new, geometrically intuitive inversion framework.