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Signature inversion of $C^1-$axial linear curves

Chong Liu, Shi Wang

Abstract

We introduce a signature inversion scheme for $C^1$-axial linear curves which are widely used in various areas. We show that in the presence of a linear coordinate function, the derivatives of the underlying curve at any point $x$ can be recovered by tracking the signature coefficients $S_{k,l}$ with $\frac{k}{k+l} \to x$. We furthermore give a quantitative estimates for the convergence rate in this inversion scheme and establish a modulus of continuity of the signature inverse $S^{-1}$ under different topologies by using this inversion procedure.

Signature inversion of $C^1-$axial linear curves

Abstract

We introduce a signature inversion scheme for -axial linear curves which are widely used in various areas. We show that in the presence of a linear coordinate function, the derivatives of the underlying curve at any point can be recovered by tracking the signature coefficients with . We furthermore give a quantitative estimates for the convergence rate in this inversion scheme and establish a modulus of continuity of the signature inverse under different topologies by using this inversion procedure.
Paper Structure (9 sections, 23 theorems, 194 equations)

This paper contains 9 sections, 23 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Inversion of axial linear curves
  3. Preliminaries on signatures
  4. The inversion scheme of axial linear curves
  5. Regularity of $S^{-1}$
  6. Preliminaries on topologies in signature theory
  7. The regularity of $S^{-1}$ for the projective norm
  8. The asymptotic norms
  9. Effective bounds and Diophantine approximation

Key Result

Theorem 1.1

For two axial monotone $C^1$-curves $\gamma_1$ and $\gamma_2$ from $[a,b]$ to $\mathbb{R}^d$, they have the same signature $S(\gamma_1) = S(\gamma_2)$ if and only if $\gamma_1$ is a reparameterization of $\gamma_2$.

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.1: a corollary of Lemma 4.6 in BGLY16
  • Theorem 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • Theorem 2.1
  • ...and 42 more