[Paper Review] Uniqueness of signature for simple curves

Horatio Boedihardjo, Hao Ni+1 more|arXiv (Cornell University)|Apr 2, 2013

Topological and Geometric Data Analysis26 references3 citations

TL;DR

This paper establishes that the signature of planar simple curves with finite $p$-variation ($1 \leq p < 2$) uniquely determines the curve, by showing that certain coefficients in the log-signature series correspond to moments of the curve's winding number. Using a topological approach, the authors prove uniqueness outside a Chordal SLE$_\kappa$ null set for $0 < \kappa \leq 4$, and express SLE $n$-point functions via expected signatures.

ABSTRACT

We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1\leq p&lt;2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLE_κ null set, where 0

Motivation & Objective

• To determine the winding number of a curve from its signature using a topological approach.
• To prove that the signature of sufficiently regular simple planar curves uniquely determines the curve.
• To establish a connection between signature coefficients and moments of the winding number.
• To express the Fourier transform of SLE $n$-point functions in terms of expected signatures.
• To provide a deterministic framework for studying SLE curves through sample path signatures.

Proposed method

• Uses Lyndon basis elements in the free Lie algebra generated by $\{\mathbf{e}_1, \mathbf{e}_2\}$ to analyze the log-signature series.
• Derives that coefficients of $\mathcal{P}_{\mathbf{e}_1^{\otimes(n+1)} \otimes \mathbf{e}_2^{\otimes(k+1)}}$ in the truncated log-signature are moments of the winding number.
• Applies the formula $\int_{\mathbb{R}^2} \frac{x^n y^k}{n!k!} \eta(\gamma - \gamma_0, (x,y)) \, dx\,dy$ to relate signature terms to winding number integrals.
• Employs the fact that all Lyndon words of degree $\leq 4$ are of the form $\mathbf{e}_1^{\otimes n} \otimes \mathbf{e}_2^{\otimes k}$, explaining why first four signature terms are determined by winding number.
• Uses the sharpness of the degree-5 Lyndon word $\mathbf{e}_1 \otimes \mathbf{e}_2 \otimes \mathbf{e}_1 \otimes \mathbf{e}_2^{\otimes 2}$ to show higher-order terms are not determined by winding number alone.
• Connects expected signature of SLE curves to the Fourier transform of $n$-point functions via complex analysis and conformal invariance.

Experimental results

Research questions

• RQ1 Can the winding number of a planar curve be recovered from its signature?
• RQ2 To what extent can the signature of a curve be reconstructed from its winding number?
• RQ3 What is the minimal order of signature terms that encode information beyond the winding number?
• RQ4 How do expected signatures of SLE curves relate to their $n$-point functions?
• RQ5 Can deterministic signature theory provide a framework for studying random SLE paths?

Key findings

• The coefficients of Lyndon basis elements $\mathcal{P}_{\mathbf{e}_1^{\otimes(n+1)} \otimes \mathbf{e}_2^{\otimes(k+1)}}$ in the log-signature are equal to $(-1)^k \int_{\mathbb{R}^2} \frac{x^n y^k}{n!k!} \eta(\gamma - \gamma_0, (x,y)) \, dx\,dy$, which are moments of the winding number.
• The first four terms of the log-signature of a closed curve with finite $p$-variation ($1 \leq p < 2$) are fully determined by the winding number function.
• The number 'four' is sharp: there exist curves with identical winding numbers but differing fifth-order signature terms, due to non-trivial Lyndon words of degree 5.
• Outside a Chordal SLE$_\kappa$ null set for $0 < \kappa \leq 4$, the signature uniquely determines the curve.
• The Fourier transform of the $n$-point function of SLE curves is expressible in terms of the expected signature of SLE paths.
• The expected signature of SLE curves provides a deterministic representation of their stochastic $n$-point functions, enabling analysis via rough path theory.