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Shortest-path recovery from signature with an optimal control approach

Marco Rauscher, Alessandro Scagliotti, Felipe Pagginelli Patricio

TL;DR

This work treats the problem of reconstructing the shortest path consistent with a given signature as a sub-Riemannian geodesic problem on the signature manifold $G^N(\mathbb{R}^d)$. The authors embed the problem into optimal control with dynamics $\dot{\xi}=\sum_{i=1}^d a_i U_i(\xi)$ and connect path-length to the control energy, establishing a $\Gamma$-convergence bridge from a soft-endpoint formulation to the hard-endpoint geodesic problem. They derive a Pontryagin Maximum Principle framework for signatures and implement a Sakawa-type iterative scheme, including variants with final-time optimization and with additional bracket vector fields, to numerically compute length-minimizing signatures and recover the associated $H^1$ paths. Numerical experiments validate the approach, showing effective recovery of shortest-signature-preserving paths and revealing the practical sparsity of higher-order bracket directions. The results provide a scalable, geometry-driven method for inverse signature problems with potential applications in stochastic-process path inference and signature-based learning, without restrictive assumptions on signature order or path dimension.

Abstract

In this paper, we consider the signature-to-path reconstruction problem from the control theoretic perspective. Namely, we design an optimal control problem whose solution leads to the minimal-length path that generates a given signature. In order to do that, we minimize a cost functional consisting of two competing terms, i.e., a weighted final-time cost combined with the $L^2$-norm squared of the controls. Moreover, we can show that, by taking the limit to infinity of the parameter that tunes the final-time cost, the problem $Γ$-converges to the problem of finding a sub-Riemannian geodesic connecting two signatures. Finally, we provide an alternative reformulation of the latter problem, which is particularly suitable for the numerical implementation.

Shortest-path recovery from signature with an optimal control approach

TL;DR

This work treats the problem of reconstructing the shortest path consistent with a given signature as a sub-Riemannian geodesic problem on the signature manifold . The authors embed the problem into optimal control with dynamics and connect path-length to the control energy, establishing a -convergence bridge from a soft-endpoint formulation to the hard-endpoint geodesic problem. They derive a Pontryagin Maximum Principle framework for signatures and implement a Sakawa-type iterative scheme, including variants with final-time optimization and with additional bracket vector fields, to numerically compute length-minimizing signatures and recover the associated paths. Numerical experiments validate the approach, showing effective recovery of shortest-signature-preserving paths and revealing the practical sparsity of higher-order bracket directions. The results provide a scalable, geometry-driven method for inverse signature problems with potential applications in stochastic-process path inference and signature-based learning, without restrictive assumptions on signature order or path dimension.

Abstract

In this paper, we consider the signature-to-path reconstruction problem from the control theoretic perspective. Namely, we design an optimal control problem whose solution leads to the minimal-length path that generates a given signature. In order to do that, we minimize a cost functional consisting of two competing terms, i.e., a weighted final-time cost combined with the -norm squared of the controls. Moreover, we can show that, by taking the limit to infinity of the parameter that tunes the final-time cost, the problem -converges to the problem of finding a sub-Riemannian geodesic connecting two signatures. Finally, we provide an alternative reformulation of the latter problem, which is particularly suitable for the numerical implementation.
Paper Structure (16 sections, 10 theorems, 80 equations, 5 figures, 1 algorithm)

This paper contains 16 sections, 10 theorems, 80 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries on Signatures
  3. Sub-Riemanian Structure
  4. Carnot-Caratheodory Metric
  5. Embedding in Optimal Control Theory
  6. Problem Formulation
  7. Relaxing the endpoint constraint through $\Gamma$-convergence
  8. Pontryagin's Maximum Principle for signatures
  9. Finding the Shortest Path
  10. Version with Final Variable Time
  11. Version with Additional Vector Fields
  12. Numerical experiments
  13. Minimizing the energy functional
  14. Solving the final variable time problem
  15. Minimizing the Version with with Additional Vector Fields
  16. ...and 1 more sections

Key Result

Theorem 2.1

For $x\in H^1([0,1],\mathbb{R}^d)$ such that $a_i := [\dot{x}]_i$, the curve of signatures, $\xi(\cdot):=S_N(x)_{0,\cdot}$, solves the following ODE where the $d$ vector fields $U_1,\ldots,U_d$ are defined as

Figures (5)

  • Figure 1: First two steps of the Algorithm \ref{['alg:loop']}
  • Figure 2: Visualization of both trajectories of a 2D simulation of the OU process and the related shortest path
  • Figure 3: Visualization of both trajectories plotted against each other of a 2D simulation of the OU process and the related shortest path
  • Figure 4: Visualization of trajectories of a 3D simulation of the OU process (left) and the related shortest path (right). The graphs show the values of the components ($y$-axis) versus time ($x$-axis).
  • Figure 5: Visualization of all trajectories of a 3D simulation of the OU process and the related shortest path, where we plot all trajectories against each other.

Theorems & Definitions (33)

  • Definition 1: Signature
  • Theorem 2.1
  • proof
  • Theorem 2.2: Chen
  • proof
  • Definition 2: Bracket
  • Definition 3: Nilpotent Lie algebra of order $N$
  • Definition 4: Exponential
  • Theorem 2.3
  • proof
  • ...and 23 more