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Higher-order Riemannian spline interpolation problems: a unified approach by gradient flows

Chun-Chi Lin, Dung The Tran

TL;DR

This work develops a unified gradient-flow framework for higher-order Riemannian spline interpolation on manifolds by modeling two linked problems as a network of curves and applying nonlinear parabolic PDEs. The authors prove global existence of classical solutions in Hölder spaces for a penalized energy $\oldsymbol{\mathcal{E}}_{k}^{\lambda,\sigma}$, and show that time-asymptotic limits provide solutions to the variational spline interpolation versus least-squares fitting, thereby resolving Problem 2 and establishing a penalty-method pathway to Problem 1. A two-stage linearization enables the use of Solonnikov’s parabolic theory to obtain local and then global results, with a careful compatibility framework ensuring regularity up to the initial time. Finally, the penalty method is shown to recover exact interpolation in the limit $\,\sigma\to 0$, providing a constructive route to compute geometric $k$-splines and informing numerical schemes for data on curved manifolds.

Abstract

This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks, providing a novel framework for tackling these challenges. This method notably extends to the variational spline interpolation problem on Lie groups, which is frequently encountered in mechanical optimal control theory. As a result, our work contributes to both geometric control theory and statistical shape data analysis. We rigorously prove the existence of global solutions in Hölder spaces for the gradient flow and demonstrate that the asymptotic limits of these solutions validate the existence of solutions to the variational spline interpolation problem. This constructive proof also offers insights into potential numerical schemes for finding such solutions, reinforcing the practical applicability of our approach.

Higher-order Riemannian spline interpolation problems: a unified approach by gradient flows

TL;DR

This work develops a unified gradient-flow framework for higher-order Riemannian spline interpolation on manifolds by modeling two linked problems as a network of curves and applying nonlinear parabolic PDEs. The authors prove global existence of classical solutions in Hölder spaces for a penalized energy , and show that time-asymptotic limits provide solutions to the variational spline interpolation versus least-squares fitting, thereby resolving Problem 2 and establishing a penalty-method pathway to Problem 1. A two-stage linearization enables the use of Solonnikov’s parabolic theory to obtain local and then global results, with a careful compatibility framework ensuring regularity up to the initial time. Finally, the penalty method is shown to recover exact interpolation in the limit , providing a constructive route to compute geometric -splines and informing numerical schemes for data on curved manifolds.

Abstract

This paper addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks, providing a novel framework for tackling these challenges. This method notably extends to the variational spline interpolation problem on Lie groups, which is frequently encountered in mechanical optimal control theory. As a result, our work contributes to both geometric control theory and statistical shape data analysis. We rigorously prove the existence of global solutions in Hölder spaces for the gradient flow and demonstrate that the asymptotic limits of these solutions validate the existence of solutions to the variational spline interpolation problem. This constructive proof also offers insights into potential numerical schemes for finding such solutions, reinforcing the practical applicability of our approach.
Paper Structure (17 sections, 17 theorems, 239 equations, 6 figures)

This paper contains 17 sections, 17 theorems, 239 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Approach
  3. Our Contribution
  4. The Structure of Article
  5. Higher-order Riemannian Spline Interpolation Problems
  6. The Spline Interpolation on Riemannian Manifolds
  7. The Spline Interpolation versus Least-Squares Fitting on Riemannian Manifolds
  8. Gradient Flow for Networks on Riemannian Manifolds
  9. The Nonlinear Parabolic Systems for Networks
  10. The Induced Linear Problem
  11. Local Solutions of Theorem \ref{['thm:Main_Thm-fitting']}
  12. Global Solutions and Proof of Theorem \ref{['thm:Main_Thm-fitting']}
  13. The Proof of Theorem \ref{['thm:Main_Thm-fitting-appl']}
  14. Appendices
  15. Notation and Terminology
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\ell \in \mathbb{N}_0$, $\alpha_j \in (0,1)$, and $\varrho_j=\ell+\alpha_j$, $j\in\{1,2\}$, such that $\alpha_1>k \alpha_2$. Assume that $M$ is a smooth $m$-dimensional Riemannian manifold, which is either compact without boundary or complete with uniformly bounded Riemannian curvature tensors Furthermore, regarding the asymptotics, there exists a pair $(\gamma_{\infty}, \chi_{1, \infty}, \l

Figures (6)

  • Figure 1: The orientation of $g$ and $h$ as $q=3$.
  • Figure 2: The matrix $\mathcal{B}(y^\ast=0, \partial_{y})$
  • Figure 3: The matrix $\mathcal{B}(y^\ast=1, \partial_{y})$
  • Figure 4: The matrix $\mathcal{B}(y^\ast=0, t, \partial_{y}, \partial_{t})$ when $q=5$.
  • Figure 5: The matrix $\mathcal{B}(y^\ast=1, t, \partial_{y}, \partial_{t})$ when $q=5$.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 3.2: The Compatibility Conditions of Order $\ell$ to the Parabolic System for Networks in \ref{['AP-q-even-fitting']} and \ref{['BC-q-even-fitting']}
  • Definition 3.3: The Geometric Compatibility Conditions of Order $\ell$ to the Intrinsic Geometrical Parabolic System for Networks in \ref{['eq:E_s-flow-fitting']}$\sim$ \ref{['eq:BC-higher-order-5-fitting']}
  • Remark 3.4: The Geometric Compatibility Conditions of Order Zero to \ref{['eq:E_s-flow-fitting']}$\sim$\ref{['eq:BC-higher-order-5-fitting']}
  • Lemma 3.5: Solutions to the linear parabolic system \ref{['eq:higher-order-linear-g_l']}
  • proof
  • ...and 37 more