A gradient flow method for smooth splines versus least-squares fitting on Riemannian manifolds

TL;DR

This paper introduces a gradient flow method for solving spline interpolation and least-squares fitting problems on smooth Riemannian manifolds, providing theoretical guarantees and potential numerical algorithms.

Contribution

It offers a rigorous proof of global solutions for the gradient flow approach, bridging geometric control theory and statistical shape analysis.

Findings

Proves existence of global solutions in Hölder spaces.

Establishes asymptotic limits for spline and least-squares solutions.

Suggests potential numerical schemes based on the constructive proof.

Abstract

This article presents a novel resolution to the problem of spline interpolation versus least-squares fitting on smooth Riemannian manifolds utilizing the method of gradient flows of networks. This approach represents a contribution to both geometric control theory and statistical shape data analysis. Our work encompasses a rigorous proof for the existence of global solutions in H\"{o}lder spaces for the gradient flow. The asymptotic limits of these solutions establish the existence of the spline interpolation versus least-squares fitting problem on smooth Riemannian manifolds, offering a comprehensive solution. Notably, the constructive nature of the proof suggests potential numerical schemes for finding solutions.