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

2024/02/28 by Lin, Chun-Chi, Tran, The Dung
#35K25 #49J20 #FOS: Mathematics #Optimization and Control (math.OC) #primary 35K52 #secondary 41A15

paper · doi:10.48550/arxiv.2402.18067

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 Hölder 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.

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