Average-distance problem with curvature penalization for data parameterization: regularity of minimizers

TL;DR

This paper introduces a new model for data parameterization that combines average-distance minimization with curvature penalization, ensuring regularity and well-behaved solutions.

Contribution

It develops a novel functional integrating distance and curvature penalties, proving existence, uniqueness, and regularity of its minimizers, including $C^{1,1}$ estimates.

Findings

Establishes existence and uniqueness of minimizers.

Proves regularity and $C^{1,1}$ estimates for minimizers.

Addresses shortcomings of previous models by ensuring regularity.

Abstract

We propose a model for finding one-dimensional structure in a given measure. Our approach is based on minimizing an objective functional which combines the average-distance functional to measure the quality of the approximation and penalizes the curvature, similarly to the elastica functional. Introducing the curvature penalization overcomes some of the shortcomings of the average-distance functional, in particular the lack of regularity of minimizers. We establish existence, uniqueness and regularity of minimizers of the proposed functional. In particular we establish $C^{1,1}$ estimates on the minimizers.