Regularity of densities in relaxed and penalized average distance problem

Xin Yang  Lu

doi:10.3934/nhm.2015.10.837

Article Contents

2015, Volume 10, Issue 4: 837-855. Doi: 10.3934/nhm.2015.10.837

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Regularity of densities in relaxed and penalized average distance problem

Xin Yang Lu^1,

1.
Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213

Received: March 2014

Revised: June 2015

Published: December 2015

Abstract / Introduction Related Papers Cited by

Abstract

The average distance problem finds application in data parameterization, which involves ``representing'' the data using lower dimensional objects. From a computational point of view it is often convenient to restrict the unknown to the family of parameterized curves. The original formulation of the average distance problem exhibits several undesirable properties. In this paper we propose an alternative variant: we minimize the functional \begin{equation*} \int_{{\mathbb{R}}^d\times \Gamma_\gamma} |x-y|^p {\,{d}}\Pi(x,y)+\lambda L_\gamma +\varepsilon\alpha(\nu) +\varepsilon' \eta(\gamma)+\varepsilon''\|\gamma'\|_{TV}, \end{equation*} where $\gamma$ varies among the family of parametrized curves, $\nu$ among probability measures on $\gamma$, and $\Pi$ among transport plans between $\mu$ and $\nu$. Here $\lambda,\varepsilon,\varepsilon',\varepsilon''$ are given parameters, $\alpha$ is a penalization term on $\mu$, $\Gamma_\gamma$ (resp. $L_\gamma$) denotes the graph (resp. length) of $\gamma$, and $\|\cdot\|_{TV}$ denotes the total variation semi-norm. We will use techniques from optimal transport theory and calculus of variations. The main aim is to prove essential boundedness, and Lipschitz continuity for Radon-Nikodym derivative of $\nu$, when $(\gamma,\nu,\Pi)$ is a minimizer.

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Mathematics Subject Classification: Primary: 49Q20, 49K10, 49Q10, 35B65.

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