CMU Campus
Center for                           Nonlinear Analysis
CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact
Publication 14-CNA-010

Regularity of Densities in Relaxed and Penalized Average Distance Problem

Xin Yang Lu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
xinyang@andrew.cmu.edu

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. However this formulation exhibits several undesirable properties. In this paper we propose an alternative variant: the average distance functional is replaced by a transport cost, and the unknown is composed both by a curve and by a "projected measure", on which an Lq penalization term is added. Moreover we will add a term penalizing non injectivity. We will use techniques from optimal transport theory and calculus of variations. The main aim is to prove essential boundedness, and a variant of Lipschitz continuity for Radon-Nikodym derivative of projected measures for minimizers.

Get the paper in its entirety as

Back to CNA Publications