Regularity of densities in relaxed and penalized average distance problem

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 L 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.