Visualizing Shape Functionals via Sinkhorn Multidimensional Scaling

Abstract: In this paper, we present Sinkhorn multidimensional scaling (Sinkhorn MDS) as a method for visualizing shape functionals in shape spaces. This approach uses the Sinkhorn divergence to map these infinite-dimensional spaces into lower-dimensional Euclidean spaces. We establish error estimates for the embedding generated by Sinkhorn MDS compared to the unregularized case. Moreover, we validate the method through numerical experiments, including visualizations of the classical Dido's problem and two newly introduced shape functionals: the double-well and Sinkhorn cone-type shape functionals. Our results demonstrate that Sinkhorn MDS effectively captures and visualizes shapes of shape functionals.