Visualizing Shape Functionals via Sinkhorn Multidimensional Scaling

TL;DR

This paper introduces Sinkhorn multidimensional scaling (Sinkhorn MDS), a novel visualization method for shape functionals that effectively maps complex shape spaces into lower dimensions using Sinkhorn divergence, supported by error analysis and numerical validation.

Contribution

The paper presents Sinkhorn MDS as a new approach for visualizing shape functionals, including error estimates and validation on classical and new shape functionals.

Findings

Effective visualization of shape functionals achieved

Error estimates for Sinkhorn MDS provided

Numerical experiments confirm method's utility

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.