Optimal Transport-Based Volumetric Shape Clustering

Abstract Clustering of data points in $n$-dimensional Euclidean space, i.~e. assigning each data point to exactly one group (cluster) in order to detect previously unseen relations within the data set, has become a standard task for (unsupervised) machine learning. In this paper, this concept is generalized to shape data consisting of three-dimensional volumetric objects. As the underlying distance on the space of such objects, the optimal transport-based Wasserstein distance is considered and different variants of such a clustering approach are presented and compared. To counteract an over-smoothing of the cluster center volumes a variational autoencoder representing these cluster centers is incorporated. Numerical experiments for three distinct, volumetric data sets are presented to validate the performance of the proposed methods.