Learning Wasserstein Isometric Embedding for Point Clouds

The Wasserstein distance has been employed for determining the distance between point clouds, which have variable numbers of points and invariance of point order. However, the high computational cost associated with the Wasserstein distance hinders its practical applications for large-scale datasets. We propose a new embedding method for point clouds, which aims to embed point clouds into a Euclidean space, isometric to the Wasserstein space defined on the point clouds. In numerical experiments, we demonstrate that the point clouds decoded from the Euclidean averages and the interpolations in the embedding space accurately mimic the Wasserstein barycenters and interpolations of the point clouds. Furthermore, we show that the embedding vectors can be utilized as inputs for machine learning models (e.g., principal component analysis and neural networks).