  Learning an effective representation of 3D point clouds requires a good
metric to measure the discrepancy between two 3D point sets, which is
non-trivial due to their irregularity. Most of the previous works resort to
using the Chamfer discrepancy or Earth Mover's distance, but those metrics are
either ineffective in measuring the differences between point clouds or
computationally expensive. In this paper, we conduct a systematic study with
extensive experiments on distance metrics for 3D point clouds. From this study,
we propose to use sliced Wasserstein distance and its variants for learning
representations of 3D point clouds. In addition, we introduce a new algorithm
to estimate sliced Wasserstein distance that guarantees that the estimated
value is close enough to the true one. Experiments show that the sliced
Wasserstein distance and its variants allow the neural network to learn a more
efficient representation compared to the Chamfer discrepancy. We demonstrate
the efficiency of the sliced Wasserstein metric and its variants on several
tasks in 3D computer vision including training a point cloud autoencoder,
generative modeling, transfer learning, and point cloud registration.
