Metric Learning for 3D Point Clouds Using Optimal Transport

Learning embeddings of any data largely depends on the ability of the target space to capture semantic relations. The widely used Euclidean space, where embeddings are represented as point vectors, is known to be lacking in its potential to exploit complex structures and relations. Contrary to standard Euclidean embeddings, in this work, we embed point clouds as discrete probability distributions in Wasserstein space. We build a contrastive learning setup to learn Wasserstein embeddings that can be used as a pre-training method with or without supervision towards any downstream task. We show that the features captured by Wasserstein embeddings are better in preserving the point cloud geometry, including both global and local information, thus resulting in improved quality embeddings. We perform exhaustive experiments and demonstrate the effectiveness of our method for point cloud classification, transfer learning, segmentation, and interpolation tasks over multiple datasets including synthetic and real-world objects. We also compare against recent methods that use Wasserstein space and show that our method outperforms them in all downstream tasks. Additionally, our study reveals a promising interpretation of capturing critical points of point clouds that makes our proposed method self-explainable.