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Generalized Sliced Wasserstein Distances
Soheil Kolouri · Kimia Nadjahi · Umut Simsekli · Roland Badeau · Gustavo Rohde

Tue Dec 10 10:45 AM -- 12:45 PM (PST) @ East Exhibition Hall B + C #76

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We further show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed proper metrics. Finally, we compare the numerical performance of the proposed distances on the generative modeling task of SW flows and report favorable results.

Author Information

Soheil Kolouri (HRL Laboratories LLC)

**Soheil Kolouri** is an Assistant Professor of Computer Science at Vanderbilt University, Nashville, TN, where he runs the Machine Intelligence and Neural Technologies (MINT) lab. Soheil is broadly interested in applied mathematics, machine learning, and computer vision. He also has a standing interest in computational optimal transport and geometry. Before joining Vanderbilt, Soheil was a research scientist and a principal investigator at HRL Laboratories, Malibu, CA. He was the PI on DARPA Learning with Less Labels (LwLL) and the Co-PI on DARPA Lifelong Learning Machines (L2M) programs. He obtained his Ph.D. in Biomedical Engineering from Carnegie Mellon University, where he received the Bertucci Fellowship Award for outstanding graduate students from the College of Engineering in 2014, and the Outstanding Dissertation Award from the Biomedical Engineering Department in 2015.

Kimia Nadjahi (Télécom ParisTech)
Umut Simsekli (Institut Polytechnique de Paris/ University of Oxford)
Roland Badeau (Télécom ParisTech)
Gustavo Rohde (University of Virginia)

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