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Statistical and Topological Properties of Sliced Probability Divergences
Kimia Nadjahi · Alain Durmus · Lénaïc Chizat · Soheil Kolouri · Shahin Shahrampour · Umut Simsekli

Thu Dec 10 09:00 AM -- 11:00 AM (PST) @ Poster Session 5 #1395

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between \emph{one-dimensional random projections} of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.

Author Information

Kimia Nadjahi (Télécom Paris)
Alain Durmus (ENS Paris Saclay)
Lénaïc Chizat (CNRS)
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.

Shahin Shahrampour (Texas A&M University)
Umut Simsekli (Inria/ENS)

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