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 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.
Approximating $f$-Divergences with Rank Statistics
Department of Mathematics, Technical University of Munich \& Munich Center for Machine Learning, Germany. The majority of the work was conducted while at the Institute of Mathematics at the Technical University of Berlin, Germany \& the Berlin Mathematical School.；Department of Signal Theory
National University of Singapore（新加坡国立大学）；FPT Software AI Center（FPT软件AI中心）；Movian AI；The Institute of Statistical Mathematics（统计数学研究所）；RIKEN AIP（理化学研究所革新智能创造领域）
