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 Activity Number: 245 - SLDS CSpeed 4 Type: Contributed Date/Time: Wednesday, August 11, 2021 : 10:00 AM to 11:50 AM Sponsor: Section on Statistical Learning and Data Science Abstract #319026 Title: Statistical Convergence Rates for Knothe-Rosenblatt Coupling Estimators Author(s): Nicholas Irons* and Zaid Harchaoui and Soumik Pal and Meyer Scetbon Companies: University of Washington and University of Washington and University of Washington and CREST, ENSAE Keywords: optimal transport; empirical processes; density estimation; generative models; coupling; convergence rate Abstract: Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^d$. The Knothe-Rosenblatt rearrangement $T$ is the unique transport map between $\mu$ and $\nu$ that satisfies a certain monotonicity property. There has been a renewed interest in the Knothe-Rosenblatt rearrangement in relation to optimal transport and statistical learning. We establish convergence rates for the statistical estimator of the Knothe-Rosenblatt of Spantini et al. (2018), which minimizes the sample average approximation of the Kullback-Leibler divergence between $S_\#\mu$ and $\nu$, where $S$ lies in the cone of monotone triangular maps on $\mathbb{R}^d$. The proof techniques involve tools from empirical process and optimal transport.

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