59 research outputs found
Minimax estimation of smooth optimal transport maps
Brenier's theorem is a cornerstone of optimal transport that guarantees the
existence of an optimal transport map between two probability distributions
and over under certain regularity conditions. The main
goal of this work is to establish the minimax estimation rates for such a
transport map from data sampled from and under additional smoothness
assumptions on . To achieve this goal, we develop an estimator based on the
minimization of an empirical version of the semi-dual optimal transport
problem, restricted to truncated wavelet expansions. This estimator is shown to
achieve near minimax optimality using new stability arguments for the semi-dual
and a complementary minimax lower bound. Furthermore, we provide numerical
experiments on synthetic data supporting our theoretical findings and
highlighting the practical benefits of smoothness regularization. These are the
first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure
High-Dimensional Statistics
These lecture notes were written for the course 18.657, High Dimensional
Statistics at MIT. They build on a set of notes that was prepared at Princeton
University in 2013-14 that was modified (and hopefully improved) over the
years.Comment: This is the 2017 version of these notes, uploaded to arXiv without
any chang
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