Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rates of Estimation of Optimal Transport Maps using Plug-in Estimators
  via Barycentric Projections   Optimal transport maps between two probability distributions $\mu$ and $\nu$
on $\mathbb{R}^d$ have found extensive applications in both machine learning
and statistics. In practice, these maps need to be estimated from data sampled
according to $\mu$ and $\nu$. Plug-in estimators are perhaps most popular in
estimating transport maps in the field of computational optimal transport. In
this paper, we provide a comprehensive analysis of the rates of convergences
for general plug-in estimators defined via barycentric projections. Our main
contribution is a new stability estimate for barycentric projections which
proceeds under minimal smoothness assumptions and can be used to analyze
general plug-in estimators. We illustrate the usefulness of this stability
estimate by first providing rates of convergence for the natural
discrete-discrete and semi-discrete estimators of optimal transport maps. We
then use the same stability estimate to show that, under additional smoothness
assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed
plug-in estimators respectively speed up the rates of convergence and
significantly mitigate the curse of dimensionality suffered by the natural
discrete-discrete/semi-discrete estimators. As a by-product of our analysis, we
also obtain faster rates of convergence for plug-in estimators of
$W_2(\mu,\nu)$, the Wasserstein distance between $\mu$ and $\nu$, under the
aforementioned smoothness assumptions, thereby complementing recent results in
Chizat et al. (2020). Finally, we illustrate the applicability of our results
in obtaining rates of convergence for Wasserstein barycenters between two
probability distributions and obtaining asymptotic detection thresholds for
some recent optimal-transport based tests of independence.

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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: physics