  This article investigates the quality of the estimator of the linear Monge
mapping between distributions. We provide the first concentration result on the
linear mapping operator and prove a sample complexity of $n^{-1/2}$ when using
empirical estimates of first and second order moments. This result is then used
to derive a generalization bound for domain adaptation with optimal transport.
As a consequence, this method approaches the performance of theoretical Bayes
predictor under mild conditions on the covariance structure of the problem. We
also discuss the computational complexity of the linear mapping estimation and
show that when the source and target are stationary the mapping is a
convolution that can be estimated very efficiently using fast Fourier
transforms. Numerical experiments reproduce the behavior of the proven bounds
on simulated and real data for mapping estimation and domain adaptation on
images.
