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This article investigates the quality of the estimator of the linear Mongemapping between distributions. We provide the first concentration result on thelinear mapping operator and prove a sample complexity of $n^{-1/2}$ when usingempirical estimates of first and second order moments. This result is then usedto derive a generalization bound for domain adaptation with optimal transport.As a consequence, this method approaches the performance of theoretical Bayespredictor under mild conditions on the covariance structure of the problem. Wealso discuss the computational complexity of the linear mapping estimation andshow that when the source and target are stationary the mapping is aconvolution that can be estimated very efficiently using fast Fouriertransforms. Numerical experiments reproduce the behavior of the proven boundson simulated and real data for mapping estimation and domain adaptation onimages.

Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation