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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.
We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hi
Domain adaptation (DA) arises as an important problem in statistical machine learning when the source data used to train a model is different from the target data used to test the model. Recent advances in DA have mainly been application-driven and h
Machine learning models have traditionally been developed under the assumption that the training and test distributions match exactly. However, recent success in few-shot learning and related problems are encouraging signs that these models can be ad
In this paper, we provide two main contributions in PAC-Bayesian theory for domain adaptation where the objective is to learn, from a source distribution, a well-performing majority vote on a different target distribution. On the one hand, we propose
In the context of K-armed stochastic bandits with distribution only assumed to be supported by [0,1], we introduce the first algorithm, called KL-UCB-switch, that enjoys simultaneously a distribution-free regret bound of optimal order $sqrt{KT}$ and