Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation
This work addresses domain adaptation challenges in machine learning by improving the theoretical understanding and efficiency of optimal transport methods, though it is incremental as it builds on existing linear mapping frameworks.
The paper tackles the problem of estimating linear Monge mappings between distributions and provides the first concentration result for the estimator, achieving a sample complexity of n^{-1/2}. This leads to a generalization bound for domain adaptation with optimal transport, approaching the performance of the theoretical Bayes predictor under mild conditions.
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.