In this paper, we propose to tackle the problem of reducing discrepancies between multiple domains referred to as multi-source domain adaptation and consider it under the target shift assumption: in all domains we aim to solve a classification problem with the same output classes, but with labels' proportions differing across them. This problem, generally ignored in the vast majority papers on domain adaptation papers, is nevertheless critical in real-world applications, and we theoretically show its impact on the adaptation success. To address this issue, we design a method based on optimal transport, a theory that has been successfully used to tackle adaptation problems in machine learning. Our method performs multi-source adaptation and target shift correction simultaneously by learning the class probabilities of the unlabeled target sample and the coupling allowing to align two (or more) probability distributions. Experiments on both synthetic and real-world data related to satellite image segmentation task show the superiority of the proposed method over the state-of-the-art.
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