Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport and the Wasserstein distance have evolved as the most natural concept to deal with such tasks, but have some computational drawbacks. In this talk, we describe an approximation framework through local linearizations that significantly reduces both the computational effort and the required training data in supervised learning settings. We also introduce LOT Wassmap, a computationally feasibly algorithm to uncover low-dimensional structures in the Wasserstein space. We provide guarantees on the embedding quality, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. The proposed algorithms are demonstrated in pattern recognition tasks in imaging and medical applications.