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We introduce LOT Wassmap, a computationally feasible algorithm to uncoverlow-dimensional structures in the Wasserstein space. The algorithm is motivatedby the observation that many datasets are naturally interpreted as probabilitymeasures rather than points in $\mathbb{R}^n$, and that finding low-dimensionaldescriptions of such datasets requires manifold learning algorithms in theWasserstein space. Most available algorithms are based on computing thepairwise Wasserstein distance matrix, which can be computationally challengingfor large datasets in high dimensions. Our algorithm leverages approximationschemes such as Sinkhorn distances and linearized optimal transport to speed-upcomputations, and in particular, avoids computing a pairwise distance matrix.We provide guarantees on the embedding quality under such approximations,including when explicit descriptions of the probability measures are notavailable and one must deal with finite samples instead. Experimentsdemonstrate that LOT Wassmap attains correct embeddings and that the qualityimproves with increased sample size. We also show how LOT Wassmap significantlyreduces the computational cost when compared to algorithms that depend onpairwise distance computations.

Linearized Wasserstein dimensionality reduction with approximation guarantees