We introduce LOT Wassmap, a computationally feasible algorithm to uncover
low-dimensional structures in the Wasserstein space. The algorithm is motivated
by the observation that many datasets are naturally interpreted as probability
measures rather than points in Rn, and that finding low-dimensional
descriptions of such datasets requires manifold learning algorithms in the
Wasserstein space. Most available algorithms are based on computing the
pairwise Wasserstein distance matrix, which can be computationally challenging
for large datasets in high dimensions. Our algorithm leverages approximation
schemes such as Sinkhorn distances and linearized optimal transport to speed-up
computations, 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 not
available and one must deal with finite samples instead. Experiments
demonstrate that LOT Wassmap attains correct embeddings and that the quality
improves with increased sample size. We also show how LOT Wassmap significantly
reduces the computational cost when compared to algorithms that depend on
pairwise distance computations.Comment: 38 pages, 10 figures. Submitte