This talk discusses computationally feasible algorithms to uncover low-dimensional structures in the Wasserstein space. This line of research is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, 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. One of our algorithms, LOT Wassmap, leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. 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.

This talk is based on joint work with Alex Cloninger, Keaton Hamm, Varun Khurana, Matthew Thorpe, and Bernhard Schmitzer.