Optimal transport (OT) and the related Wasserstein metric ($W$) are powerful and ubiquitous tools for comparing distributions. However, computing pairwise Wasserstein distances rapidly becomes intractable as cohort size grows. An attractive alternative would be to find an embedding space in which pairwise Euclidean distances map to OT distances, akin to standard multidimensional scaling (MDS). We present Wasserstein Wormhole, a transformer-based autoencoder that embeds empirical distributions into a latent space wherein Euclidean distances approximate OT distances. Extending MDS theory, we show that our objective function implies a bound on the error incurred when embedding non-Euclidean distances. Empirically, distances between Wormhole embeddings closely match Wasserstein distances, enabling linear time computation of OT distances. Along with an encoder that maps distributions to embeddings, Wasserstein Wormhole includes a decoder that maps embeddings back to distributions, allowing for operations in the embedding space to generalize to OT spaces, such as Wasserstein barycenter estimation and OT interpolation. By lending scalability and interpretability to OT approaches, Wasserstein Wormhole unlocks new avenues for data analysis in the fields of computational geometry and single-cell biology.

MOTIVATION: Transcriptional dynamics are governed by the action of regulatory proteins and are fundamental to systems ranging from normal development to disease. RNA velocity methods for tracking phenotypic dynamics ignore information on the regulatory drivers of gene expression variability through time. RESULTS: We introduce scKINETICS (Key regulatory Interaction NETwork for Inferring Cell Speed), a dynamical model of gene expression change which is fit with the simultaneous learning of per-cell transcriptional velocities and a governing gene regulatory network. Fitting is accomplished through an expectation-maximization approach designed to learn the impact of each regulator on its target genes, leveraging biologically motivated priors from epigenetic data, gene-gene coexpression, and constraints on cells' future states imposed by the phenotypic manifold. Applying this approach to an acute pancreatitis dataset recapitulates a well-studied axis of acinar-to-ductal transdifferentiation whilst proposing novel regulators of this process, including factors with previously appreciated roles in driving pancreatic tumorigenesis. In benchmarking experiments, we show that scKINETICS successfully extends and improves existing velocity approaches to generate interpretable, mechanistic models of gene regulatory dynamics. AVAILABILITY AND IMPLEMENTATION: All python code and an accompanying Jupyter notebook with demonstrations are available at http://github.com/dpeerlab/scKINETICS.