Title: A Biconvex Formulation for Transport of Mixture Models
Primary area: applications to neuroscience & cognitive science
Keywords: Optimal transport, Mixture models, Biconvex formulation, Single-cell RNA sequencing

Abstract: Optimal transport (OT) provides a principled framework for mapping between probability distributions. Despite extensive progress in the field, OT remains computationally demanding, and the resulting transport plans are often difficult to interpret. Here, we propose Optimal Mixture Transport (OMT), an efficient algorithm that leverages mixture modeling and entropic regularization to yield interpretable transport plans. We show that transport between mixtures, in particular mixtures of Gaussians which are universal approximators in $L^2$, can be formulated as a biconvex optimization problem with a unique minimizer. This formulation not only reduces computational cost, but also provides component-level correspondences, offering insights into complex distributions. We demonstrate the practicality and effectiveness of OMT across a diverse collection of synthetic benchmarks and real-world datasets, including large-scale single-cell RNA sequencing measurements.

TL;DR: This paper introduces Optimal Mixture Transport (OMT), a computationally efficient and interpretable method that uses mixture models to find meaningful, component-level mappings between complex probability distributions.