A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution

Problem
The paper addresses the computational challenges of applying optimal transport (OT) to large-scale datasets, particularly the difficulty in interpreting pointwise transport plans. Despite advancements in OT, existing methods struggle with scalability and often yield complex solutions that are not easily interpretable. The authors propose a novel approach, Optimal Mixture Transport (OMT), which reformulates the transport problem to focus on mixtures of subpopulations rather than individual samples. This work is presented as a preprint and has not yet undergone peer review.

Method
OMT introduces a biconvex optimization framework that guarantees a unique global minimizer for the transport problem. By modeling subpopulations as exponential-family distributions, OMT effectively decouples computational complexity from the sample size, allowing the method to scale with the number of mixture components instead. The authors provide theoretical guarantees that ensure stability in the OMT map, demonstrating that bounded perturbations in the underlying distributions result in bounded changes in the transport plan. The optimization is performed using standard techniques for biconvex problems, although specific details on the architecture and training compute are not disclosed.

Results
The authors validate OMT on various synthetic benchmarks and real-world datasets, including image data and large-scale single-cell RNA sequencing measurements. They report significant improvements in computational efficiency and interpretability compared to traditional OT methods. For instance, OMT achieves a reduction in computational time by an order of magnitude on large datasets while maintaining accuracy in transport plans. Specific performance metrics against named baselines are not detailed in the abstract, but the results indicate that OMT outperforms existing OT methods in both speed and stability.

Limitations
The authors acknowledge that while OMT provides a unique solution and stability guarantees, the method’s reliance on the exponential-family assumption may limit its applicability to certain types of data distributions. Additionally, the paper does not explore the impact of varying the number of mixture components on performance, which could be a critical factor in practical applications. The theoretical guarantees, while promising, may require further empirical validation across diverse datasets.

Why it matters
The introduction of OMT has significant implications for the field of optimal transport, particularly in applications involving large-scale data where interpretability and computational efficiency are paramount. By shifting the focus from individual samples to mixtures, OMT opens new avenues for research in probabilistic modeling and data analysis. This work could enhance methodologies in various domains, including image processing and genomics, where understanding the transport of distributions is crucial. The findings are available on arXiv and may inspire further exploration of scalable transport methods in machine learning.