1. Low-Rank Optimal Transport through Factor Relaxation with Latent Coupling
- Author
-
Halmos, Peter, Liu, Xinhao, Gold, Julian, and Raphael, Benjamin J
- Subjects
Computer Science - Machine Learning ,Computer Science - Artificial Intelligence ,Statistics - Machine Learning - Abstract
Optimal transport (OT) is a general framework for finding a minimum-cost transport plan, or coupling, between probability distributions, and has many applications in machine learning. A key challenge in applying OT to massive datasets is the quadratic scaling of the coupling matrix with the size of the dataset. [Forrow et al. 2019] introduced a factored coupling for the k-Wasserstein barycenter problem, which [Scetbon et al. 2021] adapted to solve the primal low-rank OT problem. We derive an alternative parameterization of the low-rank problem based on the $\textit{latent coupling}$ (LC) factorization previously introduced by [Lin et al. 2021] generalizing [Forrow et al. 2019]. The LC factorization has multiple advantages for low-rank OT including decoupling the problem into three OT problems and greater flexibility and interpretability. We leverage these advantages to derive a new algorithm $\textit{Factor Relaxation with Latent Coupling}$ (FRLC), which uses $\textit{coordinate}$ mirror descent to compute the LC factorization. FRLC handles multiple OT objectives (Wasserstein, Gromov-Wasserstein, Fused Gromov-Wasserstein), and marginal constraints (balanced, unbalanced, and semi-relaxed) with linear space complexity. We provide theoretical results on FRLC, and demonstrate superior performance on diverse applications -- including graph clustering and spatial transcriptomics -- while demonstrating its interpretability., Comment: 53 pages, 13 figures, NeurIPS 2024. Comments welcome!
- Published
- 2024