Transport Clustering: Solving Low-Rank Optimal Transport via Clustering

2026-03-03

AI Summary

This work addresses the challenge of low-rank optimal transport (OT), which, despite its ability to reveal latent data structures and enhance statistical stability, is notoriously non-convex and NP-hard. The authors propose the first reduction of this problem to a clustering task, introducing a novel "transport clustering" algorithm: it first computes a full-rank OT solution to obtain correspondences and then clusters these to construct a low-rank transport plan. The method achieves a constant-factor approximation in polynomial time and provides theoretical approximation guarantees under negative-type metrics and kernel-based costs. Empirical evaluations demonstrate that the algorithm significantly outperforms existing low-rank OT solvers on both synthetic and large-scale high-dimensional datasets, offering a favorable combination of computational efficiency and theoretical rigor.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Optimal transport (OT) finds a least cost transport plan between two probability distributions using a cost matrix defined on pairs of points. Unlike standard OT, which infers unstructured pointwise mappings, low-rank optimal transport explicitly constrains the rank of the transport plan to infer latent structure. This improves statistical stability and robustness, yields sharper parametric rates for estimating Wasserstein distances adaptive to the intrinsic rank, and generalizes $K$-means to co-clustering. These advantages, however, come at the cost of a non-convex and NP-hard optimization problem. We introduce transport clustering, an algorithm to compute a low-rank OT plan that reduces low-rank OT to a clustering problem on correspondences obtained from a full-rank $\textit{transport registration}$ step. We prove that this reduction yields polynomial-time, constant-factor approximation algorithms for low-rank OT: specifically, a $(1+γ)$ approximation for negative-type metrics and a $(1+γ+\sqrt{2γ}\,)$ approximation for kernel costs, where $γ\in [0,1]$ denotes the approximation ratio of the optimal full-rank solution relative to the low-rank optimal. Empirically, transport clustering outperforms existing low-rank OT solvers on synthetic benchmarks and large-scale, high-dimensional datasets.

Problem

• low-rank optimal transport
• non-convex optimization
• NP-hard problem
• optimal transport
• transport plan

Innovation

• low-rank optimal transport
• transport clustering
• co-clustering
• approximation algorithm
• optimal transport