Structured Matching via Cost-Regularized Unbalanced Optimal Transport

2025-11-24

AI Summary

Conventional unbalanced optimal transport (UOT) in heterogeneous spaces suffers from fixed, pre-specified ground cost functions that fail to capture intrinsic geometric structures of data. Method: We propose a cost-regularized unbalanced Gromov–Wasserstein (NR-UGW) framework that jointly optimizes the transport plan and a learnable ground cost function. The cost is parameterized via linear transformations within an inner-product family and integrated with entropic regularization for computational efficiency. NR-UGW supports mass creation/destruction and structural alignment across Euclidean spaces. Contribution/Results: Unlike standard UOT, NR-UGW dynamically adapts to geometric discrepancies between heterogeneous measures. On single-cell multi-omics data, it significantly improves contour alignment accuracy under sample-missing conditions—particularly where cell-level correspondences are absent—enabling robust cross-modal integration without explicit matching.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Unbalanced optimal transport (UOT) provides a flexible way to match or compare nonnegative finite Radon measures. However, UOT requires a predefined ground transport cost, which may misrepresent the data's underlying geometry. Choosing such a cost is particularly challenging when datasets live in heterogeneous spaces, often motivating practitioners to adopt Gromov-Wasserstein formulations. To address this challenge, we introduce cost-regularized unbalanced optimal transport (CR-UOT), a framework that allows the ground cost to vary while allowing mass creation and removal. We show that CR-UOT incorporates unbalanced Gromov-Wasserstein type problems through families of inner-product costs parameterized by linear transformations, enabling the matching of measures or point clouds across Euclidean spaces. We develop algorithms for such CR-UOT problems using entropic regularization and demonstrate that this approach improves the alignment of heterogeneous single-cell omics profiles, especially when many cells lack direct matches.

Problem

• Addresses the challenge of predefined ground transport costs in unbalanced optimal transport
• Enables matching measures across heterogeneous spaces with variable ground costs
• Improves alignment of heterogeneous single-cell omics profiles with unmatched cells

Innovation

• Cost-regularized UOT framework allows variable ground transport cost
• Incorporates unbalanced Gromov-Wasserstein via parameterized inner-product costs
• Uses entropic regularization algorithms for heterogeneous data alignment