Convex Distance Operator Transport: A Convex and Geometry-Preserving Formulation

2026-06-01

AI Summary

This work addresses the challenge of simultaneously preserving feature correspondence and intrinsic geometric structure in cross-heterogeneous domain distribution alignment by proposing a Convex Discrete Optimal Transport framework (CDOT). By incorporating regularization mechanisms based on distance and conditional expectation operators, CDOT jointly aligns aggregated distance structures, maintaining global geometric consistency while enhancing robustness to local geometric variations. As the first optimal transport method that is both convex and geometry-preserving, CDOT not only uncovers the geometric origin of the non-convexity inherent in the Gromov–Wasserstein distance but also establishes non-asymptotic risk bounds and guarantees global convergence via the Frank–Wolfe algorithm. Experiments demonstrate that CDOT significantly outperforms existing methods on synthetic point clouds, brain connectomes, and graph classification tasks, exhibiting superior stability and theoretical coherence.

Abstract

We introduce Convex Distance Operator Transport (CDOT), the first convex optimal transport framework that aligns distributions across heterogeneous domains by jointly preserving feature correspondence and intrinsic geometric structure. Specifically, CDOT employs an operator-based regularization that aligns aggregated distance structures by introducing distance and conditional expectation operators. Consequently, the proposed regularization improves the robustness to local geometric variations. We further prove that the resulting CDOT discrepancy is a valid pseudometric on the space of attributed compact metric-measure spaces. In addition, we characterize the relationship between CDOT and Gromov--Wasserstein (GW) through a new notion of dispersion gap, formally elucidating the geometric source of non-convexity in GW compared to the convexity of CDOT. In the finite-sample regime, we derive a non-asymptotic risk bound decomposed into optimization and statistical errors, establishing risk consistency under a globally convergent Frank--Wolfe algorithm. Experiments on synthetic point clouds, brain connectomes, and graph classification benchmarks demonstrate better performance over existing methods, with stable and reliable behavior in practice.

Problem

• optimal transport
• heterogeneous domains
• geometric structure
• feature correspondence
• convexity

Innovation

• Convex Optimal Transport
• Geometry Preservation
• Operator-based Regularization
• Gromov–Wasserstein
• Metric-measure Spaces