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.

翻译：我们提出凸距离算子传输（CDOT），这是首个在异质域间通过联合保留特征对应与内在几何结构来对齐分布的凸最优传输框架。具体而言，CDOT采用基于算子的正则化方法，通过引入距离算子和条件期望算子对齐聚合距离结构，从而提升对局部几何变异的鲁棒性。我们进一步证明，所得的CDOT散度在带属性的紧度量测度空间上构成有效的伪度量。此外，通过引入弥散间隙（dispersion gap）这一新概念，我们揭示了CDOT与Gromov--Wasserstein（GW）之间的关系，正式阐明了相较于CDOT的凸性，GW非凸性的几何来源。在有限样本情形下，我们推导出可分解为优化误差与统计误差的非渐近风险界，并建立了全局收敛的Frank-Wolfe算法的风险一致性。在合成点云、脑连接组与图分类基准上的实验表明，该方法在性能上优于现有方法，且在实践中表现出稳定可靠的行为。