做分布对齐或几何数据处理的团队，CDOT 解决了传统 GW 非凸优化不稳定的痛点，可以直接用这个凸框架提升效果，建议点开看理论证明和实验对比。
研究人员提出 Convex Distance Operator Transport (CDOT)，这是首个凸最优传输框架，能在异构域之间对齐分布，同时保留特征对应和内在几何结构。CDOT 通过基于算子的正则化，引入距离和条件期望算子来对齐聚合距离结构，从而提升对局部几何变化的鲁棒性。理论证明 CDOT 差异是属性紧致度量-测度空间上的有效伪度量，并揭示了其与 Gromov-Wasserstein 的非凸性差异。实验在合成点云、脑连接组和图分类基准上表现优于现有方法，且行为稳定可靠。
Convex Distance Operator Transport: A Convex and Geometry-Preserving Formulation
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.