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--Wasserstien (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.

Scientists often need to compare two collections of data that live in completely different worlds — for example, brain-connection maps from two different people, or molecules described by entirely different measurements. A popular mathematical tool called optimal transport can score how best to "morph" one collection into another, but the version built to work across such mismatched worlds is notoriously unreliable: the computation can get stuck partway, handing back different answers depending on where it happens to start. We introduce CDOT, a new method that reshapes this comparison into a "convex" problem — one shaped like a single smooth valley, so any standard solver rolls down to the same correct bottom every time. Instead of matching every pairwise distance one by one, CDOT compares how each point sits within the overall shape of its dataset, which makes it robust to small local distortions. We prove that CDOT behaves like a genuine notion of distance and that it homes in on the right answer as more data arrives. On synthetic shapes, brain networks, and graph-classification benchmarks, it is both more accurate and far more stable than existing approaches.