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

Abstract:

This paper presents Convex Distance Operator Transport (CDOT), a novel convex optimal transport framework designed to align probability distributions across diverse domains while simultaneously maintaining feature correspondence and intrinsic geometric structures. To achieve this, CDOT utilizes an operator-based regularization technique that harmonizes aggregated distance configurations through the integration of distance and conditional expectation operators. This approach significantly enhances resilience against local geometric fluctuations. We rigorously demonstrate that the resulting CDOT discrepancy functions as a valid pseudmetric within the space of attributed compact metric-measure spaces. Furthermore, we clarify the connection between CDOT and Gromov--Wasserstein (GW) distance by introducing a novel concept known as the dispersion gap, which formally identifies the geometric origins of GW’s non-convexity in contrast to the inherent convexity of CDOT. In the context of finite-sample scenarios, we establish a non-asymptotic risk bound that separates optimization errors from statistical errors, thereby confirming risk consistency when employing a globally convergent Frank--Wolfe algorithm. Empirical evaluations on synthetic point clouds, brain connectomes, and graph classification tasks indicate that CDOT outperforms existing methodologies, exhibiting stable and dependable performance in practical applications.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC