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

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.

• The paper introduces CDOT, a novel convex formulation that transforms non-convex GW-type optimal transport into a globally optimizable problem using operator-based methods.
• Empirical results on synthetic point clouds, brain connectomes, and graph benchmarks demonstrate improved accuracy, reduced MSE, and robust statistical consistency.
• CDOT advances both theoretical insights and practical applications by preserving geometric structure, enabling scalable optimization, and offering potential integration in deep learning models.

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

Introduction and Motivation

Traditional Optimal Transport (OT) methods, particularly those grounded in Wasserstein distances, facilitate pointwise matching between distributions in a shared metric space. However, in many applications—such as graph analysis, neuroscience, or structured data—data is naturally modeled as distributions over heterogeneous metric-measure (mm) spaces, with potentially incompatible geometries and sizes. The Gromov–Wasserstein (GW) framework offers a metric-based approach to compare such structures by aligning pairwise distance distributions. Still, GW and its attributed variant (FGW) are inherently non-convex due to their reliance on quadratic costs over tensor products of couplings. This non-convexity impedes global optimization and complicates theoretical analysis.

The paper introduces Convex Distance Operator Transport (CDOT), a fundamentally new OT formulation that addresses these limitations. CDOT formulates geometry-preserving OT as a convex optimization by lifting pointwise geometry (distance matrices) and matching (transport plans) into the space of linear operators on $L^2$ function spaces. This operator approach ensures convexity, efficient global optimization, and robust geometric structure preservation. Theoretical results show that CDOT defines a pseudometric over attributed compact mm spaces and provides non-asymptotic risk bounds with statistical consistency.

Operator-Based Formulation

A key innovation of CDOT is its operator-theoretic treatment of structure and matching:

• Distance Operator: For a mm space $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$, define $$D_{\mathbb{P}_X}: L^2(\mathcal{X}, \mathbb{P}_X) \to L^2(\mathcal{X}, \mathbb{P}_X)$$ as

$$(D_{\mathbb{P}_X}f)(x) = \int_{\mathcal{X}} d_{\mathcal{X}}(x, x') f(x')\, \mathbb{P}_X(dx').$$

• Conditional Expectation Operator: Given a transport plan $\pi \in \Pi(\mathbb{P}_X, \mathbb{P}_Y)$, let $$T_\pi: L^2(\mathcal{Y}, \mathbb{P}_Y) \to L^2(\mathcal{X}, \mathbb{P}_X)$$ be

$$(T_\pi g)(x) = \int_{\mathcal{Y}} g(y)\, \pi(dy \mid x).$$

• Structural Regularization: CDOT’s key geometric penalty,

$$\mathcal{R}(\pi) = \| D_{\mathbb{P}_X} T_\pi - T_\pi D_{\mathbb{P}_Y} \|_{HS}^2,$$

quantifies how well $T_\pi$ “intertwines” the geometries of $\mathcal{X}$ and $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$0.

The CDOT optimization problem jointly balances feature alignment and this structural regularization via the strictly convex objective

$(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$1

over $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$2. Here, $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$3 is typically squared feature distance.

Geometric and Theoretical Properties

• Convexity: CDOT’s operator-based formulation transforms the quadratic non-convexity of GW-type objectives into a global convex problem with guaranteed existence of optimal solutions.
• Pseudometric Structure: The CDOT discrepancy $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$4 defined by the optimal value of $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$5 is a pseudometric on attributed mm spaces—vanishing only when distance operators and features align under some coupling. While this is weaker than (FG)W’s requirement of measure-preserving isometries, it ensures robust, geometry-aware, and relaxable similarity.
• Dispersion Gap: By decomposing the (F)GW quadratic form, the authors introduce the notion of “dispersion,” which isolates the source of GW’s non-convexity:

$(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$6

where $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$7 penalizes the variance or “determinism” in the coupling. Unlike GW, CDOT completely omits this term, favoring convexity even if it admits more diffuse, fractional couplings.

• Empirical Risk and Optimization: In finite-sample regimes, the objective is a convex quadratic program over the transport polytope. The authors exploit the affine structure of the objective for efficient Frank–Wolfe optimization, with an option for post hoc hard assignment via the Hungarian method.

Figure 2: Level sets of $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$8 (left) and $(\mathcal{X}, d_{\mathcal{X}}, \mathbb{P}_X)$9 (right), visualizing their distinct optimization landscapes and illustrating how CDOT ensures global convergence.

Figure 1: Empirical transport polytope projected onto the $$D_{\mathbb{P}_X}: L^2(\mathcal{X}, \mathbb{P}_X) \to L^2(\mathcal{X}, \mathbb{P}_X)$$0 plane, showcasing the convex vertical level sets of CDOT and the non-convex diagonal level sets of GW.

Empirical Results

Extensive experimental evaluations demonstrate the practical impact of CDOT’s operator perspective:

• Synthetic Point Clouds: On structured matching tasks between 2D point clouds with labeled clusters, CDOT obtains the lowest mean-squared error (MSE) across all sample sizes and exceeds (E)FGW, IsoRank, Spectral, and COPT baselines. Increasing data size systematically reduces MSE, supporting the risk consistency result.

Figure 5: Synthetic data matching performance with $$D_{\mathbb{P}_X}: L^2(\mathcal{X}, \mathbb{P}_X) \to L^2(\mathcal{X}, \mathbb{P}_X)$$1 samples, indicating rapid convergence of CDOT’s matching accuracy as sample size increases.

• Brain Connectome Matching: On OASIS-3 structural connectome graphs, CDOT achieves highest node correspondence accuracy when using robust diffusion distances, outperforming GW-type methods and topological baselines. FGW leads with geodesic distances, but diffusion metrics, which capture smoother global geometry, yield greater robustness for CDOT.

  Figure 3: Anatomical visualization of a brain connectome from OASIS-3, showing region-based node colorings used in structural alignment experiments.

• Graph Classification: On TUDataset graph benchmarks (bioinformatics, social networks), CDOT achieves the best classification accuracy in both attributed and non-attributed settings, highlighting its ability to fuse feature and geometric signals in a metrically meaningful manner.

Implications and Future Perspectives

The introduction of CDOT advances the practical and theoretical landscape for matching and comparing heterogeneously structured data:

• Theoretical Insights: The operator-level discrepancy framework offers a new perspective on geometric alignment, rooted in functional analysis, that naturally extends classical metric geometry and reveals the inner structure of GW-type objectives.
• Practical AI Applications: CDOT provides a reliable, robust, and interpretable method for applications requiring the alignment of structured distributions, including neuroscience, network analysis, structured prediction, and scientifically interpretable AI.
• Statistical Learning: The availability of finite-sample risk bounds and consistency results open the door to statistically principled inference with geometry-preserving transport methods.
• Extensions: The paper suggests integrating CDOT as a differentiable metric layer in deep learning architectures, enabling joint learning of features and ground metrics for structure-aware domain adaptation, manifold learning, or geometric generative modeling. Furthermore, CDOT’s pseudometric nature naturally supports the construction of testing and inference frameworks on mm spaces, such as isomorphism or equivalence tests.

Conclusion

CDOT presents a rigorously justified, convex, and geometry-preserving optimal transport framework that resolves key limitations of prior GW-type distances. By aligning aggregated operator profiles rather than enforcing rigid pairwise correspondences, it achieves stable optimization, improved empirical performance, and theoretically sound statistical guarantees. While CDOT relaxes rigid point-to-point matching in favor of robust geometry aggregation, in many structured-data or scientific applications, this yields more stable, scalable, and interpretable solutions. Future investigation into learnable metrics, scalable solvers, and statistical testing paradigms on mm spaces will further broaden its foundational and applied impact.