Entropic Partial Optimal Transport and Partial Gromov--Wasserstein Distance between Gaussian Mixtures

This paper develops an entropic partial optimal transport (EPOT) framework for Gaussian mixture models and defines a partial mixture Gromov–Wasserstein distance.

The authors first formulate a finite EPOT problem by combining the dummy-point representation of partial transport with entropic regularization. For fixed ε > 0, they prove that this model recovers balanced entropic OT as λ → ∞, with quantitative estimates for the transported mass and the value, together with convergence of the optimizer. Specifically, they prove the existence and uniqueness of the minimizer (Proposition 3.2), establish the quantitative unmatched-mass estimate (Proposition 3.3), and obtain the value estimate (Theorem 3.4). In particular, the EPOT value converges to OTε(a, b) − ε as stated in Corollary 3.5, and the real–real block of the minimizer converges to the balanced entropic coupling by Corollary 3.6.

The authors then specialize EPOT to Gaussian mixtures by using squared Gaussian Wasserstein distances as component costs. The resulting partial component coupling induces a continuous Gaussian-mixture transport plan through the optimal Gaussian maps between components. They prove narrow convergence of these induced plans in the large-penalty limit (Theorem 4.1) and analyze the subsequent zero-entropy limit, where the entropic optimizer selects the maximum-entropy optimizer among the unregularized mixture OT minimizers (Proposition 4.2 and Theorem 4.3). The same componentwise construction yields the entropic partial displacement interpolation (4.18), whose convergence to the balanced and maximum-entropy interpolations is established in Proposition 4.4.

The authors also extend the partial-matching idea to Gromov–Wasserstein (GW) geometry by applying partial GW to the finite metric measure spaces of Gaussian components. This yields a partial mixture GW distance (5.2), whose basic properties follow from the partial GW theory. It admits minimizers, defines a metric on the associated component metric measure spaces modulo strong isomorphism, and coincides with the balanced mixture GW distance for sufficiently large λ (Theorem 5.1).

Finally, the authors derive barycentric projection maps that convert component couplings into pointwise correspondences. For the entropic partial barycentric projection map, the large-penalty convergence is established in Theorem 6.1. The numerical experiments on synthetic Gaussian mixtures and point clouds demonstrate that the penalty parameter λ > 0 controls the amount of matched mass, whereas the regularization parameter ε > 0 controls the concentration of the component coupling. Partial mixture GW preserves the common geometric structure in the presence of outliers, while the component-level GW formulation substantially reduces the computational burden.

Improvements to AI Systems:

1. Robust Partial Matching for Noisy Data: Enhance AI systems (e.g., clustering, domain adaptation) to handle datasets with outliers or missing parts by integrating the EPOT framework. The system can explicitly control the fraction of unmatched mass via the penalty λ, enabling it to ignore spurious points while aligning core structures—unlike standard OT which forces full matching.

2. Entropic Regularization for Scalable Gaussian Mixture Alignment: Improve generative models or shape-matching systems by using the entropic partial OT with Gaussian component costs. The AI can now compute smooth, differentiable transport plans between Gaussian mixtures with tunable concentration (ε), making gradient-based learning stable and computationally efficient for high-dimensional mixture data.

3. Maximum-Entropy Selection in Degenerate OT: When multiple optimal transport plans exist (e.g., in underdetermined problems), the AI system can automatically select the most diffuse, least-committal solution by leveraging the zero-entropy limit result (Theorem 4.3). This improves generalization and avoids overconfident, arbitrary alignments in tasks like representation learning.

4. Partial Gromov–Wasserstein for Structure-Preserving Outlier Rejection: For graph or point-cloud matching, the AI can use the partial mixture GW distance to compare metric measure spaces while ignoring outlier components. This yields robust geometric correspondence (e.g., in 3D registration or molecular alignment) without requiring pre-filtering, and reduces computational cost by operating on component-level summaries rather than raw points.

5. Barycentric Projection for Interpretable Correspondences: The derived barycentric maps allow AI systems to convert probabilistic component couplings into explicit point-to-point mappings. This improves explainability in tasks like image morphing or style transfer, where the system can output a concrete displacement field while still accounting for partial overlap and noise.

6. Hyperparameter-Guided Mass Control: The quantitative estimates (Propositions 3.3, 3.5) enable AI systems to automatically tune λ to achieve a desired matched-mass fraction (e.g., 90% of data) with provable convergence guarantees, improving reliability in applications like sensor fusion or multi-modal registration where overlap ratios are known a priori.

7. Entropic Partial Displacement Interpolation: AI systems for animation or data augmentation can generate smooth, physically plausible interpolations between Gaussian mixtures that gracefully handle missing data, using the entropic partial interpolation (4.18). This is useful for generating intermediate states in time-series forecasting or video frame interpolation with occlusions.

8. Component-Level GW for Scalable Unbalanced Matching: The AI system can now perform unbalanced matching on large point clouds by first compressing them into Gaussian components (via EM), then applying partial GW on those components. This reduces complexity from O(N2) to O(K2) (K ≪ N), enabling real-time matching in robotics or autonomous driving without sacrificing outlier robustness.

Optimal transport and Gromov--Wasserstein distances are useful tools for comparing probability measures and metric measure spaces, but their balanced formulations force all mass to be matched. This constraint is often too strong for data with outliers, missing parts, or only partial overlap. In this paper, we develop entropic partial optimal transport for Gaussian mixture models and define a partial mixture Gromov--Wasserstein distance. For the finite entropic partial optimal transport problem, we prove the existence and uniqueness of the minimizer and establish quantitative large-penalty estimates. Moreover, the resulting entropic partial component couplings induce continuous partial transport plans through Gaussian optimal maps. We analyze their large-penalty and subsequent zero-entropy limits and construct the associated displacement interpolations and barycentric projection maps. In addition, by identifying each Gaussian mixture with a finite metric measure space of Gaussian components, we establish the metric property and large-penalty limit of the partial mixture Gromov--Wasserstein distance. Finally, numerical experiments on synthetic Gaussian mixtures and point clouds illustrate the effects of the penalty and entropic regularization and the robustness of partial matching to outliers.