arXiv:2605. 14981v2 Announce Type: replace Abstract: Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system.
arXiv:2606. 30523v1 Announce Type: new Abstract: Covariance matrices serve as compact descriptors of feature distributions in many machine-learning pipelines, including domain adaptation and Gaussian embeddings.
The paper introduces a geometric framework for measuring how far empirical datasets deviate from the Gaussian family using optimal transport theory. It defines two new quantities—the relative Wasserstein angle and the orthogonal projection distance—based on the cone structure of the relative translation invariant quadratic Wasserstein space, and shows that the usual moment‑matching Gaussian is not generally the $W_2$‑nearest Gaussian. Closed‑form expressions are derived for one‑dimensional and several location–scale families, while a numerical approximation is proposed for higher dimensions, with experiments demonstrating convergence, stability, and the angle’s robustness as a non‑Gaussianity indicator.
arXiv:2608. 04234v1 Announce Type: cross Abstract: We study the problem of aligning data from multiple modalities into a shared representation space, focusing on settings where strong pretrained unimodal encoders are available but cross-modal paired data are scarce.
The paper introduces Constant‑Curvature Sliced Gromov‑Wasserstein (CCSGW), a new divergence for aligning probability distributions on heterogeneous constant‑curvature spaces such as hyperbolic and spherical manifolds. It extends sliced Gromov‑Wasserstein by adding geodesic‑based one‑dimensional projections for spherical spaces, enabling efficient and principled comparison across manifolds with different curvatures while preserving intrinsic geometric relationships. The authors provide theoretical analysis showing that CCSGW controls intrinsic geometric discrepancy and demonstrate consistent performance gains when integrated into mixed‑curvature learning tasks like graph anomaly detection, node classification, and multimodal learning.
arXiv:2407. 01718v2 Announce Type: replace-cross Abstract: Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis.
The paper studies algorithms for computing the Entropic Gromov-Wasserstein (EGW) distance, a measure of discrepancy between metric measure spaces. It introduces Averaged Mirror Descent (AMD), which averages successive Mirror Descent steps and is proven to converge for any cost function, and shows that a dual gradient method with a fixed step size also converges for arbitrary costs, even when iterations are inexact. Empirical comparisons demonstrate that both AMD and the dual gradient method succeed on cases where classical Mirror Descent fails.
arXiv:2602. 04272v2 Announce Type: replace-cross Abstract: The Importance-Weighted Evidence Lower Bound (IW-ELBO) has emerged as an effective objective for variational inference (VI), tightening the standard ELBO and mitigating the mode-seeking behaviour.
arXiv:2606. 12120v1 Announce Type: new Abstract: Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature.
arXiv:2609.38049v1 Announce Type: new Abstract: Generative models for function-valued data, such as time series and solutions of partial differential equations, must learn distributions over infinite...
arXiv:2602. 02241v2 Announce Type: replace Abstract: Entropic optimal transport (EOT) in continuous spaces with quadratic cost is a classical tool for solving the domain translation problem.
arXiv:2605. 07914v2 Announce Type: replace Abstract: Sharpness-aware and gradient-alignment methods have been shown to improve generalization, however each family of methods targets a single geometric property of the loss landscape, while ignoring the other.