arXiv:2606. 07782v1 Announce Type: cross Abstract: We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry.
arXiv:2510.26266v3 Announce Type: replace Abstract: The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a p...
The paper demonstrates that the common-geodesic condition—where every output triple shares a geodesic point in a metric—does not ensure Fisher consistency for the structured SVM with the standard coordinate-wise argmax decoder. It presents minimal counterexamples, including a four-output star and a tree-metric classification, showing that only path-shaped trees maintain argmax consistency. The study also identifies the smallest full-support counterexamples and provides exact primal-dual certificates for all optimality claims.
The paper introduces a method for approximate vector search in ultrametric (infinity-metric) spaces by using a projection operator that transforms arbitrary dissimilarity functions into ultrametric ones while preserving nearest neighbors. It extends this approach to q-metric spaces, where triangle inequalities involve q-th powers, and learns an efficient approximation of the projected distances. Experiments show that higher q values speed up search at the cost of recall, yet the proposed methods remain competitive with existing search techniques.
arXiv:2610. 22592v1 Announce Type: cross Abstract: Empirical averages are fundamental statistical primitives.
arXiv:2605. 09916v2 Announce Type: replace-cross Abstract: We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets.
Quantifying the diversity of a set of objects is a fundamental problem in machine learning and statistics. Mironov and Prokhorenkova (2025) recently showed that three axioms-monotonicity, uniqueness,...
arXiv:2511.09801v3 Announce Type: replace-cross Abstract: This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by inc...
arXiv:2603. 07221v2 Announce Type: replace Abstract: Margin-based learning, exemplified by linear and kernel methods, is one of the few classical settings where generalization guarantees are independent of the number of parameters.
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:2601.01405v3 Announce Type: replace-cross Abstract: Ball Mapper summarizes a finite metric dataset by covering the sample with closed balls centered at selected landmarks and connecting landmar...
arXiv:2610.19570v1 Announce Type: cross Abstract: Quantifying the diversity of a set of objects is a fundamental problem in machine learning and statistics. Mironov and Prokhorenkova (2025) recently...
The paper introduces EGGroW, efficient algorithms for computing geodesic Gromov-Wasserstein distances using entropic Sinkhorn-like methods, GenusSink techniques, and random features. It addresses the cubic time complexity of traditional GWD calculations on dense intra-space distance matrices, enabling scalable comparisons of probabilistic distributions on general geodesic manifolds and graph shortest‑path distances. The authors demonstrate EGGroW’s effectiveness in downstream tasks such as 3D pose estimation and partial 3D template recovery, showing accurate results where Euclidean‑based methods fail while maintaining a light computational footprint.