סטטיסטיקות ווסרשטיין למרחקי מטריצות: גרומוב-וסרשטיין

תקציר מקורי באנגליתarXiv: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. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose \emph{Distance-Matrix Wasserstein} (DMW), a hierarchy of Wasserstein statistics comparing laws of random finite distance matrices. Rather than optimizing a global point-level alignment, DMW samples $n$ points from each space, records their pairwise distances, and transports the resulting matrix laws. We prove that DMW is a relaxation and lower bound of GW, and establish a reverse approximation inequality: the GW--DMW gap is controlled by the Wasser