Gromov-Wasserstein (GW) compares distributions through relations within each space. This pointwise comparison can be too sensitive in one-to-many settings, where several target outcomes refine one source state and their mean carries the geometry of interest. We introduce a weak GW framework that compares source relations with relations between the target conditional laws induced by a coupling. For inner-product relations, we retain the conditional means $m_π(x)=\mathbb{E}_π[Y\mid X=x]$. The resulting barycentric weak inner-product GW (wIGW) satisfies $\mathrm{wIGW}_{\mathrm{bar}}^2(μ,ν)=\inf_{η\preceq_{\mathrm{cx}}ν}\mathrm{IGW}^2(μ,η)$. Here $η\preceq_{\mathrm{cx}}ν$ means that $ν$ is a mean-preserving spread of $η$. Thus wIGW searches for an intermediate target geometry that can be refined into the prescribed target law without changing conditional means. Under finite second moments, minimizers exist and martingale gluing recovers an optimal coupling. With ridge regularization, moment duality gives an $A$-$B$ min-max problem whose inner step is weak optimal transport with a quadratic cost parameterized by $A$ and $B$; the outer problem optimizes these matrices. For finitely supported measures, we give an iterative algorithm. Under a quantitative ridge condition, the reduced problem is convex--concave, and the projected outer iteration satisfies an explicit contraction bound for inexact inner solves. Point cloud and graph feature refinement experiments illustrate how mean-preserving target refinements can have zero cost. A paired peripheral blood mononuclear cell (PBMC) multiome study evaluates atlas based cell type transfer through RNA/ATAC alignment in cell to cell and prototype to cell settings, with the prototype to cell setting representing the one-to-many case.
We propose a new approach to two-sample testing for deciding whether two sets of samples are drawn from the same distribution. The test is built on a statistical discrepancy based on the zero-flow criterion, termed zero-flow discrepancy (ZFD). We prove the validity of ZFD and propose a practical testing procedure, termed the zero-flow two-sample test (ZF2ST). The key idea is to learn how samples from the two distributions are locally misaligned and use the resulting directional pattern as evidence of distributional difference. By separating witness learning from hypothesis evaluation, ZF2ST can use flexible neural networks while maintaining valid statistical calibration. We develop both regression-based and power-maximized approaches for learning the witness. Experiments on synthetic and image datasets demonstrate that ZF2ST can achieve strong testing power for structured distributional changes while maintaining well-calibrated type-I error.
We introduce the Directional Kernel Mean Difference (DKMD), a signed statistic for univariate distribution comparison that preserves the direction of distributional shifts. Unlike the squared Maximum Mean Discrepancy (MMD), which discards directional information by squaring the RKHS distance, DKMD integrates the difference of kernel mean embeddings against a fixed odd weighting function. This construction yields three structural properties: antisymmetry, immunity to symmetric distributional differences, and directional monotonicity under stochastic dominance. We derive a data-driven Riemann estimator that ensures asymptotic consistency with the continuous formulation, strictly preserving the theoretical guarantees of the signed statistic in empirical evaluations. To overcome the quadratic computational cost of kernel methods, we develop an $O(N \log N)$ prefix--suffix scanning algorithm that exploits the total order of the real line while requiring only $O(N)$ memory. Experiments on synthetic benchmarks demonstrate that DKMD correctly isolates directional shifts from symmetric perturbations, remains robust to heavy-tailed outliers that can flip the sign of the mean difference, and scales to millions of samples in seconds.
Jan Speller, Malte Luttermann, Marcel Gehrke +1cs.AI
To allow for principled comparison between two probabilistic graphical models defined over non-identical variable sets, they have to be lifted to a common measurable space. To this end, we propose an extension scheme for any two given models and establish the formal foundation: Unmatched components are completed using conditionally uniform (Laplace) extensions such that the resulting joint distributions differ from the original ones only by multiplicative constants and coincide under projection. This preserves the probabilistic semantics while enabling the application of well-defined distributional discrepancy measures. We establish the invariance of the induced joint under projection and use the extensions to provide a minimal structural extension of two factor graphs to the smalles common measurable space as well as to a common graphical structure by a deterministic algorithm. In addition, we discuss structural and measure-theoretic properties and identify promising criteria for comparison methodologies.