重心弱内积Gromov-Wasserstein
Barycentric Weak Inner-Product Gromov-Wasserstein
AI总结：
针对GW在一对多场景下逐点比较过于敏感的问题，提出重心弱内积GW（wIGW）框架，通过保均值扩张的中间目标几何比较分布，给出迭代算法与理论保证，并在多组学等实验中验证了有效性。
AI中文摘要：
Gromov-Wasserstein（GW，格罗莫夫-瓦瑟斯坦）通过各空间内部的关系来比较分布。在一对多场景下，这种逐点比较可能过于敏感——此时多个目标结果细化同一个源状态，且它们的均值承载着我们关注的几何结构。我们提出一种弱GW框架，将源关系与由耦合诱导的目标条件分布之间的关系进行比较。对于内积关系，我们保留条件均值$m_π(x)=\text{E}_π[Y\bracevert X=x]$。由此得到的重心弱内积GW（wIGW）满足$\text{wIGW}_{\text{bar}}^2(μ,ν)=\text{inf}_{η\braceceq_{\text{cx}}ν}\text{IGW}^2(μ,η)$，其中$η\braceceq_{\text{cx}}ν$表示$ν$是$η$的保均值扩张。因此wIGW寻找一个中间目标几何结构，该结构可在不改变条件均值的情况下被细化为给定的目标分布。在有限二阶矩条件下，极小值解存在，且鞅粘合可恢复最优耦合。结合岭正则化时，矩对偶性给出一个A-B极小-极大问题：其内部步骤是代价为二次型、由A和B参数化的弱最优传输，外部问题则优化这些矩阵。对于有限支撑测度，我们给出了一种迭代算法。在定量岭条件下，约简后的问题是凸-凹的，且投影外部迭代对不精确内部求解满足显式收缩界。点云和图特征细化实验展示了保均值的目标细化如何实现零代价。一项配对的外周血单核细胞（PBMC）多组学研究通过RNA/ATAC比对，在细胞-细胞和原型-细胞设置下评估了基于图谱的细胞类型转移，其中原型-细胞设置代表一对多场景。
英文摘要：
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.
