标题： 平衡测度网络和超网络的运输度量的圆锥公式 显示英文标题

标题： Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks

摘要： 广义最优传输的Gromov-Wasserstein (GW)变体，旨在比较定义在不同度量空间上的概率密度，已成为分析具有复杂结构的数据的重要工具，例如点云或网络的集合。 为了克服某些限制，例如对等质量测度比较的限制和对异常值的敏感性，最近文献中引入了几种不平衡或部分传输的GW距离松弛方法。 本文关注的是Séjourné、Vialard和Peyré引入的锥形Gromov-Wasserstein (CGW)距离。 我们提供了一种半耦合形式的新公式，并将框架扩展到度量测度空间设置之外，以比较更一般的网络和超网络结构。 通过这种新公式，我们建立了CGW度量的几个基本性质，包括在缩放下的行为、在体积增长约束极限下的变分收敛性，以及与已建立的最优传输度量的比较界限。 我们进一步推导了定量界限，以表征CGW度量对底层测度扰动的鲁棒性。 CGW的超网络形式允许一种简单且可证明收敛的块坐标上升算法进行估计，并通过在合成和现实世界高维和结构化数据集上的实验，展示了我们方法的计算可行性和可扩展性。 显示英文摘要

摘要： The Gromov-Wasserstein (GW) variant of optimal transport, designed to compare probability densities defined over distinct metric spaces, has emerged as an important tool for the analysis of data with complex structure, such as ensembles of point clouds or networks. To overcome certain limitations, such as the restriction to comparisons of measures of equal mass and sensitivity to outliers, several unbalanced or partial transport relaxations of the GW distance have been introduced in the recent literature. This paper is concerned with the Conic Gromov-Wasserstein (CGW) distance introduced by S\'{e}journ\'{e}, Vialard, and Peyr\'{e}. We provide a novel formulation in terms of semi-couplings, and extend the framework beyond the metric measure space setting, to compare more general network and hypernetwork structures. With this new formulation, we establish several fundamental properties of the CGW metric, including its scaling behavior under dilation, variational convergence in the limit of volume growth constraints, and comparison bounds with established optimal transport metrics. We further derive quantitative bounds that characterize the robustness of the CGW metric to perturbations in the underlying measures. The hypernetwork formulation of CGW admits a simple and provably convergent block coordinate ascent algorithm for its estimation, and we demonstrate the computational tractability and scalability of our approach through experiments on synthetic and real-world high-dimensional and structured datasets.