论文标题
多麦克拉莫夫 - 沃斯坦运输和barycenters
Multi-Marginal Gromov-Wasserstein Transport and Barycenters
论文作者
论文摘要
Gromov-Wasserstein（GW）距离是Gromov-Hausdorff和Wasserstein距离的组合，可以比较两个不同的度量测量空间（MM空间）。由于其在衡量和远距离变换下的不变性，它们非常适合于图形和形状分析中的许多应用。在本文中，我们介绍了一组MM空间及其正则化和不平衡版本之间多界数GW传输的概念。作为一种特殊情况，我们讨论了多界融合的变体，该变体将MM空间的结构信息与来自其他标签空间的标签信息结合在一起。为了以数值方式解决新的配方，我们考虑了多核心GW问题的双凸松弛，如果成本函数有条件地为负面，则在平衡的情况下，这是紧张的。可以通过交替的最小化来解决松弛的模型，其中每个步骤都可以通过多界线的sndhorn方案执行。我们展示了我们的多核心GW问题与（不平衡，融合）GW barycenters的关系，并呈现了各种数值结果，这表明了概念的潜力。
Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis. In this paper, we introduce the concept of multi-marginal GW transport between a set of mm-spaces as well as its regularized and unbalanced versions. As a special case, we discuss multi-marginal fused variants, which combine the structure information of an mm-space with label information from an additional label space. To tackle the new formulations numerically, we consider the bi-convex relaxation of the multi-marginal GW problem, which is tight in the balanced case if the cost function is conditionally negative definite. The relaxed model can be solved by an alternating minimization, where each step can be performed by a multi-marginal Sinkhorn scheme. We show relations of our multi-marginal GW problem to (unbalanced, fused) GW barycenters and present various numerical results, which indicate the potential of the concept.
