Gromov-Wasserstein Averaging of Kernel and Distance Matrices
Gromov-Wasserstein Averaging of Kernel and Distance Matrices
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发表时间：
2016-06
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通讯作者：
G. Peyré;Marco Cuturi;J. Solomon
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作者：
G. Peyré;Marco Cuturi;J. Solomon
本文提出了一种计算距离矩阵或核矩阵的质心的新方法。这些矩阵定义了从各个域中采样的点之间的相互关系，不需要具有相同的大小或行与行对应。我们使用软分配准则来比较这些矩阵，该准则测量从一个相似矩阵的行到另一个相似矩阵的行的概率映射所引起的最小失真；这个准则相当于度量空间之间的Gromov-Wasserstein （GW）距离的正则化版本。然后将重心定义为输入矩阵相对于该准则的Frechet平均值，使软赋值的加权和最小化。我们为由此产生的非凸优化问题提供了一个快速迭代算法，建立在最先进的正则化最优运输工具之上。我们演示了它在量子化学中形状质心计算和分子构型能级预测中的应用。
This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices. These matrices, which define the interrelationships between points sampled from individual domains, are not required to have the same size or to be in row-by-row correspondence. We compare these matrices using the softassign criterion, which measures the minimum distortion induced by a probabilistic map from the rows of one similarity matrix to the rows of another; this criterion amounts to a regularized version of the Gromov-Wasserstein (GW) distance between metric-measure spaces. The barycenter is then defined as a Frechet mean of the input matrices with respect to this criterion, minimizing a weighted sum of softassign values. We provide a fast iterative algorithm for the resulting nonconvex optimization problem, built upon state-of-the-art tools for regularized optimal transportation. We demonstrate its application to the computation of shape barycenters and to the prediction of energy levels from molecular configurations in quantum chemistry.