论文标题
公制度量空间上的经典多维缩放
Classical Multidimensional Scaling on Metric Measure Spaces
论文作者
论文摘要
我们将经典多维缩放程序推广到一般度量测量空间的设置。我们为广义CMDS运算符开发了相关的光谱理论，该理论为CMD提供了更自然和严格的数学背景。此外，我们表明，CMDS运算符的所有负特征值的总和是一个新的不变式测量度量测量空间的非平局。此外，研究了几个非典型示例度量度量空间的CMDS输出，特别是S^{d-1}球的CMD和欧几里得空间的子集的CMD。最后，我们证明了有关Gromov-Wasserstein距离的广义CMDS过程的稳定性。
We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure space. Furthermore, the cMDS output of several non-finite exemplar metric measures spaces, in particular the cMDS for spheres S^{d-1} and subsets of Euclidean space, are studied. Finally, we prove the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance.
