We introduce an estimator for manifold distances based on graph Laplacian estimates of the Laplace-Beltrami operator. We show that the estimator is consistent for suitable choices of graph Laplacians in the literature, based on an equidistributed sample of points drawn from a smooth density bounded away from zero on an unknown compact Riemannian submanifold of Euclidean space. The estimator resembles, and in fact its convergence properties are derived from, a special case of the Kontorovic dual reformulation of Wasserstein distance known as Connes' Distance Formula.

翻译：我们根据Laplace-Beltrami操作员的Laplacecian图图估算,引入了多距离估计值。我们表明,根据从零平坦密度中提取的、从零平坦密度中提取的点数的均衡样本,估计值一致,在文献中选择“Laplace-Beltrami”操作员的图解拉placian估计值中,估计值一致。该估计值相似,事实上,其趋同性来自Kontorovic双倍重置瓦瑟斯坦距离(Connes的距离公式)的一个特殊案例。