We introduce an estimator for distances in a compact Riemannian manifold based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper bound the error in the estimate of manifold distances, or more precisely an estimate of a spectrally truncated variant of manifold distance of interest in non-commutative geometry (cf. [Connes and Suijelekom, 2020]), in terms of spectral errors in the graph Laplacian estimates and, implicitly, several geometric properties of the manifold. A consequence is a proof of consistency for (untruncated) manifold distances. 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：我们提出了一种基于图拉普拉斯算子对拉普拉斯-贝尔特拉米算子的估计，用于估计紧致黎曼流形中的距离。我们以图拉普拉斯估计的谱误差以及流形的若干几何属性为隐式参数，给出了流形距离估计误差的上界，更精确地说，是对非交换几何中感兴趣的流形距离的谱截断变体（参见Connes和Suijelekom, 2020）的估计误差的上界。该结果的一个推论是证明了（非截断的）流形距离的一致性。该估计量类似于瓦瑟斯坦距离的康托罗维奇对偶公式的一个特例——即众所周知的Connes距离公式，事实上，其收敛性质也源于此。