非参数流形学习

AI总结：

本文提出一种基于图拉普拉斯估计的非参数流形距离估计器，给出误差上界并证明一致性，其收敛性源于Connes距离公式。

摘要：

我们提出了一种基于图拉普拉斯对拉普拉斯-贝尔特拉米算子估计的紧致黎曼流形中距离的估计器。我们以图拉普拉斯估计中的谱误差以及隐式地流形的若干几何性质为项，给出了流形距离估计误差的上界，更确切地说，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]）的估计误差上界。一个推论是（未截断的）流形距离一致性的证明。该估计器类似于Wasserstein距离的Kontorovic对偶重构的一个特例，即Connes距离公式，并且其收敛性质实际上源自该特例。

英文摘要：

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.

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