随机邻域图上的图拉普拉斯算子及其收敛性
Graph Laplacians and their convergence on random neighborhood graphs
摘要：
给定欧几里得空间中一个子流形上支持的概率测度的样本，可以构造一个邻域图，它可以看作是子流形的一个近似。这种图的图拉普拉斯算子用于几种机器学习方法，如半监督学习，降维和聚类。在本文中，我们确定了三个不同的图拉普拉斯算子在文献中使用的逐点极限的样本量增加和邻域大小接近零。我们表明，对于一个一致的措施，所有的图拉普拉斯算子的子流形上有相同的限制常数。然而，在子流形上的非一致测度的情况下，只有所谓的随机游动图拉普拉斯算子收敛于加权Laplace-Beltrami算子。
英文摘要：
Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.
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