A Shared Nearest Neighbor (SNN) graph is a type of graph construction using shared nearest neighbor information, which is a secondary similarity measure based on the rankings induced by a primary $k$-nearest neighbor ($k$-NN) measure. SNN measures have been touted as being less prone to the curse of dimensionality than conventional distance measures, and thus methods using SNN graphs have been widely used in applications, particularly in clustering high-dimensional data sets and in finding outliers in subspaces of high dimensional data. Despite this, the theoretical study of SNN graphs and graph Laplacians remains unexplored. In this pioneering work, we make the first contribution in this direction. We show that large scale asymptotics of an SNN graph Laplacian reach a consistent continuum limit; this limit is the same as that of a $k$-NN graph Laplacian. Moreover, we show that the pointwise convergence rate of the graph Laplacian is linear with respect to $(k/n)^{1/m}$ with high probability.

翻译：共享最近邻（SNN）图是一种利用共享最近邻信息构建的图结构，其基于由原始 $k$-近邻（$k$-NN）度量生成的排名而定义的二次相似性度量。SNN度量被认为比传统距离度量更不易受维度灾难的影响，因此基于SNN图的方法在应用中广泛使用，特别是在高维数据聚类以及高维数据子空间异常点检测中。然而，SNN图及其图拉普拉斯算子的理论研究尚属于空白领域。在这项开创性工作中，我们首次对此方向做出贡献。我们证明，SNN图拉普拉斯算子在大尺度渐近意义下趋于一致连续极限，且该极限与 $k$-NN图拉普拉斯算子的极限相同。此外，我们证明该图拉普拉斯算子的逐点收敛速率在大概率下与 $(k/n)^{1/m}$ 成线性关系。