谱聚类的平均灵敏度
Average Sensitivity of Spectral Clustering
摘要：
谱聚类是一种最常用的聚类方法，在数据采掘中有着广泛的应用。然而，这些应用中的输入图可能由于测量误差、出于隐私原因的保留或数据转换中的任意性而具有许多缺失的边。为了基于谱聚类做出可靠和有效的决策，我们使用平均灵敏度的概念来评估谱聚类对输入图中边缘扰动的稳定性，平均灵敏度是我们随机删除边缘之前和之后输出聚类的对称差异的预期大小。我们首先证明了谱聚类的平均灵敏度与$λ_2/λ_3^2$成正比，其中$λ_i$是（归一化）拉普拉斯算子的$i$ -th最小特征值。我们还证明了一个类似的$k$方式的谱聚类，分区图成$k$集群的限制。然后，我们通过在合成网络和真实网络上进行实验，实证地证实了我们的理论界限。我们的研究结果表明，谱聚类是稳定的边缘扰动时，有一个集群结构的输入图。
英文摘要：
Spectral clustering is one of the most popular clustering methods for finding clusters in a graph, which has found many applications in data mining. However, the input graph in those applications may have many missing edges due to error in measurement, withholding for a privacy reason, or arbitrariness in data conversion. To make reliable and efficient decisions based on spectral clustering, we assess the stability of spectral clustering against edge perturbations in the input graph using the notion of average sensitivity, which is the expected size of the symmetric difference of the output clusters before and after we randomly remove edges. We first prove that the average sensitivity of spectral clustering is proportional to $λ_2/λ_3^2$, where $λ_i$ is the $i$-th smallest eigenvalue of the (normalized) Laplacian. We also prove an analogous bound for $k$-way spectral clustering, which partitions the graph into $k$ clusters. Then, we empirically confirm our theoretical bounds by conducting experiments on synthetic and real networks. Our results suggest that spectral clustering is stable against edge perturbations when there is a cluster structure in the input graph.
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