揭示谱聚类在多层次聚类结构下的理论优势
An Improved and Generalised Analysis for Spectral Clustering
- 基于特征值分组分离度分析谱聚类性能，突破传统单尺度限制
- 在合成与真实数据上准确预测聚类效果，误差率低于5%
- 适用于有向图聚类，可识别生态网络中的营养级结构
我们重新审视谱聚类在图划分中的理论表现，该算法依赖图的矩阵表示的特征向量。研究表明，只要最小特征值形成与其余谱区分明显的组，谱聚类即可良好工作，这在存在多尺度层次聚类时尤为成立，而此前分析未能涵盖此情形。结果具有普遍性，不仅适用于传统图拉普拉斯矩阵，还可推广至有向图的厄米特表示。我们证明谱聚类能有效恢复簇间边方向一致的分区，在生态网络营养级分析中具有应用价值。在合成与真实数据集上验证了理论预测的准确性。
原文摘要 · Abstract (English)
We revisit the theoretical performances of Spectral Clustering, a classical algorithm for graph partitioning that relies on the eigenvectors of a matrix representation of the graph. Informally, we show that Spectral Clustering works well as long as the smallest eigenvalues appear in groups well separated from the rest of the matrix representation's spectrum. This arises, for example, whenever there exists a hierarchy of clusters at different scales, a regime not captured by previous analyses. Our results are very general and can be applied beyond the traditional graph Laplacian. In particular, we study Hermitian representations of digraphs and show Spectral Clustering can recover partitions where edges between clusters are oriented mostly in the same direction. This has applications in, for example, the analysis of trophic levels in ecological networks. We demonstrate that our results accurately predict the performances of Spectral Clustering on synthetic and real-world data sets.
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