谱聚类社区发现算法分析：一般二分环境
Analysis of spectral clustering algorithms for community detection: the general bipartite setting
摘要：
我们认为谱聚类算法的社区检测下一般的二分随机块模型（SBM）。现代谱聚类算法由三个步骤组成：（1）适当的邻接矩阵或拉普拉斯矩阵的正则化（2）谱截断的形式和（3）减少谱域中的k-means型算法。我们专注于基于邻接的谱聚类，并为第一步，提出了一种新的数据驱动的正则化，即使对于稀疏网络，也可以恢复邻接矩阵的浓度。这个结果是基于最近的随机二进制矩阵的正则化工作，但避免使用未知的人口水平参数，而是从数据中估计必要的数量。我们还提出并研究了一种新的变化的频谱截断步骤，并显示这种变化如何改变的性质的误分类率在一般的SBM。然后，我们将展示如何将一致性结果扩展到SBM之外的模型，例如具有近似聚类的非均匀随机图模型，包括图子聚类问题，以及一般的亚高斯双聚类。本文的一个主题是提供一个更好的理解分析的谱方法社区检测和建立一致性的结果，在相当一般的聚类模型和广泛的制度的程度增长，包括稀疏的情况下，平均预期的程度增长任意缓慢。
英文摘要：
We consider spectral clustering algorithms for community detection under a general bipartite stochastic block model (SBM). A modern spectral clustering algorithm consists of three steps: (1) regularization of an appropriate adjacency or Laplacian matrix (2) a form of spectral truncation and (3) a k-means type algorithm in the reduced spectral domain. We focus on the adjacency-based spectral clustering and for the first step, propose a new data-driven regularization that can restore the concentration of the adjacency matrix even for the sparse networks. This result is based on recent work on regularization of random binary matrices, but avoids using unknown population level parameters, and instead estimates the necessary quantities from the data. We also propose and study a novel variation of the spectral truncation step and show how this variation changes the nature of the misclassification rate in a general SBM. We then show how the consistency results can be extended to models beyond SBMs, such as inhomogeneous random graph models with approximate clusters, including a graphon clustering problem, as well as general sub-Gaussian biclustering. A theme of the paper is providing a better understanding of the analysis of spectral methods for community detection and establishing consistency results, under fairly general clustering models and for a wide regime of degree growths, including sparse cases where the average expected degree grows arbitrarily slowly.
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