Spectral clustering views the similarity matrix as a weighted graph, and partitions the data by minimizing a graph-cut loss. Since it minimizes the across-cluster similarity, there is no need to model the distribution within each cluster. As a result, one reduces the chance of model misspecification, which is often a risk in mixture model-based clustering. Nevertheless, compared to the latter, spectral clustering has no direct ways of quantifying the clustering uncertainty (such as the assignment probability), or allowing easy model extensions for complicated data applications. To fill this gap, we propose the Bayesian forest model as a generative graphical model for spectral clustering. This is motivated by our discovery that the posterior connecting matrix in a forest model has almost the same leading eigenvectors, as the ones used by normalized spectral clustering. To induce a distribution for the forest, we develop a ``forest process'' as a graph extension to the urn process, while we carefully characterize the differences in the partition probability. We derive a simple Markov chain Monte Carlo algorithm for posterior estimation, and demonstrate superior performance compared to existing algorithms. We illustrate several model-based extensions useful for data applications, including high-dimensional and multi-view clustering for images.

翻译：谱聚类将相似矩阵视为加权图，通过最小化图割损失对数据进行划分。由于该方法最小化跨簇相似度，因此无需为每个簇内的分布建模。这降低了模型误设的风险——而模型误设通常是基于混合模型聚类常见的隐患。然而，与后者相比，谱聚类缺乏直接量化聚类不确定性（如分配概率）的途径，也难以针对复杂数据应用进行便捷的模型扩展。为填补这一空白，我们提出贝叶斯森林模型，将其作为谱聚类的生成式图模型。这一思路源于我们的发现：森林模型中的后验连接矩阵与前几个特征向量几乎一致——这些特征向量正是归一化谱聚类所使用的。为诱导森林的分布，我们开发了一种“森林过程”，作为瓮过程的图扩展，同时细致刻画了划分概率的差异。我们推导出简单的马尔可夫链蒙特卡洛算法用于后验估计，并展示了其优于现有算法的性能。通过多个面向数据应用的模型扩展实例（包括高维图像聚类与多视图聚类），我们论证了该方法的实用性。