基于模型的张量聚类的双重增强期望最大化算法
A Doubly-Enhanced EM Algorithm for Model-Based Tensor Clustering
摘要：
现代科学研究经常以张量的形式收集数据集，这就要求创新的统计分析方法。特别是，迫切需要张量聚类方法来理解数据中的异质性。我们提出了一个张量正态混合模型（TNMM）的方法，使概率解释和计算的易处理性。我们的统计模型利用张量协方差结构来减少简约建模的参数数量，同时明确利用相关性来更好地进行变量选择和聚类。我们提出了一个双重增强的期望最大化（DEEM）算法在此模型下进行聚类。E步和M步都是为张量数据精心定制的，以考虑高维的统计精度和计算成本。理论研究证实，即使张量的每个模式的维度以样本量的指数速率增长，DEEM也能实现一致的聚类。数值研究表明，与现有的方法相比，DEEM具有良好的性能。
英文摘要：
Modern scientific studies often collect data sets in the forms of tensors, which call for innovative statistical analysis methods. In particular, there is a pressing need for tensor clustering methods to understand the heterogeneity in the data. We propose a tensor normal mixture model (TNMM) approach to enable probabilistic interpretation and computational tractability. Our statistical model leverages the tensor covariance structure to reduce the number of parameters for parsimonious modeling, and at the same time explicitly exploits the correlations for better variable selection and clustering. We propose a doubly-enhanced expectation-maximization (DEEM) algorithm to perform clustering under this model. Both the E-step and the M-step are carefully tailored for tensor data in order to account for statistical accuracy and computational cost in high dimensions. Theoretical studies confirm that DEEM achieves consistent clustering even when the dimension of each mode of the tensors grows at an exponential rate of the sample size. Numerical studies demonstrate favorable performance of DEEM in comparison to existing methods.
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