Matrix valued data has become increasingly prevalent in many applications. Most of the existing clustering methods for this type of data are tailored to the mean model and do not account for the dependence structure of the features, which can be very informative, especially in high-dimensional settings. To extract the information from the dependence structure for clustering, we propose a new latent variable model for the features arranged in matrix form, with some unknown membership matrices representing the clusters for the rows and columns. Under this model, we further propose a class of hierarchical clustering algorithms using the difference of a weighted covariance matrix as the dissimilarity measure. Theoretically, we show that under mild conditions, our algorithm attains clustering consistency in the high-dimensional setting. While this consistency result holds for our algorithm with a broad class of weighted covariance matrices, the conditions for this result depend on the choice of the weight. To investigate how the weight affects the theoretical performance of our algorithm, we establish the minimax lower bound for clustering under our latent variable model. Given these results, we identify the optimal weight in the sense that using this weight guarantees our algorithm to be minimax rate-optimal in terms of the magnitude of some cluster separation metric. The practical implementation of our algorithm with the optimal weight is also discussed. Finally, we conduct simulation studies to evaluate the finite sample performance of our algorithm and apply the method to a genomic dataset.

翻译：矩阵数据在许多应用中日益普遍。现有的大多数针对此类数据的聚类方法均基于均值模型，并未考虑特征间的依赖结构，而这种结构在尤其在高维场景中可能蕴含丰富信息。为从依赖结构中提取聚类信息，我们针对以矩阵形式排列的特征提出了一种新的潜变量模型，其中包含若干未知的隶属矩阵来表示行和列的聚类。在此模型下，我们进一步提出了一类层次聚类算法，采用加权协方差矩阵的差异作为相异性度量。理论上，我们证明了在温和条件下，该算法在高维场景中具有聚类一致性。尽管这一一致性结果适用于一大类加权协方差矩阵的算法，但该结果成立的条件依赖于权重的选择。为探究权重如何影响算法的理论性能，我们在潜变量模型下建立了聚类的极小化最优下界。基于这些结果，我们识别出最优权重——使用该权重可保证算法在某些聚类分离度量的量级上达到极小化速率最优。此外，还讨论了采用最优权重的算法在实际中的实现。最后，通过仿真研究评估了算法的有限样本性能，并将该方法应用于一个基因组数据集。