Optimal variable clustering for high-dimensional matrix valued data

Area of Science:

• Statistics
• Machine Learning
• Data Science

Background:

• Matrix-valued data is increasingly common in various applications.
• Existing clustering methods often focus on the mean model and neglect the informative feature dependence structure.
• This limitation is particularly relevant in high-dimensional settings or when mean information is insufficient.

Purpose of the Study:

• To develop a new latent variable model for matrix-valued data that utilizes the feature dependence structure for clustering.
• To propose hierarchical clustering algorithms based on a weighted covariance matrix dissimilarity measure.
• To theoretically analyze the clustering consistency and optimality of the proposed method.

Main Methods:

• A novel latent variable model for matrix-valued data is proposed, incorporating row and column membership matrices.
• A class of hierarchical clustering algorithms is developed using the difference of a weighted covariance matrix as a dissimilarity measure.
• Theoretical analysis includes establishing clustering consistency in high-dimensional settings and deriving minimax lower bounds.

Main Results:

• The proposed algorithm demonstrates clustering consistency under mild conditions in high-dimensional settings.
• An optimal weight for the covariance matrix is identified, ensuring minimax rate-optimality.
• Simulation studies show superior performance compared to existing methods, evidenced by a higher adjusted Rand index (ARI).

Conclusions:

• The developed latent variable model and hierarchical clustering algorithm effectively utilize the dependence structure of matrix-valued data.
• The method provides theoretical guarantees for high-dimensional clustering and achieves optimal performance.
• The approach offers practical advantages and yields meaningful interpretations, as demonstrated on a genomic dataset.