Sparse Matrix Graphical Models

Abstract

Matrix-variate observations are frequently encountered in many contemporary statistical problems due to a rising need to organize and analyze data with structured information. In this article, we propose a novel sparse matrix graphical model for these types of statistical problems. By penalizing, respectively, two precision matrices corresponding to the rows and columns, our method yields a sparse matrix graphical model that synthetically characterizes the underlying conditional independence structure. Our model is more parsimonious and is practically more interpretable than the conventional sparse vector-variate graphical models. Asymptotic analysis shows that our penalized likelihood estimates enjoy better convergent rates than that of the vector-variate graphical model. The finite sample performance of the proposed method is illustrated via extensive simulation studies and several real datasets analysis.

KEY WORDS:

• Conditional independence
• Matrix-variate normal distribution
• Penalized likelihood
• Sparsistency
• Sparsity

Acknowledgments

Research support from National University of Singapore research grants is gratefully acknowledged. We thank coeditor Prof. Xuming He, an associate editor, and two anonymous referees for their constructive comments that have led to a much improved article.