A dual subspace parsimonious mixture of matrix normal distributions

Abstract

We present a parsimonious dual-subspace clustering approach for a mixture of matrix-normal distributions. By assuming certain principal components of the row and column covariance matrices are equally important, we express the model in fewer parameters without sacrificing discriminatory information. We derive update rules for an ECM algorithm and set forth necessary conditions to ensure identifiability. We use simulation to demonstrate parameter recovery, and we illustrate the parsimony and competitive performance of the model through two data analyses.

A parameters used in model selection simulation

The parameters used to generate observations are as follows. The mean parameters are,

$$\begin{aligned} \textbf{M}_1= & {} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{bmatrix} \,\, \textbf{M}_2 = \begin{bmatrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ \end{bmatrix} \\ \textbf{M}_3= & {} \begin{bmatrix} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ \end{bmatrix} \,\, \textbf{M}_4 = \begin{bmatrix} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ \end{bmatrix} \\ \end{aligned}$$

The covariance parameters are the same across groups and across dimensions and are specifed as,

$$\begin{aligned} \varvec{\varPhi }_{ig}&= \begin{bmatrix} 1.5157166 &{} 0 \\ 0 &{} 1.5157166\\ \end{bmatrix}, \quad \text { and } \quad \eta _{ig} = 0.7578583, \\ \end{aligned}$$

where $i = 1,2$.