Peter Hoff 2022-08-10

The Kronecker-core decomposition

Let $Y_1,\ldots,Y_n$ be a sample of $p_1\times p_2$ matrices, and let $y_1,\ldots,y_n$ be their vectorizations. The sample covariance matrix of $y_1,\ldots, y_n$ is a $p\times p$ positive semidefinite matrix, where $p=p_1 \cdot p_2$ . The Kronecker-core decomposition of the covariance matrix is a matrix factorization of the form

$S = (K_2\otimes K_1)^{1/2} C (K_2\otimes K_1)^{1/2},$

where

The covariance matrix $K=K_2\otimes K_1$ represents the “separable part” of , whereas represents the “nonseparable part”. In particular, if is separable, than .

Reference

Hoff, McCormack and Zhang (2022). Core Shrinkage Covariance Estimation for Matrix-variate Data.

Installation

# Development version 
devtools::install_github("pdhoff/covKCD")
# CRAN-approved version 
install.packages("covKCD")

Basic usage

p1<-3 ; p2<-4 ; n<-10
# Random sample of n independent standard normal p1xp2 matrices 
Y<-array(rnorm(n*p1*p2),dim=c(n,p1,p2))
dim(Y)

## [1] 10  3  4

# Sample covariance matrix 
S<-covKCD::mcov(Y)
dim(S)

## [1] 12 12

# KCD 
KCDS<-covKCD::covKCD(S,p1,p2)
KCDS$K1

##             [,1]        [,2]        [,3]
## [1,]  1.33635120 -0.44289281 -0.06881939
## [2,] -0.44289281  0.67144330 -0.02587571
## [3,] -0.06881939 -0.02587571  1.04758540

KCDS$K2

##             [,1]       [,2]        [,3]        [,4]
## [1,]  0.94256591  0.1405612 -0.23992050  0.01091884
## [2,]  0.14056122  0.9621321 -0.25617191 -0.01655000
## [3,] -0.23992050 -0.2561719  1.15019722  0.08804809
## [4,]  0.01091884 -0.0165500  0.08804809  0.70790240

# Check decomposition 
K1h<-covKCD::msqrt(KCDS$K1)
K2h<-covKCD::msqrt(KCDS$K2)
range( kronecker(K2h,K1h) %*% KCDS$C %*% kronecker(K2h,K1h) - S )

## [1] -1.887379e-15  1.370432e-15

The Kronecker-core decomposition

Reference

Installation

Basic usage

Additional examples