Graphical model - log transformed covariance matrix

Question

It is known that the non-zero entries of the inverse covariance matrix constitute an estimate on the edges in a graphical model. I wonder if there is an analog or similar result for the log-transformed of the covariance matrix (which applies the logarithm on all eigenvalues). I'm happy for any inputs.

Answer 1

This is a super interesting question! I don't have an answer to it and interested to know if someone know literature along this line.

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Here are some numerical experiments to show that a sparse precision matrix do not usually correspond to a sparse log-transformed precision or covariance matrix. or inverse of log covariance.

Given a chain-like model, its precision matrix shall be band-like sparse. $$ C^{-1}= \begin{bmatrix} 2 & 1 & 0 & 0\\ 1 & 2 & 1 & 0\\ 0 & 1 & 2 & 1\\ 0 & 0 & 1 & 2\\ \end{bmatrix} $$ Its covariance matrix is not sparse $$ C^{-1}= \begin{bmatrix} 0.8000 & -0.6000 & 0.4000 & -0.2000\\ -0.6000 & 1.2000 & -0.8000 & 0.4000\\ 0.4000 & -0.8000 & 1.2000 & -0.6000\\ -0.2000 & 0.4000 & -0.6000 & 0.8000\\ \end{bmatrix} $$ Similarly the log transformed $C$ or $C^{-1}$ are also not sparse... same for $(\log C)^{-1}$, they are all dense matrices, I cannot read the chain graph structure from these matrices... $$ -\log C =\log C^{-1}= \begin{bmatrix} -0.5100 & -0.6456 & 0.2152 & -0.0796\\ -0.6456 & -0.2948 & -0.7252 & 0.2152\\ 0.2152 & -0.7252 & -0.2948 & -0.6456\\ -0.0796 & 0.2152 & -0.6456 & -0.5100\\ \end{bmatrix} $$

$$ (\log C)^{-1}=\begin{bmatrix} 1.4582 & -0.0526 & -0.9820 & 0.9935\\ -0.0526 & 0.4762 & 0.9409 & -0.9820\\ -0.9820 & 0.9409 & 0.4762 & -0.0526\\ 0.9935 & -0.9820 & -0.0526 & 1.4582\\ \end{bmatrix} $$