Eigen structure of a new class of covariance and inverse covariance matrices

Abstract

There is a one to one mapping between a $p$ dimensional strictly positive definite covariance matrix $\Sigma$ and its matrix logarithm $L$. We exploit this relationship to study the structure induced on $\Sigma$ through a sparsity constraint on $L$. Consider $L$ as a random matrix generated through a basis expansion, with the support of the basis coefficients taken as a simple random sample of size $s=s^{*}$ from the index set $[p(p+1)/2]=\{1,\ldots,p(p+1)/2\}$. We find that the expected number of non-unit eigenvalues of $\Sigma$, denoted $\mathbb{E}[|\mathcal{A}|]$, is approximated with near perfect accuracy by the solution of the equation

\[\frac{4p+p(p-1)}{2(p+1)}[\log (\frac{p}{p-d})-\frac{d}{2p(p-d)}]-s^{*}=0.\] Furthermore, the corresponding eigenvectors are shown to possess only ${p-|\mathcal{A}^{c}|}$ non-zero entries. We use this result to elucidate the precise structure induced on $\Sigma$ and $\Sigma^{-1}$. We demonstrate that a positive definite symmetric matrix whose matrix logarithm is sparse is significantly less sparse in the original domain. This finding has important implications in high dimensional statistics where it is important to exploit structure in order to construct consistent estimators in non-trivial norms. An estimator exploiting the structure of the proposed class is presented.