## Bernoulli

- Bernoulli
- Volume 23, Number 4B (2017), 3166-3177.

### 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.

#### Article information

**Source**

Bernoulli, Volume 23, Number 4B (2017), 3166-3177.

**Dates**

Received: December 2015

Revised: March 2016

First available in Project Euclid: 23 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1495505088

**Digital Object Identifier**

doi:10.3150/16-BEJ840

**Mathematical Reviews number (MathSciNet)**

MR3654802

**Zentralblatt MATH identifier**

06778282

**Keywords**

covariance matrix matrix logarithm precision matrix spectral theory

#### Citation

Battey, Heather. Eigen structure of a new class of covariance and inverse covariance matrices. Bernoulli 23 (2017), no. 4B, 3166--3177. doi:10.3150/16-BEJ840. https://projecteuclid.org/euclid.bj/1495505088