I was reading this article https://www.di.ens.fr/~aspremon/PDF/CovSelSIMAX.pdf, whose goal is to estimate the covariance matrix from a the sample covariance matrix drawn from a distribution $X$.
' Given a sample covariance matrix, we solve a maximum likelihood problem penalized by the number of nonzero coefficients in the inverse covariance matrix. Our objective is to find a sparse representation of the sample data and to highlight conditional independence relationships between the sample variables.'
The likelihood problem is only for the case where $X$ is a multivariate normal variable, so why did the author apply their method to a random distribution in section 4. ?
Why are we seeking for a sparse covriance matrix ? Doesn't the covariance matrix have to be unique?