I'm reading a paper in the context of high-dimensional data where the authors propose to estimate a $p\times p$ unknown covariance matrix $\Sigma$ with a $p\times p$ matrix $\tilde{\Sigma}$ that has low rank $r<p$. Despite doing this, they don't give an intuition about the motivation of this approach.
What are the consequences (or benefits) of having a low-rank estimation of $\Sigma$?
All that I can think of is the connection with PCA: computing the eigenvectors of the low-rank estimation $\tilde{\Sigma}$ I will have the principal components. But why is this useful when I can compute the classical estimator $\hat{\Sigma}=n^{-1}X^{T}X$ (for $n\times p$ centered data $X$) and then compute the eigenvectors?