matrix with diagonal entries as transformed function of the sample eigenvalues and $\tilde{U}$ is the matrix of the eigenvectors. In another interesting paper @Ledoit:17 proposed an estimator that shrinks the sample covariance matrix towards the identity matrix. In another paper, @Karoui:15 proposed a non-parametric estimation of spectrum of eigenvalues and show that his estimator is consistent in the sense of weak convergence of distributions.

The covariance matrix estimates based on eigen-spectrum shrinkage are well-conditioned in the sense that their eigenvalues are well bounded away from zero. These estimates are based on the shrinkage of the eigenvalues and therefore invariant under some orthogonal group i.e. the shrinkage estimators shrink the eigenvalues but eigenvectors remain unchanged. In other words, the basis (eigenvector) in which the data are given is not taken advantage of and therefore the methods rely on premise that one will be able to find a good estimate in any basis. In particular, it is reasonable to believe that the basis generating the data is somewhat nice. Often this translates into the assumption that the covariance matrix has particular structure that one should be able to take advantage of. In these situations, it becomes natural to perform certain form of 