Inconsistency of diagonal scaling under high-dimensional limit: a replica approach

In this note, we claim that diagonal scaling of a sample covariance matrix is asymptotically inconsistent if the ratio of the dimension to the sample size converges to a positive constant, where population is assumed to be Gaussian with a spike covariance model. Our non-rigorous proof relies on the replica method developed in statistical physics. In contrast to similar results known in literature on principal component analysis, the strong inconsistency is not observed. Numerical experiments support the derived formulas.