On the Optimal Stacking of Information-Plus-Noise Matrices

Observations of the form D + X, where D is a matrix representing information, and X is a random matrix representing noise, can be grouped into a compound observation matrix, on the same information + noise form. There are many ways the observations can be stacked into such a matrix, for instance vertically, horizontally, or quadratically. An unbiased estimator for the spectrum of D can be formulated for each stacking scenario in the case of Gaussian noise. We compare these spectrum estimators for the different stacking scenarios, and show that all kinds of stacking actually decrease the variance of the corresponding spectrum estimators when compared to just taking an average of the observations, and find which stacking is optimal in this sense. When the number of observations grow, however, it is shown that the difference between the estimators is marginal, with only the cases of vertical and horizontal stackings having a higher variance asymptotically.