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The linear minimum mean squared error (LMMSE) estimator is the best linearestimator for a Bayesian linear inverse problem with respect to the meansquared error. It arises as the solution operator to a Tikhonov-typeregularized inverse problem with a particular quadratic discrepancy term and aparticular quadratic regularization operator. To be able to evaluate the LMMSEestimator, one must know the forward operator and the first two statisticalmoments of both the prior and the noise. If such knowledge is not available,one may approximate the LMMSE estimator based on given samples. In this work,it is investigated, in a finite-dimensional setting, how many samples areneeded to reliably approximate the LMMSE estimator, in the sense that, withhigh probability, the mean squared error of the approximation is smaller than agiven multiple of the mean squared error of the LMMSE estimator.

How many samples are needed to reliably approximate the best linear estimator for a linear inverse problem?