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We consider the problem of low rank matrix recovery in a stochastically noisyhigh dimensional setting. We propose a new estimator for the low rank matrix,based on the iterative hard thresholding method, and that is computationallyefficient and simple. We prove that our estimator is efficient both in terms ofthe Frobenius risk, and in terms of the entry-wise risk uniformly over anychange of orthonormal basis. This result allows us, in the case where thedesign is Gaussian, to provide the limiting distribution of the estimator,which is of great interest for constructing tests and confidence sets for lowdimensional subsets of entries of the low rank matrix.

An iterative hard thresholding estimator for low rank matrix recovery with explicit limiting distribution