High dimensional matrix estimation with unknown variance of the noise

We propose a new pivotal method for estimating high-dimensional matrices. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A\_0$ corrupted by noise. We propose a new method for estimating $A\_0$ which does not rely on the knowledge or an estimation of the standard deviation of the noise $σ$. Our estimator achieves, up to a logarithmic factor, optimal rates of convergence under the Frobenius risk and, thus, has the same prediction performance as previously proposed estimators which rely on the knowledge of $σ$. Our method is based on the solution of a convex optimization problem which makes it computationally attractive.

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