High dimensional matrix estimation with unknown variance of the noise

Published in Statistica Sinica, 2017

S. Gaïffas and O. Klopp

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 $A_0$ that does not rely on the knowledge or on an estimation of the standard deviation of the noise $\sigma$. Our estimator achieves, up to a logarithmic factor, optimal rates of convergence under Frobenius risk and, thus, has the same prediction performance as previously proposed estimators that rely on the knowledge of $\sigma$. Some numerical experiments show the benefits of this approach.

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