Statistica Sinica 27 (2017), 115-145

HIGH DIMENSIONAL MATRIX ESTIMATION WITH

UNKNOWN VARIANCE OF THE NOISE

Stéphane Gaïffas and Olga Klopp

CMAP, École Polytechnique and Modal'X,

University Paris Ouest Nanterre la Défense

Abstract: 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 σ. 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 σ. Some numerical experiments show the benefits of this approach.

Key words and phrases: Low rank matrix estimation, matrix completion, matrix regression, unknown variance of the noise.