High dimensional matrix estimation with unknown variance of the noise

TL;DR

This paper introduces a novel pivotal estimator for high-dimensional matrix recovery that does not require prior knowledge of noise variance, achieving near-optimal convergence rates and computational efficiency.

Contribution

It presents a new convex optimization-based method for matrix estimation that is robust to unknown noise variance, improving practical applicability.

Findings

Achieves near-optimal convergence rates under Frobenius risk.

Does not require prior knowledge of noise standard deviation.

Computationally efficient convex optimization approach.

Abstract

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 $\sigma$. 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 $\sigma$. Our method is based on the solution of a convex optimization problem which makes it computationally attractive.