Matrix norm shrinkage estimators and priors

TL;DR

This paper introduces a new class of minimax estimators for normal mean matrices that shrink Schatten norms, extending classical estimators and priors, and demonstrates their effectiveness through simulations.

Contribution

It generalizes existing estimators and priors for matrix mean estimation using Schatten norms, providing minimax properties and practical performance analysis.

Findings

Proposed estimators are minimax under Frobenius loss.

New priors based on Schatten norms are superharmonic and minimax.

Simulation results show improved performance for low-rank matrices.

Abstract

We develop a class of minimax estimators for a normal mean matrix under the Frobenius loss, which generalizes the James--Stein and Efron--Morris estimators. It shrinks the Schatten norm towards zero and works well for low-rank matrices. We also propose a class of superharmonic priors based on the Schatten norm, which generalizes Stein's prior and the singular value shrinkage prior. The generalized Bayes estimators and Bayesian predictive densities with respect to these priors are minimax. We examine the performance of the proposed estimators and priors in simulation.