Estimation of Kullback-Leibler losses for noisy recovery problems within the exponential family

TL;DR

This paper develops methods to estimate Kullback-Leibler losses in noisy recovery problems within the exponential family, providing unbiased or nearly unbiased estimators inspired by Stein's method, with applications to image denoising and variable selection.

Contribution

It introduces conditions for unbiased estimation of Kullback-Leibler losses in exponential family noise models, extending Stein's unbiased risk estimator framework.

Findings

Effective estimators for Kullback-Leibler loss demonstrated in simulations.

Applications to image denoising and variable selection show practical benefits.

Kullback-Leibler loss estimation improves model selection in noisy exponential family settings.

Abstract

We address the question of estimating Kullback-Leibler losses rather than squared losses in recovery problems where the noise is distributed within the exponential family. Inspired by Stein unbiased risk estimator (SURE), we exhibit conditions under which these losses can be unbiasedly estimated or estimated with a controlled bias. Simulations on parameter selection problems in applications to image denoising and variable selection with Gamma and Poisson noises illustrate the interest of Kullback-Leibler losses and the proposed estimators.