Predictive risk estimation for the Expectation Maximization algorithm with Poisson data

TL;DR

This paper introduces a new predictive risk estimator for Poisson data tailored for the EM algorithm, enabling optimal regularization parameter selection with proven asymptotic unbiasedness and demonstrated effectiveness in image deconvolution tasks.

Contribution

It develops a Poisson-specific Stein's Lemma and a corresponding risk estimator, extending existing methods to Poisson models and applying it to improve EM algorithm regularization.

Findings

The estimator is asymptotically unbiased with increasing counts.

It effectively selects regularization parameters for EM in image deconvolution.

Performance surpasses existing methods in numerical tests.

Abstract

In this work, we introduce a novel estimator of the predictive risk with Poisson data, when the loss function is the Kullback-Leibler divergence, in order to define a regularization parameter's choice rule for the Expectation Maximization (EM) algorithm. To this aim, we prove a Poisson counterpart of the Stein's Lemma for Gaussian variables, and from this result we derive the proposed estimator showing its analogies with the well-known Stein's Unbiased Risk Estimator valid for a quadratic loss. We prove that the proposed estimator is asymptotically unbiased with increasing number of measured counts, under certain mild conditions on the regularization method. We show that these conditions are satisfied by the EM algorithm and then we apply this estimator to select its optimal reconstruction. We present some numerical tests in the case of image deconvolution, comparing the performances of the proposed estimator with other methods available in the literature, both in the inverse crime and non-inverse crime setting.