External Division of Two Bregman Proximity Operators for Poisson Inverse Problems

TL;DR

This paper introduces a new external division operator of Bregman proximity operators for Poisson inverse problems, improving sparse recovery by reducing bias and enhancing stability and performance over traditional methods.

Contribution

The paper proposes a novel external division of Bregman proximity operators and integrates it into the NoLips algorithm, providing better bias mitigation and interpretability.

Findings

More stable convergence than KL-based methods

Superior performance on synthetic and image data

Clear geometric interpretations in primal and dual spaces

Abstract

This paper presents a novel method for recovering sparse vectors from linear models corrupted by Poisson noise. The contribution is twofold. First, an operator defined via the external division of two Bregman proximity operators is introduced to promote sparse solutions while mitigating the estimation bias induced by classical $\ell_1$-norm regularization. This operator is then embedded into the already established NoLips algorithm, replacing the standard Bregman proximity operator in a plug-and-play manner. Second, the geometric structure of the proposed external-division operator is elucidated through two complementary reformulations, which provide clear interpretations in terms of the primal and dual spaces of the Poisson inverse problem. Numerical tests show that the proposed method exhibits more stable convergence behavior than conventional Kullback-Leibler (KL)-based approaches and achieves significantly superior performance on synthetic data and an image restoration problem.