External Division of Two Bregman Proximity Operators for Poisson Inverse Problems

2026-02-12

AI Summary

This work addresses sparse recovery in linear models under Poisson noise, where conventional ℓ₁ regularization often introduces estimation bias. To mitigate this issue, the authors propose a novel operator that, for the first time, combines two Bregman proximity operators via an extragradient scheme and embeds it within the NoLips algorithmic framework. Optimizing with respect to the Kullback–Leibler (KL) divergence, the method effectively reduces bias while promoting sparsity. The geometric structure of the approach is further elucidated through a primal–dual perspective. Experimental results on both synthetic data and image restoration tasks demonstrate that the proposed algorithm achieves significantly improved reconstruction performance and more stable convergence compared to standard KL-based methods.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Privacy
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Poisson inverse problems
• sparse recovery
• Poisson noise
• linear models
• sparsity

Innovation

• Bregman proximity operator
• external division
• Poisson inverse problems
• sparsity promotion
• NoLips algorithm