发表机构
INSA-Lyon（里昂国立应用科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种右Bregman近端梯度算法，通过镜像坐标重参数化实现凸复合优化，并应用于泊松逆问题，恢复并推广了Richardson-Lucy更新，证明了单调递减和次线性收敛，实验验证了其性能。
AI 中文摘要
我们通过将标准Bregman近端梯度（BPG）方法应用于目标的镜像坐标重参数化，引入了一种用于凸复合优化的新型Bregman近端算法。在原始变量中，所得算法交替进行预条件梯度步和右Bregman近端更新。我们的分析依赖于相对光滑性在镜像坐标而非原始坐标中成立。在此条件下，我们证明了在一般凸设置中目标函数的单调递减。然后，我们使用加权负熵作为势函数，将该方法专门应用于泊松逆问题。所得方案恢复了经典的Richardson-Lucy乘法更新，并将其推广到一般凸正则化器。基于最近对乘法更新的收敛性分析，我们建立了泊松设置中函数值的次线性收敛速率。最后，我们在具有泊松分布观测的多个成像逆问题上展示了正则化算法的性能。代码可在此https URL公开获取。
英文摘要
We introduce a novel Bregman proximal algorithm for convex composite optimization by applying the standard Bregman proximal gradient (BPG) method to a mirror-coordinate reparameterization of the objective. In primal variables, the resulting algorithm alternates between a preconditioned gradient step followed by a right Bregman proximal update. Our analysis relies on relative smoothness holding in the mirror coordinates instead of primal coordinates. Under this condition, we prove monotonic decrease of the objective in the general convex setting. We then specialize the method to Poisson inverse problems using weighted negative entropy as the potential. The resulting scheme recovers the classical Richardson-Lucy multiplicative updates and extends them to general convex regularizers. Building on a recent convergence analysis of multiplicative updates, we establish a sublinear convergence rate in function values for the Poisson setting. Finally, we demonstrate the performance of the regularized algorithm on several imaging inverse problems with Poisson-distributed observations. The code is publicly available at https://github.com/Tmodrzyk/MU-Bregman.
