Bregman邻近方法迭代收敛的统一框架
A Unified Framework for Iterate Convergence of Bregman Proximal Methods
AI总结：
该研究提出适用于广泛核函数与复合目标函数的BPM迭代收敛统一框架，确立了标准BPM在温和条件下的迭代收敛性，还得到镜像流无凸性假设的首个轨迹收敛结果，构建了BPM的统一轨迹收敛理论。
AI中文摘要：
Bregman邻近方法（BPMs）的迭代收敛性长期以来悬而未决，尤其是针对非凸目标函数的情况。近期，Chen等人（2026）通过所谓的缩放Kurdyka-Łojasiewicz（SKŁ）性质，在BPM的迭代收敛性研究中取得了进展，但仅适用于Shannon熵核和线性约束问题。本文提出了一种适用于广泛核函数及复合目标函数的统一迭代收敛框架。我们的方法扩展了Chen等人（2026）的分析工具，尤其是在确保生成序列收敛中起核心作用的SKŁ性质。通过引入依赖核的参数化函数，我们证明扩展后的SKŁ性质对所有连续亚解析函数成立，特别是当核具有闭域时。随后，我们验证了标准BPM在温和正则条件下满足该框架的假设，从而确立了其针对广泛目标函数的迭代收敛性。此外，基于参数化函数，我们证明连续时间BPM（镜像流）对于o-极小可定义目标函数收敛到一个平稳点，得到了无需对目标函数施加凸性假设或对平稳点施加隔离假设的镜像流首个轨迹收敛结果。综上，这些离散和连续时间的收敛结果为BPM提供了统一的轨迹收敛理论。
英文摘要：
Iterate convergence of Bregman proximal methods (BPMs) has long remained open, especially for nonconvex objectives. Recently, \citet{chen2026skl} made progress by establishing iterate convergence for a BPM via the so-called scaled Kurdyka-Łojasiewicz (SKŁ) property, but only for the Shannon entropy kernel and linearly constrained problems. In this paper, we develop a unified iterate convergence framework that applies to a broad group of kernels and composite objective functions. Our approach extends the analytical tools in \cite{chen2026skl}, in particular the SKŁ property, which plays a central role in ensuring convergence of the generated sequences. By introducing kernel-dependent parameterization functions, we show that the extended SKŁ property holds for all continuous subanalytic functions, particularly when the kernel has a closed domain. We then verify that the assumptions of the framework are satisfied by standard BPMs under mild regularity conditions, thereby establishing their iterate convergence for a wide range of objective functions. Furthermore, based on the parameterization functions, we show that the continuous-time BPM (mirror flow) converges to a stationary point for o-minimal definable objective functions, yielding the first trajectory convergence result for mirror flow without imposing convexity assumptions on the objective function or isolation assumptions on stationary points. Taken together, these discrete- and continuous-time convergence results provide a unified trajectory convergence theory for BPMs.
