发表机构
Universidad Técnica Federico Santa María; Center for Mathematical Modeling (CNRS IRL2807), Universidad de Chile; Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins（费德里科·圣塔玛丽亚技术大学; 智利大学数学建模中心（法国国家科学研究中心国际研究联系项目2807）; 奥希金斯大学工程科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究通过指数加权高斯扰动获得的随机重心估计器，利用径向优势条件得出相关重心范数上界及充分条件，得到弱凸函数随机近端估计器收敛速率等，还用于随机投影估计器，通过数值实验验证相关性质。
AI 中文摘要
我们研究了通过指数加权高斯扰动获得的近端点和度量投影的随机重心估计器。我们的主要结果是在相对于规定轮廓的径向优势条件下，关于密度与指数权重成比例的概率测度的抽象比较定理。这产生了根据一维比较测度的相关重心范数的显式上界。我们还提供了径向优势的易处理充分条件，包括强凸性、添加非负凸项和星形约束。结果，我们获得了弱凸函数的随机近端估计器的改进收敛速率以及常数的渐近尖锐性。相同框架给出了到闭凸集上的随机投影估计器的相应速率。我们进一步建立了重心逼近算子的基本结构性质，如光滑性和余强制性。数值实验说明了预测速率、常数的维数缩放及其渐近尖锐性。
英文摘要
We study stochastic barycentric estimators for proximal points and metric projections obtained by exponentially reweighting Gaussian perturbations. Our main result is an abstract comparison theorem for probability measures with densities proportional to an exponential weight, under a radial dominance condition relative to a prescribed profile. This yields an explicit upper bound for the norm of the associated barycenter in terms of a one-dimensional comparison measure. We also provide tractable sufficient conditions for radial dominance, including strong convexity, addition of nonnegative convex terms, and star-shaped constraints. As a consequence, we obtain a refined convergence rate for stochastic proximal estimators of weakly convex functions, together with asymptotic sharpness of the constant. The same framework yields a corresponding rate for stochastic projection estimators onto closed convex sets. We further establish basic structural properties of the barycentric approximation operator, such as smoothness and cocoercivity. Numerical experiments illustrate the predicted rate, the dimensional scaling of the constant, and its asymptotic sharpness.
