标题： 非线性统计逆学习问题的Tikhonov正则化收敛分析 显示英文标题

标题： Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems

摘要： 我们研究了一个非线性统计逆学习问题，其中在一些随机设计点上通过一个非线性算子观察到一个量的噪声图像。 我们考虑广泛使用的Tikhonov正则化（或正则化方法，MOR）方法来重建非线性不适定逆问题中该量的估计器。 估计器定义为Tikhonov泛函的最小值，该泛函是数据不匹配项和二次惩罚项的和。 我们使用再生核希尔伯特空间的假设对Tikhonov正则化方案的最小值进行理论分析。 我们讨论了所提出方案在通过适当源条件定义的可接受解类上的收敛最优速率。 显示英文摘要

摘要： We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.