AI 中文总结
本文针对再生核巴拿赫空间上的统计逆问题，采用Tikhonov正则化结合统计学习技术，推导了估计解的概率收敛速率，通过数值实验验证了方法的有效性。
AI 中文摘要
近年来，由于统计学习理论与泛函分析方法在机器学习和人工智能领域的重要性日益提升，统计逆问题受到了广泛关注。本文研究满足方程 $Au = g$ 的元素 $u^{\text{†}}$ 的稳定逼近，其中 $A$ 是将巴拿赫空间映射到适当函数空间的线性算子。函数 $g$ 仅通过受噪声污染、且服从未知分布 $\rho$ 的独立同分布数据点观测得到。本文采用 Tikhonov 正则化方案，结合统计学习技术与再生核巴拿赫空间（Reproducing Kernel Banach Spaces，RKBS）框架对解进行估计。当数据点数量增加时，我们建立了估计解相对于真实解的收敛性，并推导了以概率形式表示的收敛速率。数值实验进一步验证了理论结果，证明了所提方法的有效性。
英文摘要
Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $ρ$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.
Journal refJournal of Complexity, Volume 95, 2026, 102050
