AI 中文总结
本研究针对非自反巴拿赫空间上的统计逆问题，采用带任意凸泛函的Tikhonov正则化方法，通过Bregman距离分析收敛性并推导误差概率上界，相关理论经数值实验验证。
AI 中文摘要
统计框架下的逆学习具有广泛应用，在机器学习、人工智能及相关领域受到广泛关注，其目标是从间接且含噪声的观测中推断未知参数。本研究针对求解方程$Au=g$的$u^{\text{†}}$的稳定近似展开研究，其中$A$为合适向量空间间的线性算子。我们考虑定义域为非自反巴拿赫空间，陪域为度量空间$X$上的实值函数空间；函数$g$由有限个独立同分布的数据点表征，这些数据点遵循某未知概率测度$\rho$。我们采用带有任意凸泛函的Tikhonov正则化方法，以得到对应给定数据点的正则化解；基于Bregman距离开展收敛分析，以概率形式推导误差上界。理论发现随后通过数值实验得到验证。
英文摘要
Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $ρ$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.
