AI 中文总结
本文针对任意赋范线性空间中的函数拟合问题，用连续线性泛函替换点评估，提出最小二乘方法并证明解的唯一性，在病态矩阵情形下引入正则化方程，并给出阶最优误差估计。
AI 中文摘要
数学学习理论中的一个基本问题是识别一个具有特定性质的函数 $f: \Omega\to {\mathbb R}$，使其拟合给定的训练数据 $\{(x_i, \xi_i)\in \Omega\times {\mathbb R}: i=1, \ldots, n\}$，即满足 $f(x_i) = \xi_i$（$i=1, \ldots, n$），其中 $\Omega$ 是 ${\mathbb R}^d$（$d\in {\mathbb N}$）的紧子集，且要求 $f$ 具有某些指定的特征。我们处理的问题是当 $f$ 属于任意赋范线性空间 ${X}$ 时，将评估映射 $f\mapsto f(x_i)$ 替换为 $X$ 上的形如 $f\mapsto \varphi_i(f)$ 的映射，其中 $\varphi_1, \ldots, \varphi_n$ 是 ${X}$ 上的连续线性泛函。利用 ${\mathbb R}^n$ 上的内积结构，我们设计了一种最小二乘方法以获得上述问题的近似解，并确定了 ${X}$ 的一个子空间，在该子空间中最小二乘解是唯一的，进而证明这等价于求解一个矩阵方程。当所考虑的矩阵病态时，我们设计了一个正则化方程，并通过适当选择正则化参数，对精确目标数据 $\xi=(\xi_1, \ldots, \xi_n)$ 以及含噪情况，导出了阶最优的误差估计。
英文摘要
One of the basic problems in mathematical learning theory is to identify a function $f: Ω\to {\mathbb R}$ with certain specific properties which fits a given training data $\{(x_i, ξ_i)\in Ω\times {\mathbb R}: i=1, \ldots, n\}$, in the sense that, $f(x_i) = ξ_i$ for $i=1, \ldots, n$, where $Ω$ is a compact subset of ${\mathbb R}^d$ for some $d\in {\mathbb N}$ and $f$ is required to have certain specified characteristics. We address this problem when $f$ belongs to an arbitrary normed linear space ${X}$, and the evaluation maps $f\mapsto f(x_i)$ are replaced by maps of the form $ f\mapsto φ_i(f)$ on $X$, where $φ_1, \ldots, φ_n$ are continuous linear functionals on ${X}$. Using an inner product structure on ${\mathbb R}^n$, we shall device a method of least-squares for obtaining an approximate solution for the above problem and also identify a subspace of ${X}$ in which the least-square solution is unique, which in turn is shown to be equivalent to solving a matrix equation. In the context when the matrix under consideration is ill-conditioned, a regularized equation is devised, and order optimal error estimates are derived for the exact targeted data $ξ=(ξ_1, \ldots, ξ_n)$ and also when it is noisy, by choosing the regularization parameter appropriately.
