论文标题
最小加权标准和内核插值的概括误差
Generalization error of minimum weighted norm and kernel interpolation
论文作者
论文摘要
我们研究了插入规定数据点的功能的概括误差，并通过最大程度地减少加权规范来选择。在自然和一般条件下，我们证明，随着参数的数量的增加，插值剂及其概括误差会融合，并且有限的插值属于繁殖的内核希尔伯特空间。严格地建立了最小加权规范插值的隐式偏见，并解释了为什么规范最小化可能会受益或过度参数损失。作为该理论的特殊情况，我们研究了三角多项式和球形谐波的插值。我们的方法来自确定性和近似理论的观点，而不是统计或随机矩阵。
We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural and general conditions, we prove that both the interpolants and their generalization errors converge as the number of parameters grow, and the limiting interpolant belongs to a reproducing kernel Hilbert space. This rigorously establishes an implicit bias of minimum weighted norm interpolation and explains why norm minimization may either benefit or suffer from over-parameterization. As special cases of this theory, we study interpolation by trigonometric polynomials and spherical harmonics. Our approach is from a deterministic and approximation theory viewpoint, as opposed to a statistical or random matrix one.
