SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1098-1118, June 2024.
Abstract. Data sites selected from modeling high-dimensional problems often appear scattered in nonpaternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high-dimensional data, which features bounding stochastic approximation error by the spectrum of the underlying kernel matrix. Both theoretical analysis and numerical simulations show that spectra of kernel matrices are reliable and stable barometers for gauging the performance of kernel-interpolation methods for high-dimensional data.

中文翻译：

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高维离散数据的核插值

SIAM 数值分析杂志，第 62 卷，第 3 期，第 1098-1118 页，2024 年 6 月
。摘要。从高维问题建模中选择的数据站点通常以非家长式的方式分散。除了某些地点的零星聚集外，随着周围空间维度的增长，它们之间的距离变得相对较远。这些特征违背了任何需要数据站点分布的局部或全局准均匀性的理论处理。结合最近开发的积分算子理论在机器学习中的应用，我们在本文中提出并研究了一种新的框架来分析高维数据的核插值，其特征是通过底层核矩阵的频谱来限制随机逼近误差。理论分析和数值模拟都表明，核矩阵谱是衡量高维数据核插值方法性能的可靠且稳定的晴雨表。