In this work, we investigate high-dimensional kernel ridge regression (KRR) on i.i.d. Gaussian data with anisotropic power-law covariance. This setting differs fundamentally from the classical source & capacity conditions for KRR, where power-law assumptions are typically imposed on the kernel eigen-spectrum itself. Our contributions are twofold. First, we derive an explicit characterization of the kernel spectrum for polynomial inner-product kernels, giving a precise description of how the kernel eigen-spectrum inherits the data decay. Second, we provide an asymptotic analysis of the excess risk in the high-dimensional regime for a particular kernel with this spectral behavior, showing that the sample complexity is governed by the effective dimension of the data rather than the ambient dimension. These results establish a fundamental advantage of learning with power-law anisotropic data over isotropic data. To our knowledge, this is the first rigorous treatment of non-linear KRR under power-law data.

翻译：本研究探讨了在具有各向异性幂律协方差的独立同分布高斯数据上进行高维核岭回归（KRR）的问题。该设定与KRR经典的源条件及容量条件存在本质区别——传统理论通常直接对核特征谱施加幂律假设。我们的贡献包含两个方面：首先，针对多项式内积核，我们推导了核谱的显式刻画，精确描述了核特征谱如何继承数据的衰减特性；其次，针对具有此类谱特性的特定核函数，我们在高维区域中对超额风险进行了渐近分析，证明样本复杂度由数据的有效维度而非环境维度主导。这些结果从理论上确立了各向异性幂律数据相较于各向同性数据在学习过程中的根本优势。据我们所知，这是首次对幂律数据下的非线性KRR问题进行的严格理论分析。