提出一种通用方法，精准估算高维核矩阵的范数。
A general technique for approximating high-dimensional empirical kernel matrices
- 基于去耦和非交换辛钦不等式，仅用核函数统计量推导上下界。
- 在多项式相关样本与维度下，给出内积核矩阵更紧的近似结果。
- 适用于各向异性高斯数据，适合关注核方法理论分析的研究者。
我们针对一般核函数 $k(ullet,ullet)$，提出了随机核矩阵期望算子范数的简单、易用的上下界。方法结合了U-统计量的去耦结果与非交换辛钦不等式，仅依赖于核函数的标量统计量及一个对应的“相关核”矩阵。将该方法应用于一般高维数据的内积核矩阵，当样本量与数据维度呈多项式关系时，得到新的、更紧的近似结果。该方法简化了以往依赖矩方法与组合论证的证明，同时对各向异性高斯数据给出了全新的近似结果。此外，利用类似技术，我们还得到了各向异性高斯数据下核回归偏差的更紧下界。
原文摘要 · Abstract (English)
We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(\cdot,\cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative Khintchine inequality to obtain upper and lower bounds depending only on scalar statistics of the kernel function and a ``correlation kernel'' matrix corresponding to $k(\cdot,\cdot)$. We then apply our method to provide new, tighter approximations for inner-product kernel matrices on general high-dimensional data, where the sample size and data dimension are polynomially related. Our method obtains simplified proofs of existing results that rely on the moment method and combinatorial arguments while also providing novel approximation results for the case of anisotropic Gaussian data. Finally, using similar techniques to our approximation result, we show a tighter lower bound on the bias of kernel regression with anisotropic Gaussian data.
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