揭示核插值在高阶范数下必然不一致的理论边界
Sobolev norm inconsistency of kernel interpolation
- 通过Sobolev型连续范数分析核插值误差
- 证明当光滑性指标超限时，泛化误差有正下界
- 适用于研究核方法泛化性能的理论工作者
我们研究了对应于有界核的再生核希尔伯特空间中最小范数插值的一致性。主要结果给出了在$ L^2 $与假设空间之间的连续范数尺度下，核插值泛化误差的下界。这些下界表明，当范数的光滑性指数大于仅依赖于假设空间嵌入指数和特征值衰减速率的常数时，核插值始终不一致。
原文摘要 · Abstract (English)
We study the consistency of minimum-norm interpolation in reproducing kernel Hilbert spaces corresponding to bounded kernels. Our main result give lower bounds for the generalization error of the kernel interpolation measured in a continuous scale of norms that interpolate between $L^2$ and the hypothesis space. These lower bounds imply that kernel interpolation is always inconsistent, when the smoothness index of the norm is larger than a constant that depends only on the embedding index of the hypothesis space and the decay rate of the eigenvalues.
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