We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems in Hilbert scales under general source conditions. Our analysis does not require the regression function to be contained in the hypothesis class, and most notably does not employ the conventional \textit{a priori} assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of ``tight'' $L^{\infty}(\mathcal{X})$ embeddings of suitable Hilbert scales. Our analysis utilizes a new Fourier isocapacitary condition, which captures the interplay of the kernel Dirichlet capacities and small ball probabilities via the optimal Hilbert scale function.

翻译：我们针对希尔伯特尺度下一类广泛的Tikhonov正则化学习问题（在一般源条件下）推导了极小极大自适应率。该分析不要求回归函数包含于假设类中，且特别地无需采用传统的核特征衰减先验假设。通过利用插值理论，我们证明了在合适希尔伯特尺度的"紧" $L^{\infty}(\mathcal{X})$ 嵌入条件下，Mercer算子的谱可以被推断。我们的分析采用了一种新的傅里叶等容量条件，该条件通过最优希尔伯特尺度函数刻画了核狄利克雷容量与小球概率之间的相互作用。