  Recently, several theories including the replica method made predictions for
the generalization error of Kernel Ridge Regression. In some regimes, they
predict that the method has a `spectral bias': decomposing the true function
$f^*$ on the eigenbasis of the kernel, it fits well the coefficients associated
with the O(P) largest eigenvalues, where $P$ is the size of the training set.
This prediction works very well on benchmark data sets such as images, yet the
assumptions these approaches make on the data are never satisfied in practice.
To clarify when the spectral bias prediction holds, we first focus on a
one-dimensional model where rigorous results are obtained and then use scaling
arguments to generalize and test our findings in higher dimensions. Our
predictions include the classification case $f(x)=$sign$(x_1)$ with a data
distribution that vanishes at the decision boundary $p(x)\sim x_1^{\chi}$. For
$\chi>0$ and a Laplace kernel, we find that (i) there exists a cross-over ridge
$\lambda^*_{d,\chi}(P)\sim P^{-\frac{1}{d+\chi}}$ such that for $\lambda\gg
\lambda^*_{d,\chi}(P)$, the replica method applies, but not for
$\lambda\ll\lambda^*_{d,\chi}(P)$, (ii) in the ridge-less case, spectral bias
predicts the correct training curve exponent only in the limit
$d\rightarrow\infty$.
