TITLE: Regularized linear vs. RKHS-regression
QUESTION [8 upvotes]: I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two.
Given input-output pairs $(x_i,y_i)$, I want to estimate a function $f(\cdot)$ as follows
\begin{equation}f(x)\approx u(x)=\sum_{i=1}^m \alpha_i K(x,x_i),\end{equation}
where $K(\cdot,\cdot)$ is a kernel function. The coefficients $\alpha_m$ can either be found by  solving
\begin{equation}
 {\displaystyle \min _{\alpha\in R^{n}}{\frac {1}{n}}\|Y-K\alpha\|_{R^{n}}^{2}+\lambda \alpha^{T}K\alpha},\end{equation}
where, with some abuse of notation, the $i,j$'th entry of the kernel matrix $K$ is ${\displaystyle K(x_{i},x_{j})} $. This gives
\begin{equation}
\alpha^*=(K+\lambda nI)^{-1}Y.
\end{equation}
Alternatively, we could treat the problem as a normal ridge regression/linear regression problem:
\begin{equation}
 {\displaystyle \min _{\alpha\in R^{n}}{\frac {1}{n}}\|Y-K\alpha\|_{R^{n}}^{2}+\lambda \alpha^{T}\alpha},\end{equation}
with solution
\begin{equation}
{\alpha^*=(K^{T}K +\lambda nI)^{-1}K^{T}Y}.
\end{equation}
What would be the crucial difference between these two approaches and their solutions?

REPLY [2 votes]: Both of the penalties can be thought of as arising from the linear regression setting in a Bayesian framework with predictor matrix $K$ and a Gaussian prior over the vector $\alpha$, centered at zero with prior variance $V$. 
In the ridge regression case $V = n^{-1}\lambda^{-1}I$ and in the other case $V = n^{-1}\lambda^{-1}K^{-1}$ (as a kernel matrix $K$ is symmetric and PSD; I'm also assuming it is invertible). This follows just by equating terms; the posterior mean has the form $(K^tK + V^{-1})^{-1}K^tY$. Plugging in $V = n^{-1}\lambda^{-1}K^{-1}$ gives $$(K^tK + n\lambda K)^{-1}K^tY = (K^t + n\lambda I)^{-1}K^{-1}K^tY = (K + n\lambda I)^{-1}Y.$$
Anyway, this is all just definitions, but the perspective might be intuition-boosting: the RKHS version stipulates explicitly that the prior over alpha has higher precision (more regularization) along directions of high variation as defined by the kernel function.