Am studying the basic concepts of RKHS and the representer theorem: In $f(x_i)=
Notation: Representer Theorem for Reproducing kernel hilbert spaces
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0Also, I was assuming that $k(x_i,\mathbb{.})$ is a vector formed by the placeholder taking in each entry in a vector $x$ except for $x_i$. Is my understanding right about this? – 2012-10-20
1 Answers
I might be misunderstanding the question here, but it seems that you're a bit confused about what $
Maybe an example might help. Consider the space $H^1(\mathbb{R})$, which is the space of functions $\{f \in L^2: f' \in L^2\}$ which has an inner product$
Note that the vectors in this space are functions, and not just a simple column vector. This space has a reproducing kernel, namely $k(x,y) = \frac{1}{2} e^{|x-y|}$. The reproducing property tells us that we can 'sample' $f$ at any point we want by taking the inner product of $f$ with $k(x,\cdot)$. That is, if you give me a function $f$ and you want to know its value at some point, say $2$, then the reproducing property tells us that:
$f(2) =
For a reference, check out Scattered Data Approximation by Holger Wendland, chapter 10.