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Reproducing kernel Hilbert space: Difference between revisions

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Provided some clarification to the statement in the introduction about pointwise convergence not implying convergence under the norm of the reproducing kernel Hilbert space. However, the example I gave does not actually apply to reproducing kernel Hilbert spaces [I used the supremum norm as my example, but the supremum norm cannot be induced by any inner product]. It would be much better if someone had an actual Hilbert space example - I just didn't want perfect to be the enemy of the good here.
m (1) Corrected the missing scaling factor \sigma in the norm for the Laplacian kernel. (2) I changed the usage of the ReLU function as ReLU(x) was useless for because of x >= 0.
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*::<math> K(x,y) = e^{-\frac{\|x - y\|}{\sigma}}, \qquad \sigma > 0 </math>
*:The squared norm of a function <math>f</math> in the RKHS <math>H</math> with this kernel is:<ref>Berlinet, Alain and Thomas, Christine. ''[https://books.google.com/books?hl=en&lr=&id=bX3TBwAAQBAJ&oi=fnd&pg=PP11&dq=%22Reproducing+kernel+Hilbert+spaces+in+Probability+and+Statistics%22&ots=jV1gYX6vJ5&sig=um-eULpDSuKtXcYhzTYXwX8ZZzA#v=onepage&q=%22Reproducing%20kernel%20Hilbert%20spaces%20in%20Probability%20and%20Statistics%22&f=false Reproducing kernel Hilbert spaces in Probability and Statistics]'', Kluwer Academic Publishers, 2004</ref>
*::<math>\|f\|_H^2=\intint_{\mathbb R}\Big( \frac1{\sigma} f(x)^2\,dx + \intsigma f'(x)^2\,dxBig) \mathrm d x.</math>
 
===[[Bergman kernel]]s===
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This implies <math>K_y=K(\cdot, y)</math> reproduces <math>f</math>.
 
By takingMoreover the limitminimum function on <math>y X\totimes X = [0,\infty)\times [0,\infty) </math>, wehas obtainthe following representations with the ReLUReLu function,:
 
: <math> \min(x,y) = x -\operatorname{ReLU}(x-y) = y - \operatorname{ReLU}(y-x). </math>
: <math>K_\infty(x)=
\begin{cases}
x, & \text{if } x\geq 0\\
0, & \text{otherwise}
\end{cases}
=\operatorname{ReLU}(x)</math>
 
Using this formulation, we can apply the [[representer theorem]] to the RKHS, letting one prove the optimality of using ReLU activations in neural network settings.{{Citation needed|date=January 2022|reason=Optimal in what sense?}}
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