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32 lines
944 B
TeX
32 lines
944 B
TeX
5 years ago
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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With the following Theorem we will have an explicit desrctiption for the
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limes of RN as the amount of nodes is increased.
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\begin{Theorem}[Ridge weight penaltiy corresponds to adapted spline]
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\label{theo:main1}
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For arbitrary training data \(\left(x_i^{train}, y_i^{train}\right)\) it holds
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\[
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\plimn \norm{\mathcal{RN^{*, \tilde{\lambda}}} - f^{*,
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\tilde{\lambda}}_{g, \pm}}_{W^{1,\infty}(K)} = 0.
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\]
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With
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\begin{align*}
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\label{eq:1}
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\tilde{\lambda} &\coloneqq \lambda n g(0), \\
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g(x) &\coloneqq
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g_{\xi}(x)\mathbb{E}\left[ v_k^2 \vert \xi_k = x \right], \forall x
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\in \mathbb{R}
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\end{align*}
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and \(RN^{*, \tilde{\lambda}}}\), \(f^{*,\tilde{\lambda}}_{g, \pm}\)
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as defined in ??? and ??? respectively.
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\end{Theorem}
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In order to proof Theo~\ref{theo:main1} we need to proof a number of
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auxilary Lemmata first.
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