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98 lines
3.4 KiB
TeX
98 lines
3.4 KiB
TeX
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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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\begin{Definition}[Sobolev Norm]
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\label{def:sobonorm}
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The natural norm of the sobolev space is given by
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\[
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\norm{f}_{W^{k,p}(K)} =
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\begin{cases}
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\left(\sum_{\abs{\alpha} \leq k}
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\norm{f^{(\alpha)}}^p_{L^p}\right)^{\nicefrac{1}{p}},&
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\text{for } 1 \leq p < \infty \\
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max_{\abs{\alpha} \leq k}\left\{f^{(\alpha)}\right\},& \text{for
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} p = \infty
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\end{cases}
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.
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\]
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\end{Definition}
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\begin{Lemma}[Poincar\'e typed inequality]
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Let \(f:\mathbb{R} \to \mathbb{R}\) differentiable with \(f' :
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\mathbb{R} \to \mathbb{R}\) Lesbeque integrable. Then for \(K=[a,b]
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\subset \mathbb{R}\) with \(f(a)=0\) it holds that
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\begin{equation}
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\label{eq:pti1}
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\exists C_K^{\infty} \in \mathbb{R}_{>0} :
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\norm{f}_{w^{1,\infty}(K)} \leq C_K^{\infty}
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\norm{f'}_{L^{\infty}(K)}.
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\end{equation}
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If additionaly \(f'\) is differentiable with \(f'': \mathbb{R} \to
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\mathbb{R}\) Lesbeque integrable then additionally
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\begin{equation}
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\label{eq:pti2}
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\exists C_K^2 \in \mathbb{R}_{>0} : \norm{f}_{W^{1,\infty}(K)} \leq
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C_K^2 \norm{f''}_{L^2(K)}.
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\end{equation}
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\proof
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With the fundamental theorem of calculus, if
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\(\norm{f}_{L^{\infty}(K)}<\infty\) we get
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\begin{equation}
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\label{eq:f_f'}
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\norm{f}_{L^{\infty}(K)} = \sup_{x \in K}\abs{\int_a^x f'(s) ds} \leq
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\sup_{x \in K}\abs{\int_a^x \sup_{y \in K} \abs{f'(y)} ds} \leq \abs{b-a}
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\sup_{y \in K}\abs{f'(y)}.
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\end{equation}
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Using this we can bound \(\norm{f}_{w^{1,\infty}(K)}\) by
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\[
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\norm{f}_{w^{1,\infty}(K)} \stackrel{\text{Def~\ref{def:sobonorm}}}{=}
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\max\left\{\norm{f}_{L^{\infty}(K)},
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\norm{f'}_{L^{\infty}(K)}\right\}
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\stackrel{(\ref{eq:f_f'})}{\leq} max\left\{\abs{b-a},
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1\right\}\norm{f'}_{L^{\infty}(K)}.
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\]
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With \(C_k^{\infty} \coloneqq max\left\{\abs{b-a}, 1\right\}\) we
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get (\ref{eq:pti1}).
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By using the Hölder inequality, we can proof the second claim.
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\begin{align*}
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\norm{f'}_{L^{\infty}(K)} &= \sup_{x \in K} \abs{\int_a^bf''(y)
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\mathds{1}_{[a,x]}(y)dy} \leq \sup_{x \in
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K}\norm{f''\mathds{1}_{[a,x]}}_{L^1(K)}\\
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&\hspace{-6pt} \stackrel{\text{Hölder}}{\leq} sup_{x
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\in
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K}\norm{f''}_{L^2(K)}\norm{\mathds{1}_{[a,x]}}_{L^2(K)}
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= \abs{b-a}\norm{f''}_{L^2(K)}.
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\end{align*}
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Thus (\ref{eq:pti2}) follows with \(C_K^2 \coloneqq
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\abs{b-a}C_K^{\infty}\).
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\qed
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\end{Lemma} |