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375 lines
15 KiB
TeX
375 lines
15 KiB
TeX
\documentclass[a4paper, 12pt, draft=true]{article}
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\usepackage{pgfplots}
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\usepackage{filecontents}
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\usepackage{subcaption}
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\usepackage{adjustbox}
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\usepackage{xcolor}
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\usepackage{tabu}
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\usepackage{showframe}
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\usepackage{graphicx}
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\usepackage{titlecaps}
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\usepackage{amssymb}
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\usepackage{mathtools}%add-on and patches to amsmath
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\usetikzlibrary{calc, 3d}
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\usepgfplotslibrary{colorbrewer}
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\newcommand\Tstrut{\rule{0pt}{2.6ex}} % = `top' strut
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\newcommand\Bstrut{\rule[-0.9ex]{0pt}{0pt}} % = `bottom' strut
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\DeclareMathOperator*{\plim}{\mathbb{P}\text{-}\lim}
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\DeclareMathOperator{\supp}{supp}
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\DeclareMathOperator*{\argmin}{arg\,min}
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\DeclareMathOperator*{\po}{\mathbb{P}\text{-}\mathcal{O}}
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\DeclareMathOperator*{\equals}{=}
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\begin{document}
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\newcommand{\plimn}[0]{\plim\limits_{n \to \infty}}
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\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
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% \pgfplotsset{
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% compat=1.11,
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% legend image code/.code={
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% \draw[mark repeat=2,mark phase=2]
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% plot coordinates {
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% (0cm,0cm)
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% (0.3cm,0cm) %% default is (0.3cm,0cm)
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% (0.6cm,0cm) %% default is (0.6cm,0cm)
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% };%
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% }
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% }
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% \begin{figure}
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% \begin{subfigure}[h]{\textwidth}
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% \begin{tikzpicture}
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% \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
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% /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
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% height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
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% ylabel = {Test Accuracy}, cycle
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% list/Dark2, every axis plot/.append style={line width
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% =1.25pt}]
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% % \addplot [dashed] table
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% % [x=epoch, y=accuracy, col sep=comma, mark = none]
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% % {Data/adam_datagen_full.log};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_1.mean};
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% % \addplot [dashed] table
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% % [x=epoch, y=accuracy, col sep=comma, mark = none]
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% % {Data/adam_datagen_dropout_02_full.log};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_datagen_1.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_datagen_dropout_02_1.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_dropout_02_1.mean};
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% \addlegendentry{\footnotesize{G.}}
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% \addlegendentry{\footnotesize{G. + D. 0.2}}
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% \addlegendentry{\footnotesize{G. + D. 0.4}}
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% \addlegendentry{\footnotesize{D. 0.2}}
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% \addlegendentry{\footnotesize{D. 0.4}}
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% \addlegendentry{\footnotesize{Default}}
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% \end{axis}
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% \end{tikzpicture}
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% \caption{1 sample per class}
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% \vspace{0.25cm}
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% \end{subfigure}
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% \begin{subfigure}[h]{\textwidth}
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% \begin{tikzpicture}
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% \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
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% /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
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% height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
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% ylabel = {Test Accuracy}, cycle
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% list/Dark2, every axis plot/.append style={line width
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% =1.25pt}]
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_dropout_00_10.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_dropout_02_10.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_datagen_dropout_00_10.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_datagen_dropout_02_10.mean};
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% \addlegendentry{\footnotesize{G.}}
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% \addlegendentry{\footnotesize{G. + D. 0.2}}
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% \addlegendentry{\footnotesize{G. + D. 0.4}}
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% \addlegendentry{\footnotesize{D. 0.2}}
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% \addlegendentry{\footnotesize{D. 0.4}}
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% \addlegendentry{\footnotesize{Default}}
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% \end{axis}
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% \end{tikzpicture}
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% \caption{10 samples per class}
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% \end{subfigure}
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% \begin{subfigure}[h]{\textwidth}
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% \begin{tikzpicture}
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% \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
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% /pgf/number format/precision=3},tick style = {draw = none}, width = 0.9875\textwidth,
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% height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
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% xlabel = {epoch}, ylabel = {Test Accuracy}, cycle
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% list/Dark2, every axis plot/.append style={line width
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% =1.25pt}, ymin = {0.92}]
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_dropout_00_100.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_dropout_02_100.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_datagen_dropout_00_100.mean};
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% \addplot table
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% [x=epoch, y=val_accuracy, col sep=comma, mark = none]
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% {Data/adam_datagen_dropout_02_100.mean};
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% \addlegendentry{\footnotesize{G.}}
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% \addlegendentry{\footnotesize{G. + D. 0.2}}
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% \addlegendentry{\footnotesize{G. + D. 0.4}}
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% \addlegendentry{\footnotesize{D. 0.2}}
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% \addlegendentry{\footnotesize{D. 0.4}}
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% \addlegendentry{\footnotesize{Default}}
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% \end{axis}
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% \end{tikzpicture}
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% \caption{100 samples per class}
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% \vspace{.25cm}
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% \end{subfigure}
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% \caption{Accuracy for the net given in ... with Dropout (D.),
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% data generation (G.), a combination, or neither (Default) implemented and trained
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% with \textsc{Adam}. For each epoch the 60.000 training samples
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% were used, or for data generation 10.000 steps with each using
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% batches of 60 generated data points. For each configuration the
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% model was trained 5 times and the average accuracies at each epoch
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% are given in (a). Mean, maximum and minimum values of accuracy on
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% the test and training set are given in (b).}
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% \end{figure}
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% \begin{table}
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% \centering
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% \begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
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% \Tstrut \Bstrut & \textsc{Adam} & D. 0.2 & Gen & Gen.+D. 0.2 \\
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% \hline
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% &
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% \multicolumn{4}{c}{\titlecap{test accuracy for 1 sample}}\Bstrut \\
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% \cline{2-5}
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% max \Tstrut & 0.5633 & 0.5312 & 0.6704 & 0.6604 \\
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% min & 0.3230 & 0.4224 & 0.4878 & 0.5175 \\
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% mean & 0.4570 & 0.4714 & 0.5862 & 0.6014 \\
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% var & 0.0040 & 0.0012 & 0.0036 & 0.0023 \\
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% \hline
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% &
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% \multicolumn{4}{c}{\titlecap{test accuracy for 10 samples}}\Bstrut \\
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% \cline{2-5}
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% max \Tstrut & 0.8585 & 0.9423 & 0.9310 & 0.9441 \\
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% min & 0.8148 & 0.9081 & 0.9018 & 0.9061 \\
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% mean & 0.8377 & 0.9270 & 0.9185 & 0.9232 \\
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% var & 2.7e-4 & 1.3e-4 & 6e-05 & 1.5e-4 \\
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% \hline
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% &
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% \multicolumn{4}{c}{\titlecap{test accuracy for 100 samples}}\Bstrut \\
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% \cline{2-5}
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% max & 0.9637 & 0.9796 & 0.9810 & 0.9805 \\
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% min & 0.9506 & 0.9719 & 0.9702 & 0.9727 \\
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% mean & 0.9582 & 0.9770 & 0.9769 & 0.9783 \\
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% var & 2e-05 & 1e-05 & 1e-05 & 0 \\
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% \hline
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% \end{tabu}
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% \caption{Values of the test accuracy of the model trained 10 times
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% of random training sets containing 1, 10 and 100 data points per
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% class.}
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% \end{table}
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% \begin{center}
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% \begin{figure}[h]
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% \centering
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% \begin{subfigure}{\textwidth}
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% \includegraphics[width=\textwidth]{Data/cnn_fashion_fig.pdf}
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% \caption{original\\image}
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% \end{subfigure}
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% \begin{subfigure}{\textwidth}
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% \includegraphics[width=\textwidth]{Data/cnn_fashion_fig1.pdf}
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% \caption{random\\zoom}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist_gen_shear.pdf}
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% \caption{random\\shear}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist_gen_rotation.pdf}
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% \caption{random\\rotation}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist_gen_shift.pdf}
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% \caption{random\\positional shift}
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% \end{subfigure}\\
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist5.pdf}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist6.pdf}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist7.pdf}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist8.pdf}
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% \end{subfigure}
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% \begin{subfigure}{0.19\textwidth}
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% \includegraphics[width=\textwidth]{Data/mnist9.pdf}
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% \end{subfigure}
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% \caption{The MNIST data set contains 70.000 images of preprocessed handwritten
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% digits. Of these images 60.000 are used as training images, while
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% the rest are used to validate the models trained.}
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% \end{figure}
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% \end{center}
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% \begin{figure}
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% \begin{adjustbox}{width=\textwidth}
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% \begin{tikzpicture}
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% \begin{scope}[x = (0:1cm), y=(90:1cm), z=(15:-0.5cm)]
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% \node[canvas is xy plane at z=0, transform shape] at (0,0)
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% {\includegraphics[width=5cm]{Data/klammern_r.jpg}};
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% \node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
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% {\includegraphics[width=5cm]{Data/klammern_g.jpg}};
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% \node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
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% {\includegraphics[width=5cm]{Data/klammern_b.jpg}};
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% \node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
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% {\includegraphics[width=5.3cm]{Data/klammern_rgb.jpg}};
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% \end{scope}
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% \end{tikzpicture}
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% \end{adjustbox}
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% \caption{On the right the red, green and blue chanels of the picture
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% are displayed. In order to better visualize the color channes the
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% black and white picture of each channel has been colored in the
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% respective color. Combining the layers results in the image on the
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% left}
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% \end{figure}
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% \begin{figure}
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% \centering
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% \begin{subfigure}{\linewidth}
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% \centering
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% \includegraphics[width=\textwidth]{Data/convnet_fig.pdf}
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% \end{subfigure}
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% \begin{subfigure}{.45\linewidth}
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% \centering
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% \begin{tikzpicture}
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% \begin{axis}[enlargelimits=false, width=\textwidth]
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% \addplot[domain=-5:5, samples=100]{tanh(x)};
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% \end{axis}
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% \end{tikzpicture}
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% \end{subfigure}
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% \begin{subfigure}{.45\linewidth}
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% \centering
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% \begin{tikzpicture}
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% \begin{axis}[enlargelimits=false, width=\textwidth,
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% ytick={0,2,4},yticklabels={\hphantom{4.}0,2,4}, ymin=-1]
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% \addplot[domain=-5:5, samples=100]{max(0,x)};
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% \end{axis}
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% \end{tikzpicture}
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% \end{subfigure}
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% \begin{subfigure}{.45\linewidth}
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% \centering
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% \begin{tikzpicture}
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% \begin{axis}[enlargelimits=false, width=\textwidth, ymin=-1,
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% ytick={0,2,4},yticklabels={$\hphantom{-5.}0$,2,4}]
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% \addplot[domain=-5:5, samples=100]{max(0,x)+ 0.1*min(0,x)};
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% \end{axis}
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% \end{tikzpicture}
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% \end{subfigure}
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% \end{figure}
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% \begin{tikzpicture}
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% \begin{axis}[enlargelimits=false]
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% \addplot [domain=-5:5, samples=101,unbounded coords=jump]{1/(1+exp(-x)};
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% \addplot[domain=-5:5, samples=100]{tanh(x)};
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% \addplot[domain=-5:5, samples=100]{max(0,x)};
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% \end{axis}
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% \end{tikzpicture}
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% \begin{tikzpicture}
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% \begin{axis}[enlargelimits=false]
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% \addplot[domain=-2*pi:2*pi, samples=100]{cos(deg(x))};
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% \end{axis}
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% \end{tikzpicture}
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\newcommand{\abs}[1]{\ensuremath{\left\vert#1\right\vert}}
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\[
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\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
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h_{k,n} = \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
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(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
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\left(\sum_{\substack{k \in \kappa \\ \xi_k \in
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[\delta l , \delta(l+1))}} \varphi(\xi_k, v_k)
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h_{k,n}\right) \approx
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\]
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\[
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\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
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(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
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\left(\sum_{\substack{k \in \kappa \\ \xi_k \in
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[\delta l , \delta(l+1))}} \left(\varphi(\delta l, v_k)
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\frac{1}{n g_\xi (\delta l)} \pm \frac{\varepsilon}{n}\right)
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\frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
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\delta(l+1))\right\}}}{\abs{\left\{m \in \kappa : \xi_m
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\in [\delta l, \delta(l+1))\right\}}}\right)
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\]
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\[
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\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
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(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
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\left(\frac{\sum_{\substack{k \in \kappa \\ \xi_k \in
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[\delta l , \delta(l+1))}}\varphi(\delta l,
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v_k)}{\abs{\left\{m \in \kappa : \xi_m
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\in [\delta l, \delta(l+1))\right\}}}
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\frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
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\delta(l+1))\right\}}}{n g_\xi (\delta l)}\right) \pm \varepsilon
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\]
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The amount of kinks in a given interval of length $\delta$ follows a
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binomial distribution,
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\[
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\mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
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\delta(l+1))\right\}}\right] = n \int_{\delta
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l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l)
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\pm \delta \tilde{\varepsilon}),
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\]
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for any $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$, since $g_\xi$ is uniformly continuous on its
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support by Assumption..
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As the distribution of $v$ is continuous as well we get
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$\mathcal{L}(v_k) = \mathcal{L} v| \xi = \delta l) \forall k \in
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\kappa : \xi_k \in [\delta l, \delta(l+1))$ for $delta \leq
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\delta(\varepsilon, \tilde{\varepsilon})$. Thus we get with the law of
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large numbers
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\begin{align*}
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&\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
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h_{k,n} \approx\\
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&\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
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(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
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\}]}}\left(\mathbb{E}[\phi(\xi, v)|\xi=\delta l]
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\stackrel{\mathbb{P}}{\pm}\right) \delta \left(1 \pm
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\frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
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\\
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&\approx \left(\sum_{\substack{l \in \mathbb{Z} \\ [\delta
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l, \delta
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(l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
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\}]}}\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \delta
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\stackrel{\mathbb{P}}{\pm}\tilde{\tilde{\varepsilon}}
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\abs{C_{g_\xi}^u - C_{g_\xi}^l}
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\right)\\
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&\phantom{\approx}\cdot \left(1 \pm
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\frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
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\end{align*}
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\newpage
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\end{document}
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%%% Local Variables:
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