Added compiled presentation

README.md

@@ -2,14 +2,14 @@
 
 This repository contains resources related to the Bachelor-Thesis
 `Application of Optimization Algorithms for Channel Decoding`
-(the related sofware has it's own [repo](https://gitlab.kit.edu/uhhxt/ba_sw))
+(the related sofware has it's own [repo](https://git.scc.kit.edu/ba_duo/ba_sw))
 
 
 ## Access compiled documents
 
 Each document has a corresponding tag, where an already compiled pdf document can be found
 (See for example
-[this](https://gitlab.kit.edu/uhhxt/ba_thesis/-/tree/prox_07_12_22/latex/presentations/02_12_2022)
+[this](https://git.scc.kit.edu/ba_duo/ba_thesis/-/tree/prox_07_12_22/latex/presentations/02_12_2022)
 tag, for one of the presentations under `latex/presentations`)
 
 

latex/thesis/bibliography.bib

@@ -45,6 +45,15 @@
 %    eprint    = {https://doi.org/10.1080/24725854.2018.1550692}
 }
 
+@online{mackay_enc,
+    author = {MacKay, David J.C.},
+    title  = {Encyclopedia of Sparse Graph Codes},
+    date   = {2023-01},
+    url    = {http://www.inference.org.uk/mackay/codes/data.html}
+}
+
+
+
 @article{proximal_algorithms,
     author = {Parikh, Neal and Boyd, Stephen},
     title = {Proximal Algorithms},
@@ -210,25 +219,11 @@
   doi={10.1109/TIT.1962.1057683}
 }
 
-@online{lautern_channelcodes,
+@misc{lautern_channelcodes,
 	author = "Helmling, Michael and Scholl, Stefan and Gensheimer, Florian and Dietz, Tobias and Kraft, Kira and Ruzika, Stefan and Wehn, Norbert",
 	title = "{D}atabase of {C}hannel {C}odes and {ML} {S}imulation {R}esults",
+	howpublished = "\url{www.uni-kl.de/channel-codes}",
     url={https://www.uni-kl.de/channel-codes},
-	date = {2023-04}
+	year = "2023"
-}
-
-@online{mackay_enc,
-    author = {MacKay, David J.C.},
-    title  = {Encyclopedia of Sparse Graph Codes},
-    date   = {2023-04},
-    url    = {http://www.inference.org.uk/mackay/codes/data.html}
-}
-
-@article{adam,
-  title={Adam: A method for stochastic optimization},
-  author={Kingma, Diederik P and Ba, Jimmy},
-  journal={arXiv preprint arXiv:1412.6980},
-  year={2014},
-  doi={10.48550/arXiv.1412.6980}
 }
 

latex/thesis/chapters/acknowledgements.tex

@@ -1,18 +0,0 @@
-\chapter*{Acknowledgements}
-
-I would like to thank Prof. Dr.-Ing. Laurent Schmalen for granting me the
-opportunity to write my bachelor's thesis at the Communications Engineering Lab,
-as well as all other members of the institute for their help and many productive
-discussions, and for creating a very pleasant environment to do research in.
-
-I am very grateful to Dr.-Ing. Holger Jäkel
-for kindly providing me with his knowledge and many suggestions,
-and for his constructive criticism during the preparation of this work.
-
-Special thanks also to Mai Anh Vu for her invaluable feedback and support
-during the entire undertaking that is this thesis.
-
-Finally, I would like to thank my family, who have enabled me to pursue my
-studies in a field I thoroughly enjoy and who have supported me completely
-throughout my journey.
-

latex/thesis/chapters/appendix.tex

@@ -508,7 +508,7 @@ $\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,
@@ -549,7 +549,7 @@ $\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,
@@ -593,7 +593,7 @@ $\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,
@@ -647,7 +647,7 @@ $\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,
@@ -692,7 +692,7 @@ $\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,
@@ -735,7 +735,7 @@ $\gamma \in \left\{ 0.01, 0.05, 0.15 \right\}$.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,

latex/thesis/chapters/comparison.tex

@@ -2,9 +2,14 @@
 \label{chapter:comparison}
 
 In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared.
-First, the two algorithms are studied on a theoretical basis.
+First the two algorithms are compared on a theoretical basis.
 Subsequently, their respective simulation results are examined, and their
-differences are interpreted based on their theoretical structure.
+differences are interpreted on the basis of their theoretical structure.
+
+%some similarities between the proximal decoding algorithm
+%and \ac{LP} decoding using \ac{ADMM} are be pointed out.
+%The two algorithms are compared and their different computational and decoding
+%performance is interpreted on the basis of their theoretical structure.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -13,11 +18,12 @@ differences are interpreted based on their theoretical structure.
 
 \ac{ADMM} and the proximal gradient method can both be expressed in terms of
 proximal operators \cite[Sec. 4.4]{proximal_algorithms}.
-Additionally, the two algorithms show some striking similarities with
+When using \ac{ADMM} as an optimization method to solve the \ac{LP} decoding
-regard to their general structure and the way in which the minimization of the
+problem specifically, this is not quite possible because of the multiple
-respective objective functions is accomplished.
+constraints.
+In spite of that, the two algorithms still show some striking similarities.
 
-The \ac{LP} decoding problem in
+To see the first of these similarities, the \ac{LP} decoding problem in
 equation (\ref{eq:lp:relaxed_formulation}) can be slightly rewritten using the
 \textit{indicator functions} $g_j : \mathbb{R}^{d_j} \rightarrow
 \left\{ 0, +\infty \right\} \hspace{1mm}, j\in\mathcal{J}$ for the polytopes
@@ -32,13 +38,12 @@ $\mathcal{P}_{d_j}, \hspace{1mm} j\in\mathcal{J}$, defined as%
 %
 by moving the constraints into the objective function, as shown in figure
 \ref{fig:ana:theo_comp_alg:admm}.
-The objective functions of the two problems are similar in that they
-both comprise two parts: one associated to the likelihood that a given
-codeword was sent, arising from the channel model, and one associated
-to the constraints the decoding process is subjected to, arising from the
-code used.
 Both algorithms are composed of an iterative approach consisting of two
-alternating steps, each minimizing one part of the objective function.
+alternating steps.
+The objective functions of both problems are similar in that they
+both comprise two parts: one associated to the likelihood that a given
+codeword was sent and one associated to the constraints the decoding process
+is subjected to.
 %
 
 \begin{figure}[h]
@@ -109,7 +114,7 @@ return $\tilde{\boldsymbol{c}}$
     \end{subfigure}%
     
 
-    \caption{Comparison of proximal decoding and \ac{LP} decoding using \ac{ADMM}}
+    \caption{Comparison of the proximal gradient method and \ac{ADMM}}
     \label{fig:ana:theo_comp_alg}
 \end{figure}%
 %
@@ -118,11 +123,15 @@ Their major difference is that while with proximal decoding the constraints
 are regarded in a global context, considering all parity checks at the same
 time, with \ac{ADMM} each parity check is
 considered separately and in a more local context (line 4 in both algorithms).
+This difference means that while with proximal decoding the alternating
+minimization of the two parts of the objective function inevitably leads to
+oscillatory behavior (as explained in section
+\ref{subsec:prox:conv_properties}), this is not the case with \ac{ADMM}, which
+partly explains the disparate decoding performance of the two methods.
 Furthermore, while with proximal decoding the step considering the constraints
 is realized using gradient descent - amounting to an approximation -
 with \ac{ADMM} it reduces to a number of projections onto the parity polytopes
-$\mathcal{P}_{d_j}, \hspace{1mm} j\in\mathcal{J}$, which always provide exact
+$\mathcal{P}_{d_j}$ which always provide exact results.
-results.
 
 The contrasting treatment of the constraints (global and approximate with
 proximal decoding as opposed to local and exact with \ac{LP} decoding using
@@ -133,27 +142,34 @@ calculation, whereas with \ac{LP} decoding it occurs due to the approximate
 formulation of the constraints - independent of the optimization method
 itself. 
 The advantage which arises because of this when employing \ac{LP} decoding is
-the \ac{ML} certificate property: when a valid codeword is returned, it is
+that it can be easily detected \todo{Not 'easily' detected}, when the algorithm gets stuck - it
-also the \ac{ML} codeword.
+returns a solution corresponding to a pseudocodeword, the components of which
+are fractional.
+Moreover, when a valid codeword is returned, it is also the \ac{ML} codeword.
 This means that additional redundant parity-checks can be added successively
 until the codeword returned is valid and thus the \ac{ML} solution is found
 \cite[Sec. IV.]{alp}.
 
 In terms of time complexity, the two decoding algorithms are comparable.
 Each of the operations required for proximal decoding can be performed
-in $\mathcal{O}\left( n \right) $ time for \ac{LDPC} codes (see section
+in linear time for \ac{LDPC} codes (see section \ref{subsec:prox:comp_perf}).
-\ref{subsec:prox:comp_perf}).
+The same is true for the $\tilde{\boldsymbol{c}}$- and $\boldsymbol{u}$-update
-The same is true for \ac{LP} decoding using \ac{ADMM} (see section
+steps of \ac{LP} decoding using \ac{ADMM}, while
-\ref{subsec:admm:comp_perf}).
+the projection step has a worst-case time complexity of
-Additionally, both algorithms can be understood as message-passing algorithms,
+$\mathcal{O}\left( n^2 \right)$ and an average complexity of
-\ac{LP} decoding using \ac{ADMM} as similarly to
+$\mathcal{O}\left( n \right)$ (see section TODO, \cite[Sec. VIII.]{lautern}).
-\cite[Sec. III. D.]{original_admm} and
+
-\cite[Sec. II. B.]{efficient_lp_dec_admm}, and proximal decoding by starting
+Both algorithms can be understood as message-passing algorithms, \ac{LP}
-with algorithm \ref{alg:prox}, substituting for the gradient of the
+decoding using \ac{ADMM} as similarly to \cite[Sec. III. D.]{original_admm}
-code-constraint polynomial and separating the $\boldsymbol{s}$ update into two parts.
+or \cite[Sec. II. B.]{efficient_lp_dec_admm} and proximal decoding by
+starting with algorithm \ref{alg:prox},
+substituting for the gradient of the code-constraint polynomial and separating
+it into two parts.
 The algorithms in their message-passing form are depicted in figure
 \ref{fig:comp:message_passing}.
 $M_{j\to i}$ denotes a message transmitted from \ac{CN} j to \ac{VN} i.
+$M_{j\to}$ signifies the special case where a \ac{VN} transmits the same
+message to all \acp{VN}.
 %
 \begin{figure}[h]
     \centering
@@ -168,14 +184,14 @@ Initialize $\boldsymbol{r}, \boldsymbol{s}, \omega, \gamma$
 while stopping critierion unfulfilled do
     for j in $\mathcal{J}$ do
         $p_j \leftarrow \prod_{i\in N_c\left( j \right) } r_i $
-        $M_{j\to i} \leftarrow p_j^2 - p_j$|\Suppressnumber|
+        $M_{j\to} \leftarrow p_j^2 - p_j$|\Suppressnumber|
 |\vspace{0.22mm}\Reactivatenumber|
     end for
     for i in $\mathcal{I}$ do
-        $s_i\leftarrow \Pi_\eta \left( s_i + \gamma \left( 4\left( s_i^2 - 1 \right)s_i
+        $s_i \leftarrow s_i + \gamma \left[ 4\left( s_i^2 - 1 \right)s_i
-            \phantom{\frac{4}{s_i}}\right.\right.$|\Suppressnumber|
+            \phantom{\frac{4}{s_i}}\right.$|\Suppressnumber|
-                  |\Reactivatenumber|$\left.\left.+ \frac{4}{s_i}\sum_{j\in
+                     |\Reactivatenumber|$\left.+ \frac{4}{s_i}\sum_{j\in N_v\left( i \right) }
-                             N_v\left( i \right) } M_{j\to i} \right)\right) $
+                        M_{j\to} \right] $
         $r_i \leftarrow r_i + \omega \left( s_i - y_i \right)$
     end for
 end while
@@ -216,14 +232,20 @@ return $\tilde{\boldsymbol{c}}$
     \end{subfigure}%
     
 
-    \caption{Proximal decoding and \ac{LP} decoding using \ac{ADMM}
+    \caption{The proximal gradient method and \ac{LP} decoding using \ac{ADMM}
         as message passing algorithms}
     \label{fig:comp:message_passing}
 \end{figure}%
 %
-This message passing structure means that both algorithms can be implemented
+It is evident that while the two algorithms are very similar in their general
-very efficiently, as the update steps can be performed in parallel for all
+structure, with \ac{LP} decoding using \ac{ADMM}, multiple messages have to be
-\acp{CN} and for all \acp{VN}, respectively.
+computed for each check node (line 6 in figure
+\ref{fig:comp:message_passing:admm}), whereas
+with proximal decoding, the same message is transmitted to all \acp{VN}
+(line 5 of figure \ref{fig:comp:message_passing:proximal}).
+This means that while both algorithms have an average time complexity of
+$\mathcal{O}\left( n \right)$, more arithmetic operations are required in the
+\ac{ADMM} case.
 
 In conclusion, the two algorithms have a very similar structure, where the
 parts of the objective function relating to the likelihood and to the
@@ -231,8 +253,9 @@ constraints are minimized in an alternating fashion.
 With proximal decoding this minimization is performed for all constraints at once
 in an approximative manner, while with \ac{LP} decoding using \ac{ADMM} it is
 performed for each constraint individually and with exact results.
-In terms of time complexity, both algorithms are linear with
+In terms of time complexity, both algorithms are, on average, linear with
-respect to $n$ and are heavily parallelizable.
+respect to $n$, although for \ac{LP} decoding using \ac{ADMM} significantly
+more arithmetic operations are necessary in each iteration.
 
 
 
@@ -240,100 +263,24 @@ respect to $n$ and are heavily parallelizable.
 \section{Comparison of Simulation Results}%
 \label{sec:comp:res}
 
-The decoding performance of the two algorithms is compared in figure
+\begin{itemize}
-\ref{fig:comp:prox_admm_dec} in form of the \ac{FER}.
+    \item The comparison of actual implementations is always debatable /
-Shown as well is the performance of the improved proximal decoding
+        contentious, since it is difficult to separate differences in
-algorithm presented in section \ref{sec:prox:Improved Implementation}.
+        algorithm performance from differences in implementation
-The \ac{FER} resulting from decoding using \ac{BP} and,
+    \item No large difference in computational performance $\rightarrow$
-wherever available, the \ac{FER} of \ac{ML} decoding, taken from
+        Parallelism cannot come to fruition as decoding is performed on the
-\cite{lautern_channelcodes}, are plotted as a reference.
+        same number of cores for both algorithms (Multiple decodings in parallel)
-The parameters chosen for the proximal and improved proximal decoders are
+    \item Nonetheless, in realtime applications / applications where the focus
-$\gamma=0.05$, $\omega=0.05$, $K=200$, $\eta = 1.5$ and $N=12$.
+        is not the mass decoding of raw data, \ac{ADMM} has advantages, since
-The parameters chosen for \ac{LP} decoding using \ac{ADMM} are $\mu = 5$,
+        the decoding of a single codeword is performed faster
-$\rho = 1$, $K=200$, $\epsilon_\text{pri} = 10^{-5}$ and
+    \item \ac{ADMM} faster than proximal decoding $\rightarrow$
-$\epsilon_\text{dual} = 10^{-5}$.
+        Parallelism
-For all codes considered within the scope of this work, \ac{LP} decoding using
+    \item Proximal decoding faster than \ac{ADMM} $\rightarrow$ dafuq
-\ac{ADMM} consistently outperforms both proximal decoding and the improved
+        (larger number of iterations before convergence? More values to compute for ADMM?)
-version, reaching very similar performance to \ac{BP}.
+\end{itemize}
-The decoding gain heavily depends on the code, evidently becoming greater for
-codes with larger $n$ and reaching values of up to $\SI{2}{dB}$.
 
-These simulation results can be interpreted with regard to the theoretical
-structure of the decoding methods, as analyzed in section \ref{sec:comp:theo}.
-The worse performance of proximal decoding is somewhat surprising, considering
-the global treatment of the constraints in contrast to the local treatment
-in the case of \ac{LP} decoding using \ac{ADMM}.
-It may be explained, however, in the context of the nature of the
-calculations performed in each case.
-With proximal decoding, the calculations are approximate, leading
-to the constraints never being quite satisfied.
-With \ac{LP} decoding using \ac{ADMM},
-the constraints are fulfilled for each parity check individually after each
-iteration of the decoding process.
-A further contributing factor might be the structure of the optimization
-process, as the alternating minimization with respect to the same variable
-leads to oscillatory behavior, as explained in section
-\ref{subsec:prox:conv_properties}.
-It should be noted that while in this thesis proximal decoding was
-examined with respect to its performance in \ac{AWGN} channels, in
-\cite{proximal_paper} it is presented as a method applicable to non-trivial
-channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
-broadening its usefulness beyond what is shown here.
 
-The timing requirements of the decoding algorithms are visualized in figure
+\begin{figure}[H]
-\ref{fig:comp:time}.
-The datapoints have been generated by evaluating the metadata from \ac{FER}
-and \ac{BER} simulations and using the parameters mentioned earlier when
-discussing the decoding performance.
-The codes considered are the same as in sections \ref{subsec:prox:comp_perf}
-and \ref{subsec:admm:comp_perf}.
-While the \ac{ADMM} implementation seems to be faster than the proximal
-decoding and improved proximal decoding implementations, inferring some
-general behavior is difficult in this case.
-This is because of the comparison of actual implementations, making the
-results dependent on factors such as the grade of optimization of each of the
-implementations.
-Nevertheless, the run time of both the proximal decoding and the \ac{LP}
-decoding using \ac{ADMM} implementations is similar, and both are
-reasonably performant, owing to the parallelizable structure of the
-algorithms.
-
-\begin{figure}[h]
-    \centering
-
-    \begin{tikzpicture}
-        \begin{axis}[grid=both,
-                     xlabel={$n$}, ylabel={Time per frame (ms)},
-                     width=0.6\textwidth,
-                     height=0.45\textwidth,
-                     legend style={at={(0.5,-0.52)},anchor=south},
-                     legend cell align={left},]
-            \addplot[RedOrange, only marks, mark=square*]
-                table [col sep=comma, x=n, y=spf,
-                       y expr=\thisrow{spf} * 1000]
-                    {res/proximal/fps_vs_n.csv};
-            \addlegendentry{Proximal decoding}
-
-            \addplot[Gray, only marks, mark=*]
-                table [col sep=comma, x=n, y=spf,
-                       y expr=\thisrow{spf} * 1000]
-                    {res/hybrid/fps_vs_n.csv};
-            \addlegendentry{Improved proximal decoding ($N=12$)}
-
-            \addplot[NavyBlue, only marks, mark=triangle*]
-                table [col sep=comma, x=n, y=spf,
-                       y expr=\thisrow{spf} * 1000]
-                    {res/admm/fps_vs_n.csv};
-            \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
-        \end{axis}    
-    \end{tikzpicture}
-
-    \caption{Comparison of the timing requirements of the different decoder implementations}
-    \label{fig:comp:time}
-\end{figure}%
-%
-
-\begin{figure}[h]
     \centering
     
     \begin{subfigure}[t]{0.48\textwidth}
@@ -342,7 +289,7 @@ algorithms.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -352,19 +299,13 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_963965.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_963965.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         %{res/hybrid/2d_ber_fer_dfr_963965.csv};
                         {res/admm/ber_2d_963965.csv};
-                \addplot[Black, line width=1pt, mark=*]
+                \addplot[PineGreen, line width=1pt, mark=triangle]
                     table [col sep=comma, x=SNR, y=FER,]
                         {res/generic/fer_ml_9633965.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_963965.csv};
             \end{axis}
         \end{tikzpicture}
 
@@ -378,7 +319,7 @@ algorithms.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -388,20 +329,15 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_bch_31_26.csv};
-                \addplot[Black, line width=1pt, mark=*]
+                \addplot[PineGreen, line width=1pt, mark=triangle*]
                     table [x=SNR, y=FER, col sep=comma,
                            discard if gt={SNR}{5.5},
-                           discard if lt={SNR}{1},]
+                           discard if lt={SNR}{1},
+                           ]
                         {res/generic/fer_ml_bch_31_26.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_bch_31_26.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -416,7 +352,7 @@ algorithms.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -428,22 +364,15 @@ algorithms.
                            discard if not={gamma}{0.05},
                            discard if gt={SNR}{5.5}]
                         {res/proximal/2d_ber_fer_dfr_20433484.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05},
-                           discard if gt={SNR}{5.5}]
-                        {res/hybrid/2d_ber_fer_dfr_20433484.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma,
                            discard if not={mu}{3.0},
                            discard if gt={SNR}{5.5}]
                         {res/admm/ber_2d_20433484.csv};
-                \addplot[Black, line width=1pt, mark=*]
+                \addplot[PineGreen, line width=1pt, mark=triangle, solid]
                     table [col sep=comma, x=SNR, y=FER,
                            discard if gt={SNR}{5.5}]
                         {res/generic/fer_ml_20433484.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_20433484.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -457,7 +386,7 @@ algorithms.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -467,16 +396,9 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_20455187.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_20455187.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_20455187.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt,
-                          discard if gt={SNR}{5}]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_20455187.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -492,7 +414,7 @@ algorithms.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -502,16 +424,9 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_40833844.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_40833844.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_40833844.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt,
-                          discard if gt={SNR}{3}]
-                    table [x=SNR, y=FER, col sep=comma]
-                        {res/generic/bp_40833844.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -525,7 +440,7 @@ algorithms.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
+                xlabel={$E_b / N_0$}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -535,16 +450,9 @@ algorithms.
                 \addplot[RedOrange, line width=1pt, mark=*, solid]
                     table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
                         {res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
-                \addplot[RedOrange, line width=1pt, mark=triangle, densely dashed]
+                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
-                    table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
-                        {res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
-                \addplot[Turquoise, line width=1pt, mark=*]
                     table [x=SNR, y=FER, col sep=comma, discard if not={mu}{3.0}]
                         {res/admm/ber_2d_pegreg252x504.csv};
-                \addplot [RoyalPurple, mark=*, line width=1pt]
-                    table [x=SNR, y=FER, col sep=comma,
-                           discard if gt={SNR}{3}]
-                        {res/generic/bp_pegreg252x504.csv};
             \end{axis}
         \end{tikzpicture}
     
@@ -562,28 +470,50 @@ algorithms.
                          xmin=10, xmax=50,
                          ymin=0, ymax=0.4,
                          legend columns=1,
-                         legend cell align={left},
                          legend style={draw=white!15!black}]
                 \addlegendimage{RedOrange, line width=1pt, mark=*, solid}
                 \addlegendentry{Proximal decoding}
-                
-                \addlegendimage{RedOrange, line width=1pt, mark=triangle, densely dashed}
-                \addlegendentry{Improved proximal decoding}
                
-                \addlegendimage{Turquoise, line width=1pt, mark=*}
+                \addlegendimage{NavyBlue, line width=1pt, mark=triangle, densely dashed}
                 \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
                 
-                \addlegendimage{RoyalPurple, line width=1pt, mark=*, solid}
+                \addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid}
-                \addlegendentry{\acs{BP} (200 iterations)}
-                
-                \addlegendimage{Black, line width=1pt, mark=*, solid}
                 \addlegendentry{\acs{ML} decoding}
             \end{axis}
         \end{tikzpicture}
     \end{subfigure}
 
-    \caption{Comparison of the decoding performance of the different decoder
+    \caption{Comparison of decoding performance between proximal decoding and \ac{LP} decoding
-        implementations for various codes}
+        using \ac{ADMM}}
     \label{fig:comp:prox_admm_dec}
 \end{figure}
 
+\begin{figure}[h]
+    \centering
+
+    \begin{tikzpicture}
+        \begin{axis}[grid=both,
+                     xlabel={$n$}, ylabel={Time per frame (s)},
+                     width=0.6\textwidth,
+                     height=0.45\textwidth,
+                     legend style={at={(0.5,-0.42)},anchor=south},
+                     legend cell align={left},]
+
+            \addplot[RedOrange, only marks, mark=*]
+                table [col sep=comma, x=n, y=spf]
+                    {res/proximal/fps_vs_n.csv};
+            \addlegendentry{Proximal decoding}
+            \addplot[PineGreen, only marks, mark=triangle*]
+                table [col sep=comma, x=n, y=spf]
+                    {res/admm/fps_vs_n.csv};
+            \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
+        \end{axis}
+    \end{tikzpicture}
+
+    \caption{Timing requirements of the proximal decoding imlementation%
+        \protect\footnotemark{}}
+    \label{fig:comp:time}
+\end{figure}%
+%
+\footnotetext{asdf}
+%

latex/thesis/chapters/conclusion.tex

@@ -1,61 +1,8 @@
-\chapter{Conclusion and Outlook}%
+\chapter{Conclusion}%
 \label{chapter:conclusion}
 
-In the context of this thesis, two decoding algorithms were considered:
+\begin{itemize}
-proximal decoding and \ac{LP} decoding using \ac{ADMM}.
+    \item Summary of results
-The two algorithms were first analyzed individually, before comparing them
+    \item Future work
-based on simulation results as well as on their theoretical structure.
+\end{itemize}
-
-For proximal decoding, the effect of each parameter on the behavior of the
-decoder was examined, leading to an approach to optimally choose the value
-of each parameter.
-The convergence properties of the algorithm were investigated in the context
-of the relatively high decoding failure rate, to derive an approach to correct
-possibly wrong components of the estimate.
-Based on this approach, an improvement of proximal decoding was suggested,
-leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and
-the parameters considered.
-
-For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about by the
-\ac{LP} relaxation were first explored.
-The decomposable nature arising from the relocation of the constraints into
-the objective function itself was recognized as the major driver in enabling
-an efficient implementation of the decoding algorithm.
-Based on simulation results, general guidelines for choosing each parameter
-were derived.
-The decoding performance, in form of the \ac{FER}, of the algorithm was
-analyzed, observing that \ac{LP} decoding using \ac{ADMM} nearly reaches that
-of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code
-in question.
-
-Finally, strong parallels were discovered with regard to the theoretical
-structure of the two algorithms, both in the constitution of their respective
-objective functions as well as in the iterative approaches used to minimize them.
-One difference noted was the approximate nature of the minimization in the
-case of proximal decoding, leading to the constraints never being truly
-satisfied.
-In conjunction with the alternating minimization with respect to the same
-variable, leading to oscillatory behavior, this was identified as
-a possible cause of its comparatively worse decoding performance.
-Furthermore, both algorithms were expressed as message passing algorithms,
-illustrating their similar computational performance.
-
-While the modified proximal decoding algorithm presented in section
-\ref{sec:prox:Improved Implementation} shows some promising results, further
-investigation is required to determine how different choices of parameters
-affect the decoding performance.
-Additionally, a more mathematically rigorous foundation for determining the
-potentially wrong components of the estimate is desirable.
-A different method to improve proximal decoding might be to use
-moment-based optimization techniques such as \textit{Adam} \cite{adam}
-to try to mitigate the effect of local minima introduced in the objective
-function as well as the adversarial structure of the minimization when employing
-proximal decoding.
-
-Another area benefiting from future work is the expansion of the \ac{ADMM}
-based \ac{LP} decoder into a decoder approximating \ac{ML} performance,
-using \textit{adaptive \ac{LP} decoding}.
-With this method, the successive addition of redundant parity checks is used
-to mitigate the decoder becoming stuck in erroneous solutions introduced due
-the relaxation of the constraints of the \ac{LP} decoding problem \cite{alp}.
 

latex/thesis/chapters/discussion.tex

@@ -33,6 +33,13 @@ examined with respect to its performance in \ac{AWGN} channels, in
 channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
 broadening its usefulness beyond what is shown here.
 
+While the modified proximal decoding algorithm presented in section
+\ref{sec:prox:Improved Implementation} shows some promising results, further
+investigation is required to determine how different choices of parameters
+affect the decoding performance.
+Additionally, a more mathematically rigorous foundation for determining the
+potentially wrong components of the estimate is desirable.
+
 Another interesting approach might be the combination of proximal and \ac{LP}
 decoding.
 Performing an initial number of iterations using proximal decoding to obtain

latex/thesis/chapters/introduction.tex

@@ -1,51 +1,16 @@
 \chapter{Introduction}%
 \label{chapter:introduction}
 
-Channel coding using binary linear codes is a way of enhancing the reliability
-of data by detecting and correcting any errors that may occur during
-its transmission or storage.
-One class of binary linear codes, \ac{LDPC} codes, has become especially
-popular due to being able to reach arbitrarily small probabilities of error
-at code rates up to the capacity of the channel \cite[Sec. II.B.]{mackay_rediscovery},
-while retaining a structure that allows for very efficient decoding.
-While the established decoders for \ac{LDPC} codes, such as \ac{BP} and the
-\textit{min-sum algorithm}, offer good decoding performance, they are suboptimal
-in most cases and exhibit an \textit{error floor} for high \acp{SNR}
-\cite[Sec. 15.3]{ryan_lin_2009}, making them unsuitable for applications
-with extreme reliability requirements.
 
-Optimization based decoding algorithms are an entirely different way of approaching
+\begin{itemize}
-the decoding problem.
+    \item Problem definition
-The first introduction of optimization techniques as a way of decoding binary
+    \item Motivation
-linear codes was conducted in Feldman's 2003 Ph.D. thesis and a subsequent paper,
+        \begin{itemize}
-establishing the field of \ac{LP} decoding \cite{feldman_thesis}, \cite{feldman_paper}.
+            \item Error floor when decoding with BP (seems to not be persent with LP decoding
-There, the \ac{ML} decoding problem is approximated by a \textit{linear program}, i.e.,
+                \cite[Sec. I]{original_admm})
-a linear, convex optimization problem, which can subsequently be solved using
+            \item Strong theoretical guarantees that allow for better and better approximations
-several different algorithms \cite{alp}, \cite{interior_point},
+                of ML decoding \cite[Sec. I]{original_admm}
-\cite{original_admm}, \cite{pdd}.
+        \end{itemize}
-More recently, novel approaches such as \textit{proximal decoding} have been
+    \item Results summary
-introduced. Proximal decoding is based on a non-convex optimization formulation
+\end{itemize}
-of the \ac{MAP} decoding problem \cite{proximal_paper}.
 
-The motivation behind applying optimization methods to channel decoding is to
-utilize existing techniques in the broad field of optimization theory, as well
-as to find new decoding methods not suffering from the same disadvantages as
-existing message passing based approaches or exhibiting other desirable properties.
-\Ac{LP} decoding, for example, comes with strong theoretical guarantees
-allowing it to be used as a way of closely approximating \ac{ML} decoding
-\cite[Sec. I]{original_admm},
-and proximal decoding is applicable to non-trivial channel models such
-as \ac{LDPC}-coded massive \ac{MIMO} channels \cite{proximal_paper}.
-
-This thesis aims to further the analysis of optimization based decoding
-algorithms as well as to verify and complement the considerations present in
-the existing literature.
-Specifically, the proximal decoding algorithm and \ac{LP} decoding using
-the \ac{ADMM} \cite{original_admm} are explored within the context of
-\ac{BPSK} modulated \ac{AWGN} channels.
-Implementations of both decoding methods are produced, and based on simulation
-results from those implementations the algorithms are examined and compared.
-Approaches to determine the optimal value of each parameter are derived and
-the computational and decoding performance of the algorithms is examined.
-An improvement on proximal decoding is suggested, achieving up to 1 dB of gain,
-depending on the parameters chosen and the code considered.

latex/thesis/chapters/theoretical_background.tex

@@ -1,13 +1,13 @@
 \chapter{Theoretical Background}%
 \label{chapter:theoretical_background}
 
-In this chapter, the theoretical background necessary to understand the
+In this chapter, the theoretical background necessary to understand this
-decoding algorithms examined in this work is given.
+work is given.
 First, the notation used is clarified.
-The physical layer is detailed - the used modulation scheme and channel model.
+The physical aspects are detailed - the used modulation scheme and channel model.
 A short introduction to channel coding with binary linear codes and especially
 \ac{LDPC} codes is given.
-The established methods of decoding \ac{LDPC} codes are briefly explained.
+The established methods of decoding LPDC codes are briefly explained.
 Lastly, the general process of decoding using optimization techniques is described
 and an overview of the utilized optimization methods is given.
 
@@ -31,7 +31,7 @@ Additionally, a shorthand notation will be used, denoting a set of indices as%
         \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}
 .\end{align*}
 %
-In order to designate element-wise operations, in particular the \textit{Hadamard product}
+In order to designate elemen-twise operations, in particular the \textit{Hadamard product}
 and the \textit{Hadamard power}, the operator $\circ$ will be used:%
 %
 \begin{alignat*}{3}
@@ -45,7 +45,7 @@ and the \textit{Hadamard power}, the operator $\circ$ will be used:%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Channel Model and Modulation}
+\section{Preliminaries: Channel Model and Modulation}
 \label{sec:theo:Preliminaries: Channel Model and Modulation}
 
 In order to transmit a bit-word $\boldsymbol{c} \in \mathbb{F}_2^n$ of length
@@ -82,7 +82,7 @@ conducting this process, whereby \textit{data words} are mapped onto longer
 \textit{codewords}, which carry redundant information.
 \Ac{LDPC} codes have become especially popular, since they are able to
 reach arbitrarily small probabilities of error at code rates up to the capacity
-of the channel \cite[Sec. II.B.]{mackay_rediscovery}, while having a structure
+of the channel \cite[Sec. II.B.]{mackay_rediscovery} while having a structure
 that allows for very efficient decoding.
 
 The lengths of the data words and codewords are denoted by $k\in\mathbb{N}$
@@ -97,7 +97,7 @@ the number of parity-checks:%
             \boldsymbol{H}\boldsymbol{c}^\text{T} = \boldsymbol{0} \right\}
 .\end{align*}
 %
-A data word $\boldsymbol{u} \in \mathbb{F}_2^k$ can be mapped onto a codeword
+A data word $\boldsymbol{u} \in \mathbb{F}_2^k$ can be mapped onto a codword
 $\boldsymbol{c} \in \mathbb{F}_2^n$ using the \textit{generator matrix}
 $\boldsymbol{G} \in \mathbb{F}_2^{k\times n}$:%
 %
@@ -179,9 +179,9 @@ codewords:
     &= \argmax_{c\in\mathcal{C}} \frac{f_{\boldsymbol{Y} \mid \boldsymbol{C}}
         \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
         \left( \boldsymbol{c} \right)}{f_{\boldsymbol{Y}}\left( \boldsymbol{y} \right) } \\
-%    &= \argmax_{c\in\mathcal{C}} f_{\boldsymbol{Y} \mid \boldsymbol{C}}
+    &= \argmax_{c\in\mathcal{C}} f_{\boldsymbol{Y} \mid \boldsymbol{C}}
-%        \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
+        \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
-%        \left( \boldsymbol{c} \right) \\
+        \left( \boldsymbol{c} \right) \\
     &= \argmax_{c\in\mathcal{C}}f_{\boldsymbol{Y} \mid \boldsymbol{C}}
         \left( \boldsymbol{y} \mid \boldsymbol{c} \right)
 .\end{align*}
@@ -204,7 +204,7 @@ Each row of $\boldsymbol{H}$, which represents one parity-check, is viewed as a
 Each component of the codeword $\boldsymbol{c}$ is interpreted as a \ac{VN}.
 The relationship between \acp{CN} and \acp{VN} can then be plotted by noting
 which components of $\boldsymbol{c}$ are considered for which parity-check.
-Figure \ref{fig:theo:tanner_graph} shows the Tanner graph for the
+Figure \ref{fig:theo:tanner_graph} shows the tanner graph for the
 (7,4) Hamming code, which has the following parity-check matrix
 \cite[Example 5.7.]{ryan_lin_2009}:%
 %
@@ -263,7 +263,7 @@ Figure \ref{fig:theo:tanner_graph} shows the Tanner graph for the
         \draw (cn3) -- (c7);
     \end{tikzpicture}
 
-    \caption{Tanner graph for the (7,4) Hamming code}
+    \caption{Tanner graph for the (7,4)-Hamming-code}
     \label{fig:theo:tanner_graph}
 \end{figure}%
 %
@@ -285,16 +285,15 @@ Message passing algorithms are based on the notion of passing messages between
 \acp{CN} and \acp{VN}.
 \Ac{BP} is one such algorithm that is commonly used to decode \ac{LDPC} codes.
 It aims to compute the posterior probabilities
-$p_{C_i \mid \boldsymbol{Y}}\left(c_i = 1 | \boldsymbol{y} \right),\hspace{2mm} i\in\mathcal{I}$,
+$p_{C_i \mid \boldsymbol{Y}}\left(c_i = 1 | \boldsymbol{y} \right),\hspace{2mm} i\in\mathcal{I}$
-see \cite[Sec. III.]{mackay_rediscovery} and use them to calculate the estimate
+\cite[Sec. III.]{mackay_rediscovery} and use them to calculate the estimate $\hat{\boldsymbol{c}}$.
-$\hat{\boldsymbol{c}}$.
 For cycle-free graphs this goal is reached after a finite
 number of steps and \ac{BP} is equivalent to \ac{MAP} decoding.
-When the graph contains cycles, however, \ac{BP} only approximates the \ac{MAP} probabilities
+When the graph contains cycles, however, \ac{BP} only approximates the probabilities
 and is sub-optimal.
 This leads to generally worse performance than \ac{MAP} decoding for practical codes.
 Additionally, an \textit{error floor} appears for very high \acp{SNR}, making
-the use of \ac{BP} impractical for applications where a very low error rate is
+the use of \ac{BP} impractical for applications where a very low \ac{BER} is
 desired \cite[Sec. 15.3]{ryan_lin_2009}.
 Another popular decoding method for \ac{LDPC} codes is the
 \textit{min-sum algorithm}.
@@ -342,7 +341,7 @@ In contrast to the established message-passing decoding algorithms,
 the perspective then changes from observing the decoding process in its
 Tanner graph representation with \acp{VN} and \acp{CN} (as shown in figure \ref{fig:dec:tanner})
 to a spatial representation (figure \ref{fig:dec:spatial}),
-where the codewords are some of the vertices of a hypercube.
+where the codewords are some of the edges of a hypercube.
 The goal is to find the point $\tilde{\boldsymbol{c}}$,
 which minimizes the objective function $g$.
 
@@ -458,38 +457,29 @@ which minimizes the objective function $g$.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{A Short Introduction to the Proximal Gradient Method and ADMM}
+\section{An introduction to the proximal gradient method and ADMM}
 \label{sec:theo:Optimization Methods}
 
-In this section, the general ideas behind the optimization methods used in
-this work are outlined.
-The application of these optimization methods to channel decoding decoding
-will be discussed in later chapters.
-Two methods are introduced, the \textit{proximal gradient method} and
-\ac{ADMM}.
-
 \textit{Proximal algorithms} are algorithms for solving convex optimization
-problems that rely on the use of \textit{proximal operators}.
+problems, that rely on the use of \textit{proximal operators}.
 The proximal operator $\textbf{prox}_{\lambda f} : \mathbb{R}^n \rightarrow \mathbb{R}^n$
 of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by
 \cite[Sec. 1.1]{proximal_algorithms}%
 %
 \begin{align*}
-    \textbf{prox}_{\lambda f}\left( \boldsymbol{v} \right)
+    \textbf{prox}_{\lambda f}\left( \boldsymbol{v} \right) = \argmin_{\boldsymbol{x}} \left(
-        = \argmin_{\boldsymbol{x} \in \mathbb{R}^n} \left(
+        f\left( \boldsymbol{x} \right) + \frac{1}{2\lambda}\lVert \boldsymbol{x}
-            f\left( \boldsymbol{x} \right) + \frac{1}{2\lambda}\lVert \boldsymbol{x}
+            - \boldsymbol{v} \rVert_2^2 \right)
-                - \boldsymbol{v} \rVert_2^2 \right)
 .\end{align*}
 %
 This operator computes a point that is a compromise between minimizing $f$
 and staying in the proximity of $\boldsymbol{v}$.
-The parameter $\lambda$ determines how each term is weighed.
+The parameter $\lambda$ determines how heavily each term is weighed.
-The proximal gradient method is an iterative optimization method
+The \textit{proximal gradient method} is an iterative optimization method
 utilizing proximal operators, used to solve problems of the form%
 %
 \begin{align*}
-    \underset{\boldsymbol{x} \in \mathbb{R}^n}{\text{minimize}}\hspace{5mm}
+    \text{minimize}\hspace{5mm}f\left( \boldsymbol{x} \right) + g\left( \boldsymbol{x} \right) 
-        f\left( \boldsymbol{x} \right) + g\left( \boldsymbol{x} \right) 
 \end{align*}
 %
 that consists of two steps: minimizing $f$ with gradient descent
@@ -502,14 +492,14 @@ and minimizing $g$ using the proximal operator
 ,\end{align*}
 %
 Since $g$ is minimized with the proximal operator and is thus not required
-to be differentiable, it can be used to encode the constraints of the optimization problem
+to be differentiable, it can be used to encode the constraints of the problem
 (e.g., in the form of an \textit{indicator function}, as mentioned in
 \cite[Sec. 1.2]{proximal_algorithms}).
 
-\ac{ADMM} is another optimization method.
+The \ac{ADMM} is another optimization method.
 In this thesis it will be used to solve a \textit{linear program}, which
-is a special type of convex optimization problem in which the objective function
+is a special type of convex optimization problem, where the objective function
-is linear and the constraints consist of linear equalities and inequalities.
+is linear, and the constraints consist of linear equalities and inequalities.
 Generally, any linear program can be expressed in \textit{standard form}%
 \footnote{The inequality $\boldsymbol{x} \ge \boldsymbol{0}$ is to be
 interpreted componentwise.}
@@ -517,53 +507,38 @@ interpreted componentwise.}
 %
 \begin{alignat}{3}
     \begin{alignedat}{3}
-        \underset{\boldsymbol{x}\in\mathbb{R}^n}{\text{minimize }}\hspace{2mm}  
+        \text{minimize }\hspace{2mm}   && \boldsymbol{\gamma}^\text{T} \boldsymbol{x}         \\
-            && \boldsymbol{\gamma}^\text{T} \boldsymbol{x}         \\
         \text{subject to }\hspace{2mm} && \boldsymbol{A}\boldsymbol{x}   & = \boldsymbol{b}   \\
-                                       &&               \boldsymbol{x}   & \ge \boldsymbol{0},
+                                       &&               \boldsymbol{x}   & \ge \boldsymbol{0}.
     \end{alignedat}
     \label{eq:theo:admm_standard}
 \end{alignat}%
 %
-where $\boldsymbol{x}, \boldsymbol{\gamma} \in \mathbb{R}^n$, $\boldsymbol{b} \in \mathbb{R}^m$
+A technique called \textit{Lagrangian relaxation} \cite[Sec. 11.4]{intro_to_lin_opt_book}
-and $\boldsymbol{A}\in\mathbb{R}^{m \times n}$.
+can then be applied.
-A technique called \textit{Lagrangian relaxation} can then be applied
-\cite[Sec. 11.4]{intro_to_lin_opt_book}.
 First, some of the constraints are moved into the objective function itself
 and weights $\boldsymbol{\lambda}$ are introduced. A new, relaxed problem
-is formulated as
+is then formulated as
 %
 \begin{align}
     \begin{aligned}
-        \underset{\boldsymbol{x}\in\mathbb{R}^n}{\text{minimize }}\hspace{2mm}
+        \text{minimize }\hspace{2mm}   & \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
-            & \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
+            + \boldsymbol{\lambda}^\text{T}\left(\boldsymbol{b}
-            + \boldsymbol{\lambda}^\text{T}\left(
+                - \boldsymbol{A}\boldsymbol{x} \right)  \\
-                \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b}\right)  \\
         \text{subject to }\hspace{2mm} & \boldsymbol{x} \ge \boldsymbol{0},
     \end{aligned}
     \label{eq:theo:admm_relaxed}
 \end{align}%
 %
 the new objective function being the \textit{Lagrangian}%
-\footnote{
-    Depending on what literature is consulted, the definition of the Lagrangian differs
-    in the order of $\boldsymbol{A}\boldsymbol{x}$ and $\boldsymbol{b}$.
-    As will subsequently be seen, however, the only property of the Lagrangian having
-    any bearing on the optimization process is that minimizing it gives a lower bound
-    on the optimal objective of the original problem.
-    This property is satisfied no matter the order of the terms and the order
-    chosen here is the one used in the \ac{LP} decoding literature making use of
-    \ac{ADMM}.
-}%
 %
 \begin{align*}
 \mathcal{L}\left( \boldsymbol{x}, \boldsymbol{\lambda} \right)
     = \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
-        + \boldsymbol{\lambda}^\text{T}\left(
+        + \boldsymbol{\lambda}^\text{T}\left(\boldsymbol{b}
-            \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b}\right)
+            - \boldsymbol{A}\boldsymbol{x} \right)
 .\end{align*}%
 %
-
 This problem is not directly equivalent to the original one, as the
 solution now depends on the choice of the \textit{Lagrange multipliers}
 $\boldsymbol{\lambda}$.
@@ -587,12 +562,12 @@ Furthermore, for uniquely solvable linear programs \textit{strong duality}
 always holds \cite[Theorem 4.4]{intro_to_lin_opt_book}.
 This means that not only is it a lower bound, the tightest lower
 bound actually reaches the value itself:
-in other words, with the optimal choice of $\boldsymbol{\lambda}$,
+In other words, with the optimal choice of $\boldsymbol{\lambda}$,
 the optimal objectives of the problems (\ref{eq:theo:admm_relaxed})
-and (\ref{eq:theo:admm_standard}) have the same value, i.e.,
+and (\ref{eq:theo:admm_standard}) have the same value.
 %
 \begin{align*}
-    \max_{\boldsymbol{\lambda}\in\mathbb{R}^m} \, \min_{\boldsymbol{x} \ge \boldsymbol{0}}
+    \max_{\boldsymbol{\lambda}} \, \min_{\boldsymbol{x} \ge \boldsymbol{0}}
         \mathcal{L}\left( \boldsymbol{x}, \boldsymbol{\lambda} \right) 
     = \min_{\substack{\boldsymbol{x} \ge \boldsymbol{0} \\ \boldsymbol{A}\boldsymbol{x}
             = \boldsymbol{b}}}
@@ -602,7 +577,7 @@ and (\ref{eq:theo:admm_standard}) have the same value, i.e.,
 Thus, we can define the \textit{dual problem} as the search for the tightest lower bound:%
 %
 \begin{align}
-    \underset{\boldsymbol{\lambda}\in\mathbb{R}^m}{\text{maximize }}\hspace{2mm}
+    \underset{\boldsymbol{\lambda}}{\text{maximize }}\hspace{2mm}
         & \min_{\boldsymbol{x} \ge \boldsymbol{0}} \mathcal{L}
         \left( \boldsymbol{x}, \boldsymbol{\lambda} \right)
     \label{eq:theo:dual}
@@ -625,7 +600,7 @@ using equation (\ref{eq:theo:admm_obtain_primal}); then, update $\boldsymbol{\la
 using gradient descent \cite[Sec. 2.1]{distr_opt_book}:%
 %
 \begin{align*}
-    \boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x} \ge \boldsymbol{0}} \mathcal{L}\left(
+    \boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x}} \mathcal{L}\left(
         \boldsymbol{x}, \boldsymbol{\lambda} \right) \\
     \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
         + \alpha\left( \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b} \right),
@@ -633,12 +608,12 @@ using gradient descent \cite[Sec. 2.1]{distr_opt_book}:%
 .\end{align*}
 %
 The algorithm can be improved by observing that when the objective function
-$g: \mathbb{R}^n \rightarrow \mathbb{R}$ is separable into a sum of
+$g: \mathbb{R}^n \rightarrow \mathbb{R}$ is separable into a number
-$N \in \mathbb{N}$ sub-functions
+$N \in \mathbb{N}$ of sub-functions
 $g_i: \mathbb{R}^{n_i} \rightarrow \mathbb{R}$,
 i.e., $g\left( \boldsymbol{x} \right) = \sum_{i=1}^{N} g_i
 \left( \boldsymbol{x}_i \right)$,
-where $\boldsymbol{x}_i\in\mathbb{R}^{n_i},\hspace{1mm} i\in [1:N]$ are subvectors of
+where $\boldsymbol{x}_i,\hspace{1mm} i\in [1:N]$ are subvectors of
 $\boldsymbol{x}$, the Lagrangian is as well:
 %
 \begin{align*}
@@ -649,18 +624,18 @@ $\boldsymbol{x}$, the Lagrangian is as well:
 \begin{align*}
     \mathcal{L}\left( \left( \boldsymbol{x}_i \right)_{i=1}^N, \boldsymbol{\lambda} \right)
         = \sum_{i=1}^{N} g_i\left( \boldsymbol{x}_i \right) 
-            + \boldsymbol{\lambda}^\text{T} \left(
+            + \boldsymbol{\lambda}^\text{T} \left( \boldsymbol{b}
-            \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x_i} - \boldsymbol{b}\right) 
+            - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x_i} \right) 
 .\end{align*}%
 %
-The matrices $\boldsymbol{A}_i \in \mathbb{R}^{m \times n_i}, \hspace{1mm} i \in [1:N]$
+The matrices $\boldsymbol{A}_i, \hspace{1mm} i \in [1:N]$ are partitions of
-form a partition of $\boldsymbol{A}$, corresponding to
+the matrix $\boldsymbol{A}$, corresponding to
 $\boldsymbol{A} = \begin{bmatrix}
     \boldsymbol{A}_1 &
     \ldots &
     \boldsymbol{A}_N
 \end{bmatrix}$.
-The minimization of each term can happen in parallel, in a distributed
+The minimization of each term can then happen in parallel, in a distributed
 fashion \cite[Sec. 2.2]{distr_opt_book}.
 In each minimization step, only one subvector $\boldsymbol{x}_i$ of
 $\boldsymbol{x}$ is considered, regarding all other subvectors as being
@@ -668,7 +643,7 @@ constant.
 This modified version of dual ascent is called \textit{dual decomposition}:
 %
 \begin{align*}
-    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i \ge \boldsymbol{0}}\mathcal{L}\left(
+    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i}\mathcal{L}\left(
         \left( \boldsymbol{x}_i \right)_{i=1}^N, \boldsymbol{\lambda}\right) 
         \hspace{5mm} \forall i \in [1:N]\\
     \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
@@ -682,15 +657,14 @@ This modified version of dual ascent is called \textit{dual decomposition}:
 It only differs in the use of an \textit{augmented Lagrangian}
 $\mathcal{L}_\mu\left( \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda} \right)$
 in order to strengthen the convergence properties.
-The augmented Lagrangian extends the classical one with an additional penalty term
+The augmented Lagrangian extends the ordinary one with an additional penalty term
-with the penalty parameter $\mu$:
+with the penaly parameter $\mu$:
 %
 \begin{align*}
     \mathcal{L}_\mu \left( \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda} \right)
         = \underbrace{\sum_{i=1}^{N} g_i\left( \boldsymbol{x_i} \right) 
-            + \boldsymbol{\lambda}^\text{T}\left(\sum_{i=1}^{N}
+            + \boldsymbol{\lambda}^\text{T}\left( \boldsymbol{b}
-                \boldsymbol{A}_i\boldsymbol{x}_i - \boldsymbol{b}\right)}
+        - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i \right)}_{\text{Ordinary Lagrangian}}
-                _{\text{Classical Lagrangian}}
             + \underbrace{\frac{\mu}{2}\left\Vert \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i
             - \boldsymbol{b} \right\Vert_2^2}_{\text{Penalty term}},
         \hspace{5mm} \mu > 0
@@ -700,20 +674,21 @@ The steps to solve the problem are the same as with dual decomposition, with the
 condition that the step size be $\mu$:%
 %
 \begin{align*}
-    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i \ge \boldsymbol{0}}\mathcal{L}_\mu\left(
+    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i}\mathcal{L}_\mu\left(
         \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda}\right) 
         \hspace{5mm} \forall i \in [1:N]\\
     \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
         + \mu\left( \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i
             - \boldsymbol{b} \right),
         \hspace{5mm} \mu > 0
+%    \boldsymbol{x}_1 &\leftarrow \argmin_{\boldsymbol{x}_1}\mathcal{L}_\mu\left(
+%        \boldsymbol{x}_1, \boldsymbol{x_2}, \boldsymbol{\lambda}\right) \\
+%    \boldsymbol{x}_2 &\leftarrow \argmin_{\boldsymbol{x}_2}\mathcal{L}_\mu\left(
+%        \boldsymbol{x}_1, \boldsymbol{x_2}, \boldsymbol{\lambda}\right) \\
+%    \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
+%        + \mu\left( \boldsymbol{A}_1\boldsymbol{x}_1 + \boldsymbol{A}_2\boldsymbol{x}_2
+%            - \boldsymbol{b} \right),
+%        \hspace{5mm} \mu > 0
 .\end{align*}
 %
 
-In subsequent chapters, the decoding problem will be reformulated as an
-optimization problem using two different methodologies.
-In chapter \ref{chapter:proximal_decoding}, a non-convex optimization approach
-is chosen and addressed using the proximal gradient method.
-In chapter \ref{chapter:lp_dec_using_admm}, an \ac{LP} based optimization problem is
-formulated and solved using \ac{ADMM}.
-

latex/thesis/res/admm/dfr_20433484.csv

@@ -1,9 +0,0 @@
-SNR,BER,FER,DFR,num_iterations
-1.0,0.06409994553376906,0.7013888888888888,0.7013888888888888,144.0
-1.5,0.03594771241830065,0.45495495495495497,0.47297297297297297,222.0
-2.0,0.014664163537755528,0.2148936170212766,0.2297872340425532,470.0
-2.5,0.004731522238525039,0.07634164777021919,0.08163265306122448,1323.0
-3.0,0.000911436423803915,0.016994783779236074,0.01749957933703517,5943.0
-3.5,0.00011736135227863285,0.002537369677176234,0.002587614621278734,39805.0
-4.0,1.0686274509803922e-05,0.00024,0.00023,100000.0
-4.5,4.411764705882353e-07,1e-05,1e-05,100000.0

latex/thesis/res/admm/dfr_20433484_metadata.json

@@ -1,10 +0,0 @@
-{
-    "duration": 46.20855645602569,
-    "name": "ber_20433484",
-    "platform": "Linux-6.2.10-arch1-1-x86_64-with-glibc2.37",
-    "K": 200,
-    "epsilon_pri": 1e-05,
-    "epsilon_dual": 1e-05,
-    "max_frame_errors": 100,
-    "end_time": "2023-04-22 19:15:37.252176"
-}

latex/thesis/res/admm/fer_epsilon_20433484.csv

@@ -1,31 +0,0 @@
-SNR,epsilon,BER,FER,DFR,num_iterations,k_avg
-1.0,1e-06,0.294328431372549,0.722,0.0,1000.0,0.278
-1.0,1e-05,0.294328431372549,0.722,0.0,1000.0,0.278
-1.0,0.0001,0.3059313725490196,0.745,0.0,1000.0,0.254
-1.0,0.001,0.3059558823529412,0.748,0.0,1000.0,0.254
-1.0,0.01,0.30637254901960786,0.786,0.0,1000.0,0.254
-1.0,0.1,0.31590686274509805,0.9,0.0,1000.0,0.925
-2.0,1e-06,0.11647058823529412,0.298,0.0,1000.0,0.702
-2.0,1e-05,0.11647058823529412,0.298,0.0,1000.0,0.702
-2.0,0.0001,0.11647058823529412,0.298,0.0,1000.0,0.702
-2.0,0.001,0.11652450980392157,0.306,0.0,1000.0,0.702
-2.0,0.01,0.09901470588235294,0.297,0.0,1000.0,0.75
-2.0,0.1,0.10512745098039215,0.543,0.0,1000.0,0.954
-3.0,1e-06,0.004563725490196078,0.012,0.0,1000.0,0.988
-3.0,1e-05,0.004563725490196078,0.012,0.0,1000.0,0.988
-3.0,0.0001,0.005730392156862745,0.015,0.0,1000.0,0.985
-3.0,0.001,0.005730392156862745,0.015,0.0,1000.0,0.985
-3.0,0.01,0.007200980392156863,0.037,0.0,1000.0,0.981
-3.0,0.1,0.009151960784313726,0.208,0.0,1000.0,0.995
-4.0,1e-06,0.0,0.0,0.0,1000.0,1.0
-4.0,1e-05,0.0,0.0,0.0,1000.0,1.0
-4.0,0.0001,0.0,0.0,0.0,1000.0,1.0
-4.0,0.001,0.0002598039215686275,0.001,0.0,1000.0,0.999
-4.0,0.01,4.901960784313725e-05,0.01,0.0,1000.0,1.0
-4.0,0.1,0.0006862745098039216,0.103,0.0,1000.0,1.0
-5.0,1e-06,0.0,0.0,0.0,1000.0,1.0
-5.0,1e-05,0.0,0.0,0.0,1000.0,1.0
-5.0,0.0001,0.0,0.0,0.0,1000.0,1.0
-5.0,0.001,0.0,0.0,0.0,1000.0,1.0
-5.0,0.01,9.803921568627451e-06,0.002,0.0,1000.0,1.0
-5.0,0.1,0.0005245098039215687,0.097,0.0,1000.0,1.0

latex/thesis/res/admm/fer_epsilon_20433484_metadata.json

@@ -1,10 +0,0 @@
-{
-    "duration": 15.407989402010571,
-    "name": "rho_kavg_20433484",
-    "platform": "Linux-6.2.10-arch1-1-x86_64-with-glibc2.37",
-    "K": 200,
-    "rho": 1,
-    "mu": 5,
-    "max_frame_errors": 100000,
-    "end_time": "2023-04-23 06:39:23.561294"
-}

latex/thesis/res/generic/bp_20433484.csv

@@ -1,9 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0315398614535635,0.598802395209581,334
-1.5,0.0172476397966594,0.352733686067019,567
-2,0.00668670591018522,0.14194464158978,1409
-2.5,0.00168951075575861,0.0388349514563107,5150
-3,0.000328745799468201,0.00800288103717338,24991
-3.5,4.21796065456326e-05,0.00109195935727272,183157
-4,4.95098039215686e-06,0.000134,500000
-4.5,2.94117647058824e-07,1.2e-05,500000

latex/thesis/res/generic/bp_20455187.csv

@@ -1,10 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0592497868712702,0.966183574879227,207
-1.5,0.0465686274509804,0.854700854700855,234
-2,0.0326898561282098,0.619195046439629,323
-2.5,0.0171613765211425,0.368324125230203,543
-3,0.00553787541776455,0.116346713205352,1719
-3.5,0.00134027952469441,0.0275900124155056,7249
-4,0.000166480738027721,0.0034201480924124,58477
-4.5,1.19607843137255e-05,0.000252,500000
-5,9.50980392156863e-07,2e-05,500000

latex/thesis/res/generic/bp_40833844.csv

@@ -1,6 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0303507766743061,0.649350649350649,308
-1.5,0.0122803195352215,0.296296296296296,675
-2,0.00284899516991547,0.0733137829912024,2728
-2.5,0.000348582879119279,0.00916968502131952,21811
-3,2.52493009763634e-05,0.000760274154860243,263063

latex/thesis/res/generic/bp_963965.csv

@@ -1,9 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0352267331433998,0.56980056980057,351
-1.5,0.0214331413947537,0.383877159309021,521
-2,0.0130737396538751,0.225733634311512,886
-2.5,0.00488312520012808,0.0960614793467819,2082
-3,0.00203045475576707,0.0396589331746976,5043
-3.5,0.000513233833401836,0.010275380189067,19464
-4,0.000107190497363908,0.0025408117893667,78715
-4.5,3.2074433522095e-05,0.00092605883251763,215969

latex/thesis/res/generic/bp_bch_31_26.csv

@@ -1,11 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0584002878042931,0.743494423791822,269
-1.5,0.0458084740620892,0.626959247648903,319
-2,0.0318872821653689,0.459770114942529,435
-2.5,0.0248431459534825,0.392927308447937,509
-3,0.0158453558113146,0.251256281407035,796
-3.5,0.0115615186982586,0.176991150442478,1130
-4,0.00708558642550811,0.115606936416185,1730
-4.5,0.00389714705036542,0.0614817091915155,3253
-5,0.00221548053104734,0.0345125107851596,5795
-5.5,0.00106888328072207,0.0172131852999398,11619

latex/thesis/res/generic/bp_pegreg252x504.csv

@@ -1,6 +0,0 @@
-SNR,BER,FER,num_iterations
-1,0.0294665331073098,0.647249190938511,309
-1.5,0.0104905437352246,0.265957446808511,752
-2,0.0021089358705197,0.05616399887672,3561
-2.5,0.000172515567971134,0.00498840196543037,40093
-3,6.3531746031746e-06,0.0002,500000

latex/thesis/thesis.tex

@@ -9,12 +9,12 @@
 \thesisTitle{Application of Optimization Algorithms for Channel Decoding}
 \thesisType{Bachelor's Thesis}
 \thesisAuthor{Andreas Tsouchlos}
-\thesisAdvisor{Prof. Dr.-Ing. Laurent Schmalen}
+%\thesisAdvisor{Prof. Dr.-Ing. Laurent Schmalen}
 %\thesisHeadOfInstitute{Prof. Dr.-Ing. Laurent Schmalen}
-\thesisSupervisor{Dr.-Ing. Holger Jäkel}
+\thesisSupervisor{Name of assistant}
 \thesisStartDate{24.10.2022}
 \thesisEndDate{24.04.2023}
-\thesisSignatureDate{24.04.2023} % TODO: Signature date
+\thesisSignatureDate{Signature date} % TODO: Signature date
 \thesisLanguage{english}
 \setlanguage
 
@@ -35,7 +35,6 @@
 \usetikzlibrary{spy}
 \usetikzlibrary{shapes.geometric}
 \usetikzlibrary{arrows.meta,arrows}
-\tikzset{>=latex}
 
 \pgfplotsset{compat=newest}
 \usepgfplotslibrary{colorbrewer}
@@ -210,7 +209,6 @@
     %
     % 6. Conclusion
 
-    \include{chapters/acknowledgements}
 
     \tableofcontents
     \cleardoublepage % make sure multipage TOCs are numbered correctly
@@ -220,9 +218,9 @@
     \include{chapters/proximal_decoding}
     \include{chapters/lp_dec_using_admm}
     \include{chapters/comparison}
-%    \include{chapters/discussion}
+    \include{chapters/discussion}
     \include{chapters/conclusion}
-%    \include{chapters/appendix}
+    \include{chapters/appendix}
 
 
     %\listoffigures