Update README.md links to project to new gitlab instance

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://git.scc.kit.edu/ba_duo/ba_sw))
+(the related sofware has it's own [repo](https://gitlab.kit.edu/uhhxt/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://git.scc.kit.edu/ba_duo/ba_thesis/-/tree/prox_07_12_22/latex/presentations/02_12_2022)
+[this](https://gitlab.kit.edu/uhhxt/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

@@ -45,15 +45,6 @@
 %    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},
@@ -219,11 +210,25 @@
   doi={10.1109/TIT.1962.1057683}
 }
 
-@misc{lautern_channelcodes,
+@online{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},
-	year = "2023"
+	date = {2023-04}
+}
+
+@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

@@ -0,0 +1,18 @@
+\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={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, 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={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, 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={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, 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={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, 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={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, 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={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 legend columns=1,
                 legend pos=outer north east,

latex/thesis/chapters/comparison.tex

@@ -2,14 +2,9 @@
 \label{chapter:comparison}
 
 In this chapter, proximal decoding and \ac{LP} Decoding using \ac{ADMM} are compared.
-First the two algorithms are compared on a theoretical basis.
+First, the two algorithms are studied on a theoretical basis.
 Subsequently, their respective simulation results are examined, and their
-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.
+differences are interpreted based on their theoretical structure.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -18,12 +13,11 @@ differences are interpreted on the basis of 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}.
-When using \ac{ADMM} as an optimization method to solve the \ac{LP} decoding
-problem specifically, this is not quite possible because of the multiple
-constraints.
-In spite of that, the two algorithms still show some striking similarities.
+Additionally, the two algorithms show some striking similarities with
+regard to their general structure and the way in which the minimization of the
+respective objective functions is accomplished.
 
-To see the first of these similarities, the \ac{LP} decoding problem in
+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
@@ -38,12 +32,13 @@ $\mathcal{P}_{d_j}, \hspace{1mm} j\in\mathcal{J}$, defined as%
 %
 by moving the constraints into the objective function, as shown in figure
 \ref{fig:ana:theo_comp_alg:admm}.
-Both algorithms are composed of an iterative approach consisting of two
-alternating steps.
-The objective functions of both problems are similar in that they
+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 and one associated to the constraints the decoding process
-is subjected to.
+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.
 %
 
 \begin{figure}[h]
@@ -114,7 +109,7 @@ return $\tilde{\boldsymbol{c}}$
     \end{subfigure}%
     
 
-    \caption{Comparison of the proximal gradient method and \ac{ADMM}}
+    \caption{Comparison of proximal decoding and \ac{LP} decoding using \ac{ADMM}}
     \label{fig:ana:theo_comp_alg}
 \end{figure}%
 %
@@ -123,15 +118,11 @@ 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
-$\mathcal{P}_{d_j}$ which always provide exact results.
+$\mathcal{P}_{d_j}, \hspace{1mm} j\in\mathcal{J}$, which always provide exact
+results.
 
 The contrasting treatment of the constraints (global and approximate with
 proximal decoding as opposed to local and exact with \ac{LP} decoding using
@@ -142,34 +133,27 @@ 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
-that it can be easily detected \todo{Not 'easily' detected}, when the algorithm gets stuck - it
-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.
+the \ac{ML} certificate property: 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 linear time for \ac{LDPC} codes (see section \ref{subsec:prox:comp_perf}).
-The same is true for the $\tilde{\boldsymbol{c}}$- and $\boldsymbol{u}$-update
-steps of \ac{LP} decoding using \ac{ADMM}, while
-the projection step has a worst-case time complexity of
-$\mathcal{O}\left( n^2 \right)$ and an average complexity of
-$\mathcal{O}\left( n \right)$ (see section TODO, \cite[Sec. VIII.]{lautern}).
-
-Both algorithms can be understood as message-passing algorithms, \ac{LP}
-decoding using \ac{ADMM} as similarly to \cite[Sec. III. D.]{original_admm}
-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.
+in $\mathcal{O}\left( n \right) $ time for \ac{LDPC} codes (see section
+\ref{subsec:prox:comp_perf}).
+The same is true for \ac{LP} decoding using \ac{ADMM} (see section
+\ref{subsec:admm:comp_perf}).
+Additionally, both algorithms can be understood as message-passing algorithms,
+\ac{LP} decoding using \ac{ADMM} as similarly to
+\cite[Sec. III. D.]{original_admm} and
+\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 the $\boldsymbol{s}$ update 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
@@ -184,14 +168,14 @@ 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 $
-        $M_{j\to} \leftarrow p_j^2 - p_j$|\Suppressnumber|
+        $M_{j\to i} \leftarrow p_j^2 - p_j$|\Suppressnumber|
 |\vspace{0.22mm}\Reactivatenumber|
     end for
     for i in $\mathcal{I}$ do
-        $s_i \leftarrow s_i + \gamma \left[ 4\left( s_i^2 - 1 \right)s_i
-            \phantom{\frac{4}{s_i}}\right.$|\Suppressnumber|
-                     |\Reactivatenumber|$\left.+ \frac{4}{s_i}\sum_{j\in N_v\left( i \right) }
-                        M_{j\to} \right] $
+        $s_i\leftarrow \Pi_\eta \left( s_i + \gamma \left( 4\left( s_i^2 - 1 \right)s_i
+            \phantom{\frac{4}{s_i}}\right.\right.$|\Suppressnumber|
+                  |\Reactivatenumber|$\left.\left.+ \frac{4}{s_i}\sum_{j\in
+                             N_v\left( i \right) } M_{j\to i} \right)\right) $
         $r_i \leftarrow r_i + \omega \left( s_i - y_i \right)$
     end for
 end while
@@ -232,20 +216,14 @@ return $\tilde{\boldsymbol{c}}$
     \end{subfigure}%
     
 
-    \caption{The proximal gradient method and \ac{LP} decoding using \ac{ADMM}
+    \caption{Proximal decoding and \ac{LP} decoding using \ac{ADMM}
         as message passing algorithms}
     \label{fig:comp:message_passing}
 \end{figure}%
 %
-It is evident that while the two algorithms are very similar in their general
-structure, with \ac{LP} decoding using \ac{ADMM}, multiple messages have to be
-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.
+This message passing structure means that both algorithms can be implemented
+very efficiently, as the update steps can be performed in parallel for all
+\acp{CN} and for all \acp{VN}, respectively.
 
 In conclusion, the two algorithms have a very similar structure, where the
 parts of the objective function relating to the likelihood and to the
@@ -253,9 +231,8 @@ 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, on average, linear with
-respect to $n$, although for \ac{LP} decoding using \ac{ADMM} significantly
-more arithmetic operations are necessary in each iteration.
+In terms of time complexity, both algorithms are linear with
+respect to $n$ and are heavily parallelizable.
 
 
 
@@ -263,24 +240,100 @@ more arithmetic operations are necessary in each iteration.
 \section{Comparison of Simulation Results}%
 \label{sec:comp:res}
 
-\begin{itemize}
-    \item The comparison of actual implementations is always debatable /
-        contentious, since it is difficult to separate differences in
-        algorithm performance from differences in implementation
-    \item No large difference in computational performance $\rightarrow$
-        Parallelism cannot come to fruition as decoding is performed on the
-        same number of cores for both algorithms (Multiple decodings in parallel)
-    \item Nonetheless, in realtime applications / applications where the focus
-        is not the mass decoding of raw data, \ac{ADMM} has advantages, since
-        the decoding of a single codeword is performed faster
-    \item \ac{ADMM} faster than proximal decoding $\rightarrow$
-        Parallelism
-    \item Proximal decoding faster than \ac{ADMM} $\rightarrow$ dafuq
-        (larger number of iterations before convergence? More values to compute for ADMM?)
-\end{itemize}
+The decoding performance of the two algorithms is compared in figure
+\ref{fig:comp:prox_admm_dec} in form of the \ac{FER}.
+Shown as well is the performance of the improved proximal decoding
+algorithm presented in section \ref{sec:prox:Improved Implementation}.
+The \ac{FER} resulting from decoding using \ac{BP} and,
+wherever available, the \ac{FER} of \ac{ML} decoding, taken from
+\cite{lautern_channelcodes}, are plotted as a reference.
+The parameters chosen for the proximal and improved proximal decoders are
+$\gamma=0.05$, $\omega=0.05$, $K=200$, $\eta = 1.5$ and $N=12$.
+The parameters chosen for \ac{LP} decoding using \ac{ADMM} are $\mu = 5$,
+$\rho = 1$, $K=200$, $\epsilon_\text{pri} = 10^{-5}$ and
+$\epsilon_\text{dual} = 10^{-5}$.
+For all codes considered within the scope of this work, \ac{LP} decoding using
+\ac{ADMM} consistently outperforms both proximal decoding and the improved
+version, reaching very similar performance to \ac{BP}.
+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.
 
-\begin{figure}[H]
+The timing requirements of the decoding algorithms are visualized in figure
+\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}
@@ -289,7 +342,7 @@ more arithmetic operations are necessary in each iteration.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -299,13 +352,19 @@ more arithmetic operations are necessary in each iteration.
                 \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[NavyBlue, line width=1pt, mark=triangle, densely dashed]
+                \addplot[RedOrange, 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[PineGreen, line width=1pt, mark=triangle]
+                \addplot[Black, line width=1pt, mark=*]
                     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}
 
@@ -319,7 +378,7 @@ more arithmetic operations are necessary in each iteration.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -329,15 +388,20 @@ more arithmetic operations are necessary in each iteration.
                 \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[NavyBlue, line width=1pt, mark=triangle, densely dashed]
+                \addplot[RedOrange, 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[PineGreen, line width=1pt, mark=triangle*]
+                \addplot[Black, line width=1pt, mark=*]
                     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}
     
@@ -352,7 +416,7 @@ more arithmetic operations are necessary in each iteration.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -364,15 +428,22 @@ more arithmetic operations are necessary in each iteration.
                            discard if not={gamma}{0.05},
                            discard if gt={SNR}{5.5}]
                         {res/proximal/2d_ber_fer_dfr_20433484.csv};
-                \addplot[NavyBlue, line width=1pt, mark=triangle, densely dashed]
+                \addplot[RedOrange, 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[PineGreen, line width=1pt, mark=triangle, solid]
+                \addplot[Black, line width=1pt, mark=*]
                     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}
     
@@ -386,7 +457,7 @@ more arithmetic operations are necessary in each iteration.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -396,9 +467,16 @@ more arithmetic operations are necessary in each iteration.
                 \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[NavyBlue, line width=1pt, mark=triangle, densely dashed]
+                \addplot[RedOrange, 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}
     
@@ -414,7 +492,7 @@ more arithmetic operations are necessary in each iteration.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -424,9 +502,16 @@ more arithmetic operations are necessary in each iteration.
                 \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[NavyBlue, line width=1pt, mark=triangle, densely dashed]
+                \addplot[RedOrange, 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}
     
@@ -440,7 +525,7 @@ more arithmetic operations are necessary in each iteration.
         \begin{tikzpicture}
             \begin{axis}[
                 grid=both,
-                xlabel={$E_b / N_0$}, ylabel={FER},
+                xlabel={$E_b / N_0$ (dB)}, ylabel={FER},
                 ymode=log,
                 ymax=1.5, ymin=8e-5,
                 width=\textwidth,
@@ -450,9 +535,16 @@ more arithmetic operations are necessary in each iteration.
                 \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[NavyBlue, line width=1pt, mark=triangle, densely dashed]
+                \addplot[RedOrange, 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}
     
@@ -470,50 +562,28 @@ more arithmetic operations are necessary in each iteration.
                          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{NavyBlue, line width=1pt, mark=triangle, densely dashed}
+                \addlegendimage{RedOrange, line width=1pt, mark=triangle, densely dashed}
+                \addlegendentry{Improved proximal decoding}
+               
+                \addlegendimage{Turquoise, line width=1pt, mark=*}
                 \addlegendentry{\acs{LP} decoding using \acs{ADMM}}
                 
-                \addlegendimage{PineGreen, line width=1pt, mark=triangle*, solid}
+                \addlegendimage{RoyalPurple, line width=1pt, mark=*, 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 decoding performance between proximal decoding and \ac{LP} decoding
-        using \ac{ADMM}}
+    \caption{Comparison of the decoding performance of the different decoder
+        implementations for various codes}
     \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

@@ -1,8 +1,61 @@
-\chapter{Conclusion}%
+\chapter{Conclusion and Outlook}%
 \label{chapter:conclusion}
 
-\begin{itemize}
-    \item Summary of results
-    \item Future work
-\end{itemize}
+In the context of this thesis, two decoding algorithms were considered:
+proximal decoding and \ac{LP} decoding using \ac{ADMM}.
+The two algorithms were first analyzed individually, before comparing them
+based on simulation results as well as on their theoretical structure.
+
+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

@@ -33,13 +33,6 @@ 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

@@ -1,16 +1,51 @@
 \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.
 
-\begin{itemize}
-    \item Problem definition
-    \item Motivation
-        \begin{itemize}
-            \item Error floor when decoding with BP (seems to not be persent with LP decoding
-                \cite[Sec. I]{original_admm})
-            \item Strong theoretical guarantees that allow for better and better approximations
-                of ML decoding \cite[Sec. I]{original_admm}
-        \end{itemize}
-    \item Results summary
-\end{itemize}
+Optimization based decoding algorithms are an entirely different way of approaching
+the decoding problem.
+The first introduction of optimization techniques as a way of decoding binary
+linear codes was conducted in Feldman's 2003 Ph.D. thesis and a subsequent paper,
+establishing the field of \ac{LP} decoding \cite{feldman_thesis}, \cite{feldman_paper}.
+There, the \ac{ML} decoding problem is approximated by a \textit{linear program}, i.e.,
+a linear, convex optimization problem, which can subsequently be solved using
+several different algorithms \cite{alp}, \cite{interior_point},
+\cite{original_admm}, \cite{pdd}.
+More recently, novel approaches such as \textit{proximal decoding} have been
+introduced. Proximal decoding is based on a non-convex optimization formulation
+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 this
-work is given.
+In this chapter, the theoretical background necessary to understand the
+decoding algorithms examined in this work is given.
 First, the notation used is clarified.
-The physical aspects are detailed - the used modulation scheme and channel model.
+The physical layer is 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 LPDC codes are briefly explained.
+The established methods of decoding \ac{LDPC} 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 elemen-twise operations, in particular the \textit{Hadamard product}
+In order to designate element-wise 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{Preliminaries: Channel Model and Modulation}
+\section{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 codword
+A data word $\boldsymbol{u} \in \mathbb{F}_2^k$ can be mapped onto a codeword
 $\boldsymbol{c} \in \mathbb{F}_2^n$ 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}}
-        \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
-        \left( \boldsymbol{c} \right) \\
+%    &= \argmax_{c\in\mathcal{C}} f_{\boldsymbol{Y} \mid \boldsymbol{C}}
+%        \left( \boldsymbol{y} \mid \boldsymbol{c} \right) p_{\boldsymbol{C}}
+%        \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}%
 %
@@ -285,15 +285,16 @@ 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
-$p_{C_i \mid \boldsymbol{Y}}\left(c_i = 1 | \boldsymbol{y} \right),\hspace{2mm} i\in\mathcal{I}$
-\cite[Sec. III.]{mackay_rediscovery} and use them to calculate the estimate $\hat{\boldsymbol{c}}$.
+$p_{C_i \mid \boldsymbol{Y}}\left(c_i = 1 | \boldsymbol{y} \right),\hspace{2mm} i\in\mathcal{I}$,
+see \cite[Sec. III.]{mackay_rediscovery} and use them to calculate the estimate
+$\hat{\boldsymbol{c}}$.
 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 probabilities
+When the graph contains cycles, however, \ac{BP} only approximates the \ac{MAP} 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 \ac{BER} is
+the use of \ac{BP} impractical for applications where a very low error rate is
 desired \cite[Sec. 15.3]{ryan_lin_2009}.
 Another popular decoding method for \ac{LDPC} codes is the
 \textit{min-sum algorithm}.
@@ -341,7 +342,7 @@ 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 edges of a hypercube.
+where the codewords are some of the vertices of a hypercube.
 The goal is to find the point $\tilde{\boldsymbol{c}}$,
 which minimizes the objective function $g$.
 
@@ -457,29 +458,38 @@ which minimizes the objective function $g$.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{An introduction to the proximal gradient method and ADMM}
+\section{A Short 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) = \argmin_{\boldsymbol{x}} \left(
-        f\left( \boldsymbol{x} \right) + \frac{1}{2\lambda}\lVert \boldsymbol{x}
-            - \boldsymbol{v} \rVert_2^2 \right)
+    \textbf{prox}_{\lambda f}\left( \boldsymbol{v} \right)
+        = \argmin_{\boldsymbol{x} \in \mathbb{R}^n} \left(
+            f\left( \boldsymbol{x} \right) + \frac{1}{2\lambda}\lVert \boldsymbol{x}
+                - \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 heavily each term is weighed.
-The \textit{proximal gradient method} is an iterative optimization method
+The parameter $\lambda$ determines how each term is weighed.
+The proximal gradient method is an iterative optimization method
 utilizing proximal operators, used to solve problems of the form%
 %
 \begin{align*}
-    \text{minimize}\hspace{5mm}f\left( \boldsymbol{x} \right) + g\left( \boldsymbol{x} \right) 
+    \underset{\boldsymbol{x} \in \mathbb{R}^n}{\text{minimize}}\hspace{5mm}
+        f\left( \boldsymbol{x} \right) + g\left( \boldsymbol{x} \right) 
 \end{align*}
 %
 that consists of two steps: minimizing $f$ with gradient descent
@@ -492,14 +502,14 @@ 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 problem
+to be differentiable, it can be used to encode the constraints of the optimization problem
 (e.g., in the form of an \textit{indicator function}, as mentioned in
 \cite[Sec. 1.2]{proximal_algorithms}).
 
-The \ac{ADMM} is another optimization method.
+\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, where the objective function
-is linear, and the constraints consist of linear equalities and inequalities.
+is a special type of convex optimization problem in which the objective function
+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.}
@@ -507,38 +517,53 @@ interpreted componentwise.}
 %
 \begin{alignat}{3}
     \begin{alignedat}{3}
-        \text{minimize }\hspace{2mm}   && \boldsymbol{\gamma}^\text{T} \boldsymbol{x}         \\
+        \underset{\boldsymbol{x}\in\mathbb{R}^n}{\text{minimize }}\hspace{2mm}  
+            && \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}%
 %
-A technique called \textit{Lagrangian relaxation} \cite[Sec. 11.4]{intro_to_lin_opt_book}
-can then be applied.
+where $\boldsymbol{x}, \boldsymbol{\gamma} \in \mathbb{R}^n$, $\boldsymbol{b} \in \mathbb{R}^m$
+and $\boldsymbol{A}\in\mathbb{R}^{m \times n}$.
+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 then formulated as
+is formulated as
 %
 \begin{align}
     \begin{aligned}
-        \text{minimize }\hspace{2mm}   & \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
-            + \boldsymbol{\lambda}^\text{T}\left(\boldsymbol{b}
-                - \boldsymbol{A}\boldsymbol{x} \right)  \\
+        \underset{\boldsymbol{x}\in\mathbb{R}^n}{\text{minimize }}\hspace{2mm}
+            & \boldsymbol{\gamma}^\text{T}\boldsymbol{x}
+            + \boldsymbol{\lambda}^\text{T}\left(
+                \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{b}
-            - \boldsymbol{A}\boldsymbol{x} \right)
+        + \boldsymbol{\lambda}^\text{T}\left(
+            \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b}\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}
 $\boldsymbol{\lambda}$.
@@ -562,12 +587,12 @@ 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.
+and (\ref{eq:theo:admm_standard}) have the same value, i.e.,
 %
 \begin{align*}
-    \max_{\boldsymbol{\lambda}} \, \min_{\boldsymbol{x} \ge \boldsymbol{0}}
+    \max_{\boldsymbol{\lambda}\in\mathbb{R}^m} \, \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}}}
@@ -577,7 +602,7 @@ and (\ref{eq:theo:admm_standard}) have the same value.
 Thus, we can define the \textit{dual problem} as the search for the tightest lower bound:%
 %
 \begin{align}
-    \underset{\boldsymbol{\lambda}}{\text{maximize }}\hspace{2mm}
+    \underset{\boldsymbol{\lambda}\in\mathbb{R}^m}{\text{maximize }}\hspace{2mm}
         & \min_{\boldsymbol{x} \ge \boldsymbol{0}} \mathcal{L}
         \left( \boldsymbol{x}, \boldsymbol{\lambda} \right)
     \label{eq:theo:dual}
@@ -600,7 +625,7 @@ using equation (\ref{eq:theo:admm_obtain_primal}); then, update $\boldsymbol{\la
 using gradient descent \cite[Sec. 2.1]{distr_opt_book}:%
 %
 \begin{align*}
-    \boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x}} \mathcal{L}\left(
+    \boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x} \ge \boldsymbol{0}} \mathcal{L}\left(
         \boldsymbol{x}, \boldsymbol{\lambda} \right) \\
     \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
         + \alpha\left( \boldsymbol{A}\boldsymbol{x} - \boldsymbol{b} \right),
@@ -608,12 +633,12 @@ using gradient descent \cite[Sec. 2.1]{distr_opt_book}:%
 .\end{align*}
 %
 The algorithm can be improved by observing that when the objective function
-$g: \mathbb{R}^n \rightarrow \mathbb{R}$ is separable into a number
-$N \in \mathbb{N}$ of sub-functions
+$g: \mathbb{R}^n \rightarrow \mathbb{R}$ is separable into a sum of
+$N \in \mathbb{N}$ 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 $\boldsymbol{x}_i,\hspace{1mm} i\in [1:N]$ are subvectors of
+where $\boldsymbol{x}_i\in\mathbb{R}^{n_i},\hspace{1mm} i\in [1:N]$ are subvectors of
 $\boldsymbol{x}$, the Lagrangian is as well:
 %
 \begin{align*}
@@ -624,18 +649,18 @@ $\boldsymbol{x}$, 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{b}
-            - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x_i} \right) 
+            + \boldsymbol{\lambda}^\text{T} \left(
+            \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x_i} - \boldsymbol{b}\right) 
 .\end{align*}%
 %
-The matrices $\boldsymbol{A}_i, \hspace{1mm} i \in [1:N]$ are partitions of
-the matrix $\boldsymbol{A}$, corresponding to
+The matrices $\boldsymbol{A}_i \in \mathbb{R}^{m \times n_i}, \hspace{1mm} i \in [1:N]$
+form a partition of $\boldsymbol{A}$, corresponding to
 $\boldsymbol{A} = \begin{bmatrix}
     \boldsymbol{A}_1 &
     \ldots &
     \boldsymbol{A}_N
 \end{bmatrix}$.
-The minimization of each term can then happen in parallel, in a distributed
+The minimization of each term can 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
@@ -643,7 +668,7 @@ constant.
 This modified version of dual ascent is called \textit{dual decomposition}:
 %
 \begin{align*}
-    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i}\mathcal{L}\left(
+    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i \ge \boldsymbol{0}}\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}
@@ -657,14 +682,15 @@ 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 ordinary one with an additional penalty term
-with the penaly parameter $\mu$:
+The augmented Lagrangian extends the classical one with an additional penalty term
+with the penalty 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( \boldsymbol{b}
-        - \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i \right)}_{\text{Ordinary Lagrangian}}
+            + \boldsymbol{\lambda}^\text{T}\left(\sum_{i=1}^{N}
+                \boldsymbol{A}_i\boldsymbol{x}_i - \boldsymbol{b}\right)}
+                _{\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
@@ -674,21 +700,20 @@ 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}\mathcal{L}_\mu\left(
+    \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i \ge \boldsymbol{0}}\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

@@ -0,0 +1,9 @@
+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

@@ -0,0 +1,10 @@
+{
+    "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

@@ -0,0 +1,31 @@
+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

@@ -0,0 +1,10 @@
+{
+    "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

@@ -0,0 +1,9 @@
+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

@@ -0,0 +1,10 @@
+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

@@ -0,0 +1,6 @@
+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

@@ -0,0 +1,9 @@
+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

@@ -0,0 +1,11 @@
+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

@@ -0,0 +1,6 @@
+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{Name of assistant}
+\thesisSupervisor{Dr.-Ing. Holger Jäkel}
 \thesisStartDate{24.10.2022}
 \thesisEndDate{24.04.2023}
-\thesisSignatureDate{Signature date} % TODO: Signature date
+\thesisSignatureDate{24.04.2023} % TODO: Signature date
 \thesisLanguage{english}
 \setlanguage
 
@@ -35,6 +35,7 @@
 \usetikzlibrary{spy}
 \usetikzlibrary{shapes.geometric}
 \usetikzlibrary{arrows.meta,arrows}
+\tikzset{>=latex}
 
 \pgfplotsset{compat=newest}
 \usepgfplotslibrary{colorbrewer}
@@ -209,6 +210,7 @@
     %
     % 6. Conclusion
 
+    \include{chapters/acknowledgements}
 
     \tableofcontents
     \cleardoublepage % make sure multipage TOCs are numbered correctly
@@ -218,9 +220,9 @@
     \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