Added discussion and fixed bibliography

latex/thesis/chapters/discussion.tex

@@ -1,8 +1,39 @@
 \chapter{Discussion}%
 \label{chapter:discussion}
     
-\begin{itemize}
-    \item Proximal decoding improvement limitations
-\end{itemize}
-%   - Improvement pitfalls
+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.
+
+As mentioned in section \ref{subsec:prox:conv_properties}, the alternating
+minimization of the two gradients in the proximal decoding algorithm leads to
+an oscillation after a number of iterations.
+One approach to alleviate this problem might be to use \ac{ADMM} instead of
+the proximal gradient method to solve the optimization problem.
+This is because due to the introduction of the dual variable, the minimization
+of each part of the objective function would no longer take place with regard
+to the same exact variable.
+Additionally, ``\ac{ADMM} will converge even when the x- and z-minimization
+steps are not carried out exactly [\ldots]''
+\cite[Sec. 3.4.4]{distr_opt_book}, which is advantageous, as the
+constraints are never truly satisfied; not even after the minimization step
+dealing with the constraint part of the objective function.
+Despite this, an initial examination by Yanxia Lu in
+\cite[Sec. 4.2.4.]{yanxia_lu_thesis} shows only limited success.
+
+Another interesting approach might be the combination of proximal and \ac{LP}
+decoding.
+Performing an initial number of iterations using proximal decoding to obtain
+a rough first estimate and subsequently using \ac{LP} decoding with only the
+violated constraints may be a way to achieve a shorter running time, because
+of the low-complexity nature of proximal decoding.
+This could be usefull, for example, to mitigate the slow convergence of
+\ac{ADMM} \cite[3.2.2]{distr_opt_book}.
+Subsequently introducing additional parity checks might be a way of combining
+the best properties of proximal decoding, \ac{LP} decoding using \ac{ADMM} and
+\textit{adaptive \ac{LP} decoding} \cite{alp} to obtain a decoder relatively
+efficiently approximating \ac{ML} performance.
 

latex/thesis/chapters/lp_dec_using_admm.tex

@@ -527,6 +527,8 @@ The resulting formulation of the relaxed optimization problem becomes%
     \end{aligned} \label{eq:lp:relaxed_formulation}
 \end{align}%
 
+\todo{Mention ML certificate property}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Decoding Algorithm}%

latex/thesis/chapters/theoretical_background.tex

@@ -570,7 +570,7 @@ Thus, we can define the \textit{dual problem} as the search for the tightest low
 %
 and recover the solution $\boldsymbol{x}_{\text{opt}}$ to problem (\ref{eq:theo:admm_standard})
 from the solution $\boldsymbol{\lambda}_\text{opt}$ to problem (\ref{eq:theo:dual})
-by computing \cite[Sec. 2.1]{admm_distr_stats}%
+by computing \cite[Sec. 2.1]{distr_opt_book}%
 %
 \begin{align}
     \boldsymbol{x}_{\text{opt}} = \argmin_{\boldsymbol{x} \ge \boldsymbol{0}}
@@ -582,7 +582,7 @@ by computing \cite[Sec. 2.1]{admm_distr_stats}%
 The dual problem can then be solved iteratively using \textit{dual ascent}: starting with an
 initial estimate for $\boldsymbol{\lambda}$, calculate an estimate for $\boldsymbol{x}$
 using equation (\ref{eq:theo:admm_obtain_primal}); then, update $\boldsymbol{\lambda}$
-using gradient descent \cite[Sec. 2.1]{admm_distr_stats}:%
+using gradient descent \cite[Sec. 2.1]{distr_opt_book}:%
 %
 \begin{align*}
     \boldsymbol{x} &\leftarrow \argmin_{\boldsymbol{x}} \mathcal{L}\left(
@@ -621,7 +621,7 @@ $\boldsymbol{A} = \begin{bmatrix}
     \boldsymbol{A}_N
 \end{bmatrix}$.
 The minimization of each term can then happen in parallel, in a distributed
-fashion \cite[Sec. 2.2]{admm_distr_stats}.
+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
 constant.