Replaced splicing with tuple

latex/thesis/chapters/lp_dec_using_admm.tex

@@ -562,7 +562,7 @@ complexity has been demonstrated to compare favorably to \ac{BP} \cite{original_
 
 The \ac{LP} decoding problem in (\ref{eq:lp:relaxed_formulation}) can be
 slightly rewritten using the auxiliary variables
-$\boldsymbol{z}_{[1:m]}$:%
+$\boldsymbol{z}_1, \ldots, \boldsymbol{z}_m$:%
 %
 \begin{align}
     \begin{aligned}
@@ -592,8 +592,8 @@ The multiple constraints can be addressed by introducing additional terms in the
 augmented lagrangian:%
 %
 \begin{align*}
-    \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]},
-        \boldsymbol{\lambda}_{[1:m]} \right) 
+    \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m,
+        \left( \boldsymbol{\lambda} \right)_{j=1}^m \right) 
     = \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
         + \sum_{j\in\mathcal{J}} \boldsymbol{\lambda}^\text{T}_j
         \left( \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \boldsymbol{z}_j \right) 
@@ -606,18 +606,21 @@ The additional constraints remain in the dual optimization problem:%
 \begin{align*}
     \text{maximize } \min_{\substack{\tilde{\boldsymbol{c}} \\
             \boldsymbol{z}_j \in \mathcal{P}_{d_j}\,\forall\,j\in\mathcal{J}}}
-    \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]},
-        \boldsymbol{\lambda}_{[1:m]} \right)
+    \mathcal{L}_{\mu}\left( \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m,
+    \left( \boldsymbol{\lambda} \right)_{j=1}^m \right)
 .\end{align*}%
 %
 The steps to solve the dual problem then become:
 %
 \begin{alignat*}{3}
-    \tilde{\boldsymbol{c}} &\leftarrow \argmin_{\tilde{\boldsymbol{c}}} \mathcal{L}_{\mu} \left( 
-        \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}, \boldsymbol{\lambda}_{[1:m]} \right) \\
+    \tilde{\boldsymbol{c}} &\leftarrow \argmin_{\tilde{\boldsymbol{c}}}
+        \mathcal{L}_{\mu} \left( 
+            \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m,
+            \left( \boldsymbol{\lambda}\right)_{j=1}^m \right) \\
     \boldsymbol{z}_j &\leftarrow \argmin_{\boldsymbol{z}_j \in \mathcal{P}_{d_j}}
         \mathcal{L}_{\mu} \left( 
-            \tilde{\boldsymbol{c}}, \boldsymbol{z}_{[1:m]}, \boldsymbol{\lambda}_{[1:m]} \right)
+            \tilde{\boldsymbol{c}}, \left( \boldsymbol{z} \right)_{j=1}^m,
+            \left( \boldsymbol{\lambda} \right)_{j=1}^m \right)
         \hspace{3mm} &&\forall j\in\mathcal{J} \\
     \boldsymbol{\lambda}_j &\leftarrow \boldsymbol{\lambda}_j
         + \mu\left( \boldsymbol{T}_j\tilde{\boldsymbol{c}}
@@ -794,7 +797,7 @@ Defining%
 the $\tilde{\boldsymbol{c}}$ update can then be rewritten as%
 %
 \begin{align*}
-    \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ
+    \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ \left(-1\right)} \circ
         \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right)  
 .\end{align*}
 %
@@ -820,7 +823,7 @@ while $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}}
             \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right) $
     end for
     for $i$ in $\mathcal{I}$ do
-        $\tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ
+        $\tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ \left( -1\right)} \circ
             \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right) $
     end for
 end while

latex/thesis/chapters/theoretical_background.tex

@@ -24,13 +24,11 @@ For example:%
     c \in \mathbb{F}_2           &\to \tilde{c} \in \left[ 0, 1 \right] \subseteq \mathbb{R}
 .\end{align*}
 %
-Additionally, a shorthand notation will be used to denote series of indices and series
-of indexed variables:%
+Additionally, a shorthand notation will be used to denote a set of indices:%
 %
 \begin{align*}
     \left[ m:n \right]      &:= \left\{ m, m+1, \ldots, n-1, n \right\},
-        \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}\\
-    x_{\left[ m:n \right] } &:= \left\{ x_m, x_{m+1}, \ldots, x_{n-1}, x_n \right\} 
+        \hspace{5mm} m < n, \hspace{2mm} m,n\in\mathbb{Z}
 .\end{align*}
 \todo{Not really slicing. How should it be denoted?}
 %
@@ -619,7 +617,7 @@ $\boldsymbol{x}$, the Lagrangian is as well:
         = \boldsymbol{b}
 \end{align*}
 \begin{align*}
-    \mathcal{L}\left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right)
+    \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) 
@@ -641,7 +639,7 @@ 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}_{[1:N]}, \boldsymbol{\lambda}\right) 
+        \left( \boldsymbol{x}_i \right)_{i=1}^N, \boldsymbol{\lambda}\right) 
         \hspace{5mm} \forall i \in [1:N]\\
     \boldsymbol{\lambda} &\leftarrow \boldsymbol{\lambda}
         + \alpha\left( \sum_{i=1}^{N} \boldsymbol{A}_i\boldsymbol{x}_i
@@ -652,13 +650,13 @@ This modified version of dual ascent is called \textit{dual decomposition}:
 
 \ac{ADMM} works the same way as dual decomposition.
 It only differs in the use of an \textit{augmented Lagrangian}
-$\mathcal{L}_\mu\left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right)$
+$\mathcal{L}_\mu\left( \left( \boldsymbol{x} \right)_{i=1}^N, \boldsymbol{\lambda} \right)$
 in order to strengthen the convergence properties.
 The augmented Lagrangian extends the ordinary one with an additional penalty term
 with the penaly parameter $\mu$:
 %
 \begin{align*}
-    \mathcal{L}_\mu \left( \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda} \right)
+    \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}}
@@ -672,7 +670,7 @@ condition that the step size be $\mu$:%
 %
 \begin{align*}
     \boldsymbol{x}_i &\leftarrow \argmin_{\boldsymbol{x}_i}\mathcal{L}_\mu\left(
-        \boldsymbol{x}_{[1:N]}, \boldsymbol{\lambda}\right) 
+        \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