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Fixed mistake in implementation details; |N_v(i)| -> d_i

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Andreas Tsouchlos 2023-04-15 18:56:55 +02:00
parent ff0e2beea0
commit bcf05d26af
1 changed files with 11 additions and 12 deletions
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23
latex/thesis/chapters/lp_dec_using_admm.tex
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@ -134,8 +134,8 @@ examplary code, which is described by the generator and parity-check matrices%
and has only two possible codewords: and has only two possible codewords:
% %
\begin{align*} \begin{align*}
\mathcal{C} = \left\{ \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}, \mathcal{C} = \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},
\begin{bmatrix} 0 & 1 & 1 \end{bmatrix} \right\} \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\}
.\end{align*} .\end{align*}
% %
Figure \ref{fig:lp:poly:exact_ilp} shows the domain of exact \ac{ML} decoding. Figure \ref{fig:lp:poly:exact_ilp} shows the domain of exact \ac{ML} decoding.
@ -631,7 +631,7 @@ The same is true for $\left( \boldsymbol{\lambda}_j \right)_i$.}
\cite[Sec. III. B.]{original_admm}:% \cite[Sec. III. B.]{original_admm}:%
% %
\begin{alignat*}{3} \begin{alignat*}{3}
\tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( \tilde{c}_i &\leftarrow \frac{1}{d_i} \left(
\sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
- \frac{1}{\mu} \left( \boldsymbol{\lambda}_j \right)_i \Big) - \frac{1}{\mu} \left( \boldsymbol{\lambda}_j \right)_i \Big)
- \frac{\gamma_i}{\mu} \right) - \frac{\gamma_i}{\mu} \right)
@ -652,7 +652,7 @@ This representation can be slightly simplified by substituting
$\boldsymbol{\lambda}_j = \mu \cdot \boldsymbol{u}_j \,\forall\,j\in\mathcal{J}$:% $\boldsymbol{\lambda}_j = \mu \cdot \boldsymbol{u}_j \,\forall\,j\in\mathcal{J}$:%
% %
\begin{alignat*}{3} \begin{alignat*}{3}
\tilde{c}_i &\leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( \tilde{c}_i &\leftarrow \frac{1}{d_i} \left(
\sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i \sum_{j\in N_v\left( i \right) } \Big( \left( \boldsymbol{z}_j \right)_i
- \left( \boldsymbol{u}_j \right)_i \Big) - \left( \boldsymbol{u}_j \right)_i \Big)
- \frac{\gamma_i}{\mu} \right) - \frac{\gamma_i}{\mu} \right)
@ -692,7 +692,7 @@ while $\sum_{j\in\mathcal{J}} \lVert \boldsymbol{T}_j\tilde{\boldsymbol{c}} - \b
- \boldsymbol{z}_j$ - \boldsymbol{z}_j$
end for end for
for $i$ in $\mathcal{I}$ do for $i$ in $\mathcal{I}$ do
$\tilde{c}_i \leftarrow \frac{1}{\left| N_v\left( i \right) \right|} \left( $\tilde{c}_i \leftarrow \frac{1}{d_i} \left(
\sum_{j\in N_v\left( i \right) } \Big( \sum_{j\in N_v\left( i \right) } \Big(
\left( \boldsymbol{z}_j \right)_i - \left( \boldsymbol{u}_j \left( \boldsymbol{z}_j \right)_i - \left( \boldsymbol{u}_j
\right)_i \right)_i
@ -724,7 +724,7 @@ The method chosen here is the one presented in \cite{lautern}.
\section{Implementation Details}% \section{Implementation Details}%
\label{sec:lp:Implementation Details} \label{sec:lp:Implementation Details}
The development process used to implement this decoding algorithm is the same The development process used to implement this decoding algorithm was the same
as outlined in section as outlined in section
\ref{sec:prox:Implementation Details} for proximal decoding. \ref{sec:prox:Implementation Details} for proximal decoding.
At first, an initial version was implemented in Python, before repeating the At first, an initial version was implemented in Python, before repeating the
@ -752,11 +752,10 @@ Using this observation, the sum can be written as%
- \boldsymbol{u}_j \right) \right)_i - \boldsymbol{u}_j \right) \right)_i
.\end{align*} .\end{align*}
Further noticing that the vectors Further noticing that the vectors
$\boldsymbol{T}_j^\text{T}\left( \boldsymbol{z}_j - \boldsymbol{u}_j \right), $\boldsymbol{T}_j^\text{T}\left( \boldsymbol{z}_j - \boldsymbol{u}_j \right)$
\hspace{1mm} j\in\mathcal{J} $
unrelated to \ac{VN} $i$ have $0$ as the $i$th component, the set of indices unrelated to \ac{VN} $i$ have $0$ as the $i$th component, the set of indices
the summation takes place over can be extended to $\mathcal{J}$, allowing the the summation takes place over can be extended to $\mathcal{J}$, allowing the
expression to be rewritten to% expression to be rewritten as%
% %
\begin{align*} \begin{align*}
\sum_{j\in \mathcal{J}}\left( \boldsymbol{T}_j^\text{T} \left( \boldsymbol{z}_j \sum_{j\in \mathcal{J}}\left( \boldsymbol{T}_j^\text{T} \left( \boldsymbol{z}_j
@ -771,12 +770,12 @@ Defining%
\boldsymbol{D} := \begin{bmatrix} \boldsymbol{D} := \begin{bmatrix}
d_1 \\ d_1 \\
\vdots \\ \vdots \\
d_m d_n
\end{bmatrix}% \end{bmatrix}%
\hspace{5mm}% \hspace{5mm}%
\text{and}% \text{and}%
\hspace{5mm}% \hspace{5mm}%
\boldsymbol{M} := \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T} \boldsymbol{s} := \sum_{j\in\mathcal{J}} \boldsymbol{T}_j^\text{T}
\left( \boldsymbol{z}_j - \boldsymbol{u}_j \right) \left( \boldsymbol{z}_j - \boldsymbol{u}_j \right)
\end{align*}% \end{align*}%
% %
@ -784,7 +783,7 @@ the $\tilde{\boldsymbol{c}}$ update can then be rewritten as%
% %
\begin{align*} \begin{align*}
\tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ \tilde{\boldsymbol{c}} \leftarrow \boldsymbol{D}^{\circ -1} \circ
\left( \boldsymbol{M} - \frac{1}{\mu}\boldsymbol{\gamma} \right) \left( \boldsymbol{s} - \frac{1}{\mu}\boldsymbol{\gamma} \right)
.\end{align*} .\end{align*}
% %