an.tsouchlos/hiwi-update-presentations
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 3 Wiki Activity

Fix homotopy definition; Fix simulation results

Browse Source
This commit is contained in:
Andreas Tsouchlos 2025-05-12 23:06:02 +02:00
parent 1499c65525
commit 4b3d39e605
2 changed files with 260 additions and 208 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

4
Makefile
View File

@ -4,7 +4,7 @@ all:
latexmk src/2024-12-03/presentation.tex latexmk src/2024-12-03/presentation.tex
mv build/presentation.pdf build/presentation_2024-12-03.pdf mv build/presentation.pdf build/presentation_2024-12-03.pdf
latexmk src/2025-01-07/presentation.tex latexmk src/2025-03-28/presentation.tex
mv build/presentation.pdf build/presentation_2025-01-07.pdf mv build/presentation.pdf build/presentation_2025-03-28.pdf
clean: clean:
rm -rf build rm -rf build

464
src/2025-03-28/presentation.tex
View File

@ -1,5 +1,5 @@
\documentclass[10pt, aspectratio=169, usenames, dvipsnames]{beamer} \documentclass[10pt, aspectratio=169, usenames, dvipsnames]{beamer}
\usepackage{tikz} \usepackage{tikz}
\usepackage{tikz-3dplot} \usepackage{tikz-3dplot}
\usetikzlibrary{spy, external, intersections} \usetikzlibrary{spy, external, intersections}
@ -9,7 +9,6 @@
\pgfplotsset{compat=newest} \pgfplotsset{compat=newest}
\usepgfplotslibrary{fillbetween} \usepgfplotslibrary{fillbetween}
\usepackage{listings} \usepackage{listings}
\usepackage{subcaption} \usepackage{subcaption}
\usepackage{bbm} \usepackage{bbm}
@ -22,27 +21,23 @@
%\definecolor{gruvbox-bg}{HTML}{282828} %\definecolor{gruvbox-bg}{HTML}{282828}
\definecolor{gruvbox-bg}{HTML}{f2e5bc} \definecolor{gruvbox-bg}{HTML}{f2e5bc}
% %
% %
% Custom commands % Custom commands
% %
% %
\input{lib/latex-common/common.tex} \input{lib/latex-common/common.tex}
\pgfplotsset{colorscheme/rocket} \pgfplotsset{colorscheme/rocket}
\newcommand{\res}{src/2025-03-28/res} \newcommand{\res}{src/2025-03-28/res}
% %
% %
% CEL Template % CEL Template
% %
% %
\newcommand{\templates}{lib/cel-template} \newcommand{\templates}{lib/cel-template}
\input{\templates/packages.tex} \input{\templates/packages.tex}
@ -62,44 +57,38 @@
% \AtBeginSubsubsection[]{} % \AtBeginSubsubsection[]{}
% \AtBeginSubsection[]{} % \AtBeginSubsection[]{}
% %
% %
% Set up document % Set up document
% %
% %
\title{HiWi Notes: Minimization of the Code Constraint Polynomial using \title{HiWi Notes: Minimization of the Code Constraint Polynomial using
Homotopy Continuation Methods} Homotopy Continuation Methods}
\subtitle{\small 07.01.2025} \subtitle{\small 28.03.2025}
\author{\vspace{1.5mm} Andreas Tsouchlos} \author{\vspace{1.5mm} Andreas Tsouchlos}
\date{ } \date{ }
\institute{Karlsruhe Institute of Technology (KIT), \institute{Karlsruhe Institute of Technology (KIT),
\\ Communications Engineering Lab (CEL) } \\ Communications Engineering Lab (CEL) }
\tikzstyle{every node}=[font=\small] \tikzstyle{every node}=[font=\small]
\captionsetup[sub]{font=small} \captionsetup[sub]{font=small}
% %
% %
% Document body % Document body
% %
% %
\begin{document} \begin{document}
\begin{frame}[plain] \begin{frame}[plain]
\maketitle \maketitle
\end{frame} \end{frame}
\newcommand{\largecitereference}[1] \newcommand{\largecitereference}[1]
{\textcolor{kit-green100}{ \large \textbf{{[#1]}} }} {\textcolor{kit-green100}{ \large \textbf{{[#1]}} }}
% %
% - The decoding problem as the search for a solution of polynomial system % - The decoding problem as the search for a solution of polynomial system
@ -108,7 +97,6 @@
% - Implementation % - Implementation
% %
% TODO: Check example homotopy % TODO: Check example homotopy
% TODO: Add the name of the homotopy construction (t-1)G + tF and note that it % TODO: Add the name of the homotopy construction (t-1)G + tF and note that it
% is not the only possibility % is not the only possibility
@ -117,27 +105,35 @@
\begin{minipage}[c]{0.65\textwidth} \begin{minipage}[c]{0.65\textwidth}
\begin{itemize} \begin{itemize}
\item Goal: Solve system of equations $F(\bm{x}) = \bm{0}, \hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ \item Goal: Solve system of equations $F(\bm{x}) = \bm{0},
\item Problem: Depending on $F$, solving this directly may be difficult \hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$
\item Problem: Depending on $F$, solving this directly
may be difficult
\item Solution: Define \emph{homotopy function} $H(\bm{x}, t)$ with \item Solution: Define \emph{homotopy function} $H(\bm{x}, t)$ with
\begin{gather*} \begin{gather*}
H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x}, 1) = F(\bm{x}) H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x},
,\end{gather*} 1) = F(\bm{x})
i.e., a deformation between two systems $G(\bm{x})$ and $F(\bm{x})$ ,
\end{gather*}
i.e., a deformation between two systems $G(\bm{x})$
and $F(\bm{x})$
(where the zeros of $G$ can be easily obtained); E.g., (where the zeros of $G$ can be easily obtained); E.g.,
\begin{gather*} \begin{gather*}
H(\bm{x}, t) = (t-1)G(\bm{x}) + tF(\bm{x}) H(\bm{x}, t) = (t-1)G(\bm{x}) + tF(\bm{x})
.\end{gather*} .
Then, compute $(\bm{x}_0, 0)$ such that $G(\bm{x}_0) = \bm{0}$ and trace path to $(\bm{x}_1, 1)$ with $F(\bm{x}_1) = \bm{0}$ \end{gather*}
Then, compute $(\bm{x}_0, 0)$ such that $G(\bm{x}_0) =
\bm{0}$ and trace path to $(\bm{x}_1, 1)$ with
$F(\bm{x}_1) = \bm{0}$
\end{itemize} \end{itemize}
\vspace{5mm} \vspace{5mm}
\addreferences \addreferences
{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
continuation method for solving systems of nonlinear and continuation method for solving systems of nonlinear and
polynomial equations}. Communications in Information and polynomial equations}. Communications in Information and
Systems 15.2 (2015): 119-307.} Systems 15.2 (2015): 119-307.}
\stopreferences \stopreferences
\end{minipage}% \end{minipage}%
\hfill \hfill
@ -151,70 +147,70 @@
\begin{tikzpicture} \begin{tikzpicture}
\begin{axis}[ \begin{axis}[
xmin=-1,xmax=1, xmin=-1,xmax=1,
ymin=-1,ymax=1, ymin=-1,ymax=1,
width=\figlength, width=\figlength,
height=\figlength, height=\figlength,
ticks=none, ticks=none,
view={0}{90}, view={0}{90},
title={$t=0$}, title={$t=0$},
title style={yshift=-1mm}, title style={yshift=-1mm},
% xlabel={$x_1$}, % xlabel={$x_1$},
% ylabel={$x_2$}, % ylabel={$x_2$},
] ]
\addplot3[point meta=\thisrow{Hmag}, \addplot3[point meta=\thisrow{Hmag},
point meta min=0, point meta min=0,
point meta max=2.5, point meta max=2.5,
quiver={u=\thisrow{H1}, quiver={u=\thisrow{H1},
v=\thisrow{H2}, v=\thisrow{H2},
scale arrows=.25, scale arrows=.25,
every arrow/.append style={% every arrow/.append style={%
line width=.5 line width=.5
+\pgfplotspointmetatransformed/1000, +\pgfplotspointmetatransformed/1000,
-{Latex[length=0pt 5,width=0pt 3]} -{Latex[length=0pt 5,width=0pt 3]}
}, },
}, },
quiver/colored = {mapped color}, quiver/colored = {mapped color},
-stealth, -stealth,
] ]
table[col sep=comma, discard if not={t}{0.0}] table[col sep=comma, discard if not={t}{0.0}]
{\res/H.csv}; {\res/H.csv};
\addplot[mark=*] coordinates {(0,0)} node[above] \addplot[mark=*] coordinates {(0,0)} node[above]
{$\bm{x}_0$}; {$\bm{x}_0$};
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\begin{tikzpicture} \begin{tikzpicture}
\begin{axis}[ \begin{axis}[
xmin=-1,xmax=1, xmin=-1,xmax=1,
ymin=-1,ymax=1, ymin=-1,ymax=1,
width=\figlength, width=\figlength,
height=\figlength, height=\figlength,
ticks=none, ticks=none,
view={0}{90}, view={0}{90},
title={$t=0.5$}, title={$t=0.5$},
title style={yshift=-1mm}, title style={yshift=-1mm},
% xlabel={$x_1$}, % xlabel={$x_1$},
% ylabel={$x_2$}, % ylabel={$x_2$},
] ]
\addplot3[point meta=\thisrow{Hmag}, \addplot3[point meta=\thisrow{Hmag},
point meta min=0, point meta min=0,
point meta max=2.5, point meta max=2.5,
quiver={u=\thisrow{H1}, quiver={u=\thisrow{H1},
v=\thisrow{H2}, v=\thisrow{H2},
scale arrows=.25, scale arrows=.25,
every arrow/.append style={% every arrow/.append style={%
line width=.5 line width=.5
+\pgfplotspointmetatransformed/1000, +\pgfplotspointmetatransformed/1000,
-{Latex[length=0pt 5,width=0pt 3]} -{Latex[length=0pt 5,width=0pt 3]}
}, },
}, },
quiver/colored = {mapped color}, quiver/colored = {mapped color},
-stealth, -stealth,
] ]
table[col sep=comma, discard if not={t}{0.5}] table[col sep=comma, discard if not={t}{0.5}]
{\res/H.csv}; {\res/H.csv};
\draw[line width=1pt] (0,0) -- (0.25,-0.25); \draw[line width=1pt] (0,0) -- (0.25,-0.25);
\addplot[mark=*] coordinates {(0.25, -0.25)}; \addplot[mark=*] coordinates {(0.25, -0.25)};
@ -225,48 +221,47 @@
\begin{tikzpicture} \begin{tikzpicture}
\begin{axis}[ \begin{axis}[
xmin=-1,xmax=1, xmin=-1,xmax=1,
ymin=-1,ymax=1, ymin=-1,ymax=1,
width=\figlength, width=\figlength,
height=\figlength, height=\figlength,
ticks=none, ticks=none,
view={0}{90}, view={0}{90},
title={$t=1$}, title={$t=1$},
title style={yshift=-1mm}, title style={yshift=-1mm},
% xlabel={$x_1$}, % xlabel={$x_1$},
% ylabel={$x_2$}, % ylabel={$x_2$},
] ]
\addplot3[point meta=\thisrow{Hmag}, \addplot3[point meta=\thisrow{Hmag},
point meta min=0, point meta min=0,
point meta max=2.5, point meta max=2.5,
quiver={u=\thisrow{H1}, quiver={u=\thisrow{H1},
v=\thisrow{H2}, v=\thisrow{H2},
scale arrows=.25, scale arrows=.25,
every arrow/.append style={% every arrow/.append style={%
line width=.5 line width=.5
+\pgfplotspointmetatransformed/1000, +\pgfplotspointmetatransformed/1000,
-{Latex[length=0pt 5,width=0pt 3]} -{Latex[length=0pt 5,width=0pt 3]}
}, },
}, },
quiver/colored = {mapped color}, quiver/colored = {mapped color},
-stealth, -stealth,
] ]
table[col sep=comma, discard if not={t}{1.0}] table[col sep=comma, discard if not={t}{1.0}]
{\res/H.csv}; {\res/H.csv};
\draw[line width=1pt] (0,0) -- (0.5,-0.5); \draw[line width=1pt] (0,0) -- (0.5,-0.5);
\addplot[mark=*] coordinates {(0.5,-0.5)} node[below right] \addplot[mark=*] coordinates {(0.5,-0.5)} node[below right]
{$\bm{x}_1$}; {$\bm{x}_1$};
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\caption{Visualization of ``snapshots'' of $H$ (e.g., $F, G$) as \caption{Visualization of ``snapshots'' of $H$ (e.g., $F, G$) as
vector fields} vector fields}
\end{figure} \end{figure}
\end{minipage} \end{minipage}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Path Tracing} \frametitle{Path Tracing}
@ -275,10 +270,13 @@
\begin{itemize} \begin{itemize}
\item Reminder: We are trying to trace the solution curve \item Reminder: We are trying to trace the solution curve
$H(\bm{x}, t) = \bm{0}$ from $t = 0$ to $t = 1$ $H(\bm{x}, t) = \bm{0}$ from $t = 0$ to $t = 1$
\item We can express the solution curve as a system of differential equations \citereference{CL15}: \item We can express the solution curve as a system of
differential equations \citereference{CL15}:
\begin{align*} \begin{align*}
DH(\bm{y}(s))\cdot \dot{\bm{y}}(s) &= 0 \\ DH(\bm{y}(s))\cdot \dot{\bm{y}}(s) &= 0 \\
\text{det}\left(\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)\end{array}\right) &= \sigma_0 \\ \text{det}\left(
\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)
\end{array}\right) &= \sigma_0 \\
\lVert \dot{\bm{y}}(s) \rVert &= 1 \\ \lVert \dot{\bm{y}}(s) \rVert &= 1 \\
\bm{y}(0) &= (\bm{x}_0, 0) \bm{y}(0) &= (\bm{x}_0, 0)
,% ,%
@ -290,21 +288,23 @@
predictor-corrector scheme, e.g., Euler's predictor and Newton's predictor-corrector scheme, e.g., Euler's predictor and Newton's
corrector \citereference{CL15}: corrector \citereference{CL15}:
\begin{align*} \begin{align*}
\hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot {\bm{y}}(\bm{s})\\ \hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot
\bm{y} &= \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm} \mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} - (DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}}) {\bm{y}}(\bm{s})\\
\bm{y} &= \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm}
\mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} -
(DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}})
.% .%
\end{align*} \end{align*}
\end{itemize} \end{itemize}
\addreferences \addreferences
{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
continuation method for solving systems of nonlinear and continuation method for solving systems of nonlinear and
polynomial equations}. Communications in Information and polynomial equations}. Communications in Information and
Systems 15.2 (2015): 119-307.} Systems 15.2 (2015): 119-307.}
\stopreferences \stopreferences
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Channel Decoding and Polynomial Equations} \frametitle{Channel Decoding and Polynomial Equations}
@ -314,36 +314,49 @@
\item To describe the decoding problem we can use the code constraint \item To describe the decoding problem we can use the code constraint
polynomial \citereference{WT22} polynomial \citereference{WT22}
\begin{align*} \begin{align*}
%h(\bm{x}) = \underbrace{\sum_{i=1}^{n}\left(1-x_i^2\right)^2}_{\text{Bipolar constraint}} + \underbrace{\sum_{j=1}^{m}\left(1 - \left(\prod_{i\in A(j)}x_i\right)\right)^2}_{\text{Parity constraint}} %h(\bm{x}) =
h(\bm{x}) = \sum_{i=1}^{n}\left(1-x_i^2\right)^2 + \sum_{j=1}^{m}\left(1 - \left(\prod_{i\in A(j)}x_i\right)\right)^2 % \underbrace{\sum_{i=1}^{n}\left(1-x_i^2\right)^2}_{\text{Bipolar
% constraint}} + \underbrace{\sum_{j=1}^{m}\left(1 -
% \left(\prod_{i\in
% A(j)}x_i\right)\right)^2}_{\text{Parity constraint}}
h(\bm{x}) = \sum_{i=1}^{n}\left(1-x_i^2\right)^2 +
\sum_{j=1}^{m}\left(1 - \left(\prod_{i\in
A(j)}x_i\right)\right)^2
.% .%
\end{align*} \end{align*}
where $A(j) = \left\{i \in [1:n]: \bm{H}_{j,i} = 1\right\}, \hspace{3mm} j\in [1:m]$ where $A(j) = \left\{i \in [1:n]: \bm{H}_{j,i} = 1\right\},
\hspace{3mm} j\in [1:m]$
represents the set of variables involved in parity check $j$. represents the set of variables involved in parity check $j$.
\item In a similar vein, we can define a polynomial system whose zeros \item In a similar vein, we can define a polynomial system whose zeros
correspond to codewords as correspond to codewords as
\begin{align*} \begin{align*}
F(\bm{x}) = \left[\begin{array}{c}1 - x_1^2 \\ \vdots\\ 1 - x_n^2 \\ 1 - \prod_{i \in A(1)}x_i \\ \vdots\\ 1 - \prod_{i \in A(m)}x_i\end{array}\right] \overset{!}{=} \bm{0} F(\bm{x}) = \left[
\begin{array}{c}1 - x_1^2 \\ \vdots\\ 1 - x_n^2 \\ 1 -
\prod_{i \in A(1)}x_i \\ \vdots\\ 1 -
\prod_{i \in A(m)}x_i
\end{array}\right] \overset{!}{=} \bm{0}
.% .%
\end{align*} \end{align*}
\end{itemize} \end{itemize}
\addreferences \addreferences
{WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC Codes. IEICE Transactions on Fundamentals of Electronics, Communi- {WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC
cations and Computer Sciences advpub (2022), 2022TAP0002.} Codes. IEICE Transactions on Fundamentals of Electronics, Communi-
cations and Computer Sciences advpub (2022), 2022TAP0002.}
\stopreferences \stopreferences
\end{frame} \end{frame}
\begin{frame}[fragile] \begin{frame}[fragile]
\frametitle{Defining Homotopies for Channel Codes} \frametitle{Defining Homotopies for Channel Codes}
\begin{itemize} \begin{itemize}
\item Problem: Homotopy continuation algorithms / existing frameworks \item Problem: Homotopy continuation algorithms / existing frameworks
only really support square systems, i.e., \# equations = \# variables. The only really support square systems, i.e., \# equations =
\# variables. The
system $F(\bm{x}) = \bm{0}$ we previously considered is overdefined system $F(\bm{x}) = \bm{0}$ we previously considered is overdefined
\item \textit{Gröbner bases} allow us to ``[...] transform F into \item \textit{Gröbner bases} allow us to ``[...] transform F into
another set G of polynomials [...] such that F and G are equivalent'' another set G of polynomials [...] such that F and G are
equivalent''
\citereference{B01}, i.e., they have the same zeros \citereference{B01}, i.e., they have the same zeros
\item Limited tests indicate that, for the systems we are interested in, \item Limited tests indicate that, for the systems we are interested in,
finding a Gröbner basis yields a square system finding a Gröbner basis yields a square system
@ -352,8 +365,11 @@
\begin{minipage}{0.45\textwidth} \begin{minipage}{0.45\textwidth}
\begin{align*} \begin{align*}
\overbrace{\bm{H}}^{\text{Parity chek matrix}} &= \left[ \begin{array}{cc} 1 & 1 \end{array}\right] \\ \overbrace{\bm{H}}^{\text{Parity check matrix}} &= \left[
F(\bm{x}) &= \left[ \begin{array}{c} \begin{array}{cc} 1 & 1
\end{array}\right] \\
F(\bm{x}) &= \left[
\begin{array}{c}
1 - x_1^2 \\ 1 - x_1^2 \\
1 - x_2^2\\ 1 - x_2^2\\
1 - x_1x_2 1 - x_1x_2
@ -367,15 +383,17 @@
\end{minipage}% \end{minipage}%
\begin{minipage}{0.45\textwidth} \begin{minipage}{0.45\textwidth}
\begin{align*} \begin{align*}
\tilde{F}(\bm{x}) &= \left[ \begin{array}{c} \tilde{F}(\bm{x}) &= \left[
\begin{array}{c}
x_1 - x_2 \\ x_1 - x_2 \\
x_2^2 - 1 x_2^2 - 1
\end{array}\right] \\ \end{array}\right] \\
G(\bm{x}) &= \left[\begin{array}{c} G(\bm{x}) &= \left[
x_1\\ \begin{array}{c}
x_2 x_1 - y_1\\
x_2 - y_2
\end{array}\right]\\ \end{array}\right]\\
H(\bm{x}, t) &= (t-1)G(\bm{x}) + tF(\bm{x}) H(\bm{x}, t) &= (1-t)G(\bm{x}) + tF(\bm{x})
\end{align*} \end{align*}
\end{minipage} \\ \end{minipage} \\
@ -384,11 +402,10 @@
\addreferences \addreferences
{B01}{Buchberger, Bruno. "Gröbner bases: A short introduction for systems {B01}{Buchberger, Bruno. "Gröbner bases: A short introduction for systems
theorists." International Conference on Computer Aided Systems Theory. theorists." International Conference on Computer Aided Systems Theory.
Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.} Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.}
\stopreferences \stopreferences
\end{frame} \end{frame}
\begin{frame}[fragile] \begin{frame}[fragile]
\frametitle{Path Tracker Implementation (Pseudo Code)} \frametitle{Path Tracker Implementation (Pseudo Code)}
@ -409,17 +426,17 @@
baselinestretch=1, % Line spacing baselinestretch=1, % Line spacing
breaklines, % Allow line breaking breaklines, % Allow line breaking
tabsize=4, % Tab width tabsize=4, % Tab width
]{python} ]{c}
func perform_prediction_step(y) {...} func perform_prediction_step(y, step_size) {...}
func perform_correction_step(y, step_size) {...} func perform_correction_step(y) {...}
func perform_step(y0) { func perform_step(y0) {
for i in range(max_retries): for i in range(max_retries):
step_size = step_size / 2 step_size = step_size / 2
y = perform_prediction_step(y0) y = perform_prediction_step(y0, step_size)
for k in range(max_corrector_iterations): for k in range(max_corrector_iterations):
y = perform_correction_step(y, step_size) y = perform_correction_step(y)
if (corrector converged) break if (corrector converged) break
if (corrector converged) break if (corrector converged) break
@ -429,15 +446,14 @@ func perform_step(y0) {
\vspace{1mm} \vspace{1mm}
\addreferences \addreferences
{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
continuation method for solving systems of nonlinear and continuation method for solving systems of nonlinear and
polynomial equations}. Communications in Information and polynomial equations}. Communications in Information and
Systems 15.2 (2015): 119-307.} Systems 15.2 (2015): 119-307.}
\stopreferences \stopreferences
\end{frame} \end{frame}
\begin{frame}[fragile] \begin{frame}[fragile]
\frametitle{Decoding Algorithm Implementation (Pseudo Code)} \frametitle{Decoding Algorithm Implementation (Pseudo Code)}
@ -453,7 +469,7 @@ func perform_step(y0) {
baselinestretch=1, % Line spacing baselinestretch=1, % Line spacing
breaklines, % Allow line breaking breaklines, % Allow line breaking
tabsize=4, % Tab width tabsize=4, % Tab width
]{python} ]{c}
func decode(y) { func decode(y) {
for i in range(max_iterations): for i in range(max_iterations):
y = perform_step(y) y = perform_step(y)
@ -466,87 +482,123 @@ func decode(y) {
\end{minted} \end{minted}
\end{frame} \end{frame}
\begin{frame}[fragile]
\begin{frame}
\frametitle{Simulation results} \frametitle{Simulation results}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\begin{subfigure}{0.5\textwidth} \begin{subfigure}[c]{0.48\textwidth}
\centering \centering
\vskip 0pt
\begin{tikzpicture} \begin{tikzpicture}
\begin{axis}[ \begin{axis}[
domain=-5:5, width=\textwidth,
width=\textwidth, height=0.75\textwidth,
height=0.75\textwidth, ylabel={FER (-\,-\,-), BER (---)},
] ymode=log,
\addplot+[mark=none, line width=1pt] legend,
{x^2}; xlabel = {$Eb/N0$ (dB)},
\addplot+[mark=none, line width=1pt] legend pos= south west,
{x^3}; xmax=6
\addplot+[mark=none, line width=1pt] ]
{x^4}; % \addplot+[scol0, densely dashed, mark=none, line
% width=1pt, forget plot]
% table[col sep=comma, x=SNR, y=FER]
% {\res/bch_31_26_hc.csv};
% \addplot+[scol0, mark=none, line width=1pt]
% table[col sep=comma, x=SNR, y=BER]
% {\res/bch_31_26_hc.csv};
% \addlegendentry{Homotopy continuation}
%
\addplot+[scol0, densely dashed, mark=none, line width=1pt,
forget plot]
table[col sep=comma, x=SNR, y=FER]
{\res/bch_31_26_hc.csv};
\addplot+[scol0, mark=none, line width=1pt]
table[col sep=comma, x=SNR, y=BER]
{\res/bch_31_26_hc.csv};
\addlegendentry{Homotopy continuation}
\addplot+[scol2, densely dashed, mark=none, line width=1pt,
forget plot]
table[col sep=comma, x=SNR, y=FER, discard if
not={gamma}{0.05}]
{\res/bch_31_26_proximal.csv};
\addplot+[scol2, mark=none, line width=1pt]
table[col sep=comma, x=SNR, y=BER, discard if
not={gamma}{0.05}]
{\res/bch_31_26_proximal.csv};
\addlegendentry{Proximal decoding}
% \addplot+[scol2, densely dashed, mark=none, line
% width=1pt, forget plot]
% table[col sep=comma, x=SNR, y=FER]
% {\res/bch_31_26_ml.csv};
% \addplot+[scol2, mark=none, line width=1pt]
% table[col sep=comma, x=SNR, y=BER]
% {\res/bch_31_26_ml.csv};
\addlegendentry{ML}
% \addplot+[mark=none, line width=1pt]
% table[col sep=comma, x=SNR, y=DFR]
% {\res/bch_31_11.csv};
% \addlegendentry{DFR}
\end{axis} \end{axis}
\end{tikzpicture} \end{tikzpicture}
\caption{BCH(31,26) Code}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}{0.5\textwidth} \begin{subfigure}[c]{0.48\textwidth}
\centering \centering
\begin{tikzpicture} \vspace*{-20mm}
\begin{axis}[
domain=-5:5, \begin{tabular}{rl|ccccc}
width=\textwidth, Parameter & Value \\ \hline
height=0.75\textwidth, $n_\text{iter}$ & for homotopy continuation & 20 \\
] $n_\text{iter}$ & for Newton corrector & 5 \\
\addplot+[mark=none, line width=1pt] $\delta_\text{max}$ & for Newton corrector & 0.01 \\
{x}; $\Delta s$ & for Euler predictor & 0.05 \\
\addplot+[mark=none, line width=1pt] $n_\text{retries}$ & for Euler predictor & 5
{2*x}; \end{tabular}
\addplot+[mark=none, line width=1pt]
{3*x}; \bigskip
\end{axis}
\end{tikzpicture} \begin{itemize}
\item Newton homotopy:
\begin{align*}
G(\bm{x}) &= F(x) - F(\bm{y}) \\
H(\bm{x}) &= (1-t)G(\bm{x}) - tF(\bm{x})\\
&= F(\bm{x}) - (1 - t) F(\bm{y})
\end{align*}
\end{itemize}
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
% \begin{figure}[H]
% \centering
%
% \begin{tikzpicture}
% \begin{axis}[
% width=\textwidth,
% height=0.75\textwidth,
% ]
% \addplot+[mark=none, line width=1pt]
% table[col sep=comma, x=x, y=y]
% {};
% \end{axis}
% \end{tikzpicture}
% \end{figure}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Next steps} \frametitle{Next steps}
\begin{itemize} \begin{itemize}
\item Simulations for other codes
\item Thorough investigation into parameter choice \item Thorough investigation into parameter choice
\item Find more mathematical background / guarantees \begin{itemize} \item Find more mathematical background / guarantees
\item How do we have to choose $\sigma_0$? \begin{itemize}
\item Guarantees for convergence? (i.e., what is the cause for \item How do we have to choose $\sigma_0$?
decoding failures?) \item Guarantees for convergence? (i.e., what is the cause for
\item When do we actually get square systems using the Gröbner basis? decoding failures?)
\end{itemize} \item When do we actually get square systems using
\item Other ideas: \begin{itemize} the Gröbner basis?
\item Generate more candidates by moving further along the \end{itemize}
solution curve (if this is possible) and then performing \item Other ideas:
choosing from this list \begin{itemize}
\end{itemize} \item Generate more candidates by moving further along the
solution curve (if this is possible) and then performing
choosing from this list
\end{itemize}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\end{document} \end{document}