Fix homotopy definition; Fix simulation results
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4
Makefile
4
Makefile
@ -4,7 +4,7 @@ all:
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latexmk src/2024-12-03/presentation.tex
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mv build/presentation.pdf build/presentation_2024-12-03.pdf
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latexmk src/2025-01-07/presentation.tex
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mv build/presentation.pdf build/presentation_2025-01-07.pdf
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latexmk src/2025-03-28/presentation.tex
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mv build/presentation.pdf build/presentation_2025-03-28.pdf
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clean:
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rm -rf build
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@ -9,7 +9,6 @@
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\pgfplotsset{compat=newest}
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\usepgfplotslibrary{fillbetween}
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\usepackage{listings}
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\usepackage{subcaption}
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\usepackage{bbm}
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@ -22,27 +21,23 @@
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%\definecolor{gruvbox-bg}{HTML}{282828}
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\definecolor{gruvbox-bg}{HTML}{f2e5bc}
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%
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%
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% Custom commands
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%
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%
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\input{lib/latex-common/common.tex}
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\pgfplotsset{colorscheme/rocket}
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\newcommand{\res}{src/2025-03-28/res}
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%
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%
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% CEL Template
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%
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%
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\newcommand{\templates}{lib/cel-template}
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\input{\templates/packages.tex}
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@ -62,44 +57,38 @@
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% \AtBeginSubsubsection[]{}
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% \AtBeginSubsection[]{}
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%
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%
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% Set up document
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%
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%
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\title{HiWi Notes: Minimization of the Code Constraint Polynomial using
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Homotopy Continuation Methods}
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\subtitle{\small 07.01.2025}
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Homotopy Continuation Methods}
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\subtitle{\small 28.03.2025}
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\author{\vspace{1.5mm} Andreas Tsouchlos}
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\date{ }
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\institute{Karlsruhe Institute of Technology (KIT),
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\\ Communications Engineering Lab (CEL) }
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\\ Communications Engineering Lab (CEL) }
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\tikzstyle{every node}=[font=\small]
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\captionsetup[sub]{font=small}
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%
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%
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% Document body
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%
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%
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\begin{document}
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\begin{frame}[plain]
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\maketitle
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\maketitle
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\end{frame}
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\newcommand{\largecitereference}[1]
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{\textcolor{kit-green100}{ \large \textbf{{[#1]}} }}
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{\textcolor{kit-green100}{ \large \textbf{{[#1]}} }}
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%
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% - The decoding problem as the search for a solution of polynomial system
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@ -108,7 +97,6 @@
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% - Implementation
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%
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% TODO: Check example homotopy
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% TODO: Add the name of the homotopy construction (t-1)G + tF and note that it
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% is not the only possibility
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@ -117,27 +105,35 @@
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\begin{minipage}[c]{0.65\textwidth}
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\begin{itemize}
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\item Goal: Solve system of equations $F(\bm{x}) = \bm{0}, \hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$
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\item Problem: Depending on $F$, solving this directly may be difficult
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\item Goal: Solve system of equations $F(\bm{x}) = \bm{0},
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\hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$
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\item Problem: Depending on $F$, solving this directly
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may be difficult
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\item Solution: Define \emph{homotopy function} $H(\bm{x}, t)$ with
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\begin{gather*}
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H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x}, 1) = F(\bm{x})
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,\end{gather*}
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i.e., a deformation between two systems $G(\bm{x})$ and $F(\bm{x})$
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H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x},
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1) = F(\bm{x})
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,
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\end{gather*}
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i.e., a deformation between two systems $G(\bm{x})$
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and $F(\bm{x})$
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(where the zeros of $G$ can be easily obtained); E.g.,
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\begin{gather*}
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H(\bm{x}, t) = (t-1)G(\bm{x}) + tF(\bm{x})
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.\end{gather*}
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Then, compute $(\bm{x}_0, 0)$ such that $G(\bm{x}_0) = \bm{0}$ and trace path to $(\bm{x}_1, 1)$ with $F(\bm{x}_1) = \bm{0}$
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.
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\end{gather*}
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Then, compute $(\bm{x}_0, 0)$ such that $G(\bm{x}_0) =
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\bm{0}$ and trace path to $(\bm{x}_1, 1)$ with
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$F(\bm{x}_1) = \bm{0}$
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\end{itemize}
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\vspace{5mm}
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\addreferences
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{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
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{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
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continuation method for solving systems of nonlinear and
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polynomial equations}. Communications in Information and
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Systems 15.2 (2015): 119-307.}
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polynomial equations}. Communications in Information and
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Systems 15.2 (2015): 119-307.}
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\stopreferences
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\end{minipage}%
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\hfill
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@ -151,70 +147,70 @@
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\begin{tikzpicture}
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\begin{axis}[
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xmin=-1,xmax=1,
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ymin=-1,ymax=1,
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width=\figlength,
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height=\figlength,
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ticks=none,
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view={0}{90},
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title={$t=0$},
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title style={yshift=-1mm},
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% xlabel={$x_1$},
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% ylabel={$x_2$},
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xmin=-1,xmax=1,
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ymin=-1,ymax=1,
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width=\figlength,
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height=\figlength,
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ticks=none,
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view={0}{90},
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title={$t=0$},
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title style={yshift=-1mm},
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% xlabel={$x_1$},
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% ylabel={$x_2$},
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]
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\addplot3[point meta=\thisrow{Hmag},
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point meta min=0,
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point meta max=2.5,
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quiver={u=\thisrow{H1},
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v=\thisrow{H2},
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scale arrows=.25,
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every arrow/.append style={%
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line width=.5
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+\pgfplotspointmetatransformed/1000,
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-{Latex[length=0pt 5,width=0pt 3]}
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},
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},
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quiver/colored = {mapped color},
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-stealth,
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]
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table[col sep=comma, discard if not={t}{0.0}]
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{\res/H.csv};
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point meta min=0,
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point meta max=2.5,
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quiver={u=\thisrow{H1},
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v=\thisrow{H2},
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scale arrows=.25,
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every arrow/.append style={%
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line width=.5
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+\pgfplotspointmetatransformed/1000,
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-{Latex[length=0pt 5,width=0pt 3]}
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},
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},
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quiver/colored = {mapped color},
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-stealth,
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]
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table[col sep=comma, discard if not={t}{0.0}]
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{\res/H.csv};
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\addplot[mark=*] coordinates {(0,0)} node[above]
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{$\bm{x}_0$};
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{$\bm{x}_0$};
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\end{axis}
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\end{tikzpicture}
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\begin{tikzpicture}
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\begin{axis}[
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xmin=-1,xmax=1,
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ymin=-1,ymax=1,
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width=\figlength,
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height=\figlength,
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ticks=none,
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view={0}{90},
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title={$t=0.5$},
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title style={yshift=-1mm},
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% xlabel={$x_1$},
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% ylabel={$x_2$},
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xmin=-1,xmax=1,
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ymin=-1,ymax=1,
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width=\figlength,
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height=\figlength,
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ticks=none,
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view={0}{90},
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title={$t=0.5$},
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title style={yshift=-1mm},
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% xlabel={$x_1$},
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% ylabel={$x_2$},
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]
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\addplot3[point meta=\thisrow{Hmag},
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point meta min=0,
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point meta max=2.5,
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quiver={u=\thisrow{H1},
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v=\thisrow{H2},
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scale arrows=.25,
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every arrow/.append style={%
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line width=.5
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+\pgfplotspointmetatransformed/1000,
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-{Latex[length=0pt 5,width=0pt 3]}
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},
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},
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quiver/colored = {mapped color},
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-stealth,
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]
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table[col sep=comma, discard if not={t}{0.5}]
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{\res/H.csv};
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point meta min=0,
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point meta max=2.5,
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quiver={u=\thisrow{H1},
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v=\thisrow{H2},
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scale arrows=.25,
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every arrow/.append style={%
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line width=.5
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+\pgfplotspointmetatransformed/1000,
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-{Latex[length=0pt 5,width=0pt 3]}
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},
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},
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quiver/colored = {mapped color},
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-stealth,
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]
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table[col sep=comma, discard if not={t}{0.5}]
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{\res/H.csv};
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\draw[line width=1pt] (0,0) -- (0.25,-0.25);
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\addplot[mark=*] coordinates {(0.25, -0.25)};
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@ -225,48 +221,47 @@
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\begin{tikzpicture}
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\begin{axis}[
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xmin=-1,xmax=1,
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ymin=-1,ymax=1,
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width=\figlength,
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height=\figlength,
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ticks=none,
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view={0}{90},
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title={$t=1$},
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title style={yshift=-1mm},
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% xlabel={$x_1$},
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% ylabel={$x_2$},
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xmin=-1,xmax=1,
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ymin=-1,ymax=1,
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width=\figlength,
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height=\figlength,
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ticks=none,
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view={0}{90},
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title={$t=1$},
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title style={yshift=-1mm},
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% xlabel={$x_1$},
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% ylabel={$x_2$},
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]
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\addplot3[point meta=\thisrow{Hmag},
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point meta min=0,
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point meta max=2.5,
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quiver={u=\thisrow{H1},
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v=\thisrow{H2},
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scale arrows=.25,
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every arrow/.append style={%
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line width=.5
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+\pgfplotspointmetatransformed/1000,
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-{Latex[length=0pt 5,width=0pt 3]}
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},
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},
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quiver/colored = {mapped color},
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-stealth,
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]
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table[col sep=comma, discard if not={t}{1.0}]
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{\res/H.csv};
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point meta min=0,
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point meta max=2.5,
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quiver={u=\thisrow{H1},
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v=\thisrow{H2},
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scale arrows=.25,
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every arrow/.append style={%
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line width=.5
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+\pgfplotspointmetatransformed/1000,
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-{Latex[length=0pt 5,width=0pt 3]}
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},
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},
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quiver/colored = {mapped color},
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-stealth,
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]
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table[col sep=comma, discard if not={t}{1.0}]
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{\res/H.csv};
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\draw[line width=1pt] (0,0) -- (0.5,-0.5);
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\addplot[mark=*] coordinates {(0.5,-0.5)} node[below right]
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{$\bm{x}_1$};
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{$\bm{x}_1$};
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\end{axis}
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\end{tikzpicture}
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\caption{Visualization of ``snapshots'' of $H$ (e.g., $F, G$) as
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vector fields}
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vector fields}
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\end{figure}
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\end{minipage}
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\end{frame}
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\begin{frame}
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\frametitle{Path Tracing}
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@ -275,10 +270,13 @@
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\begin{itemize}
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\item Reminder: We are trying to trace the solution curve
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$H(\bm{x}, t) = \bm{0}$ from $t = 0$ to $t = 1$
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\item We can express the solution curve as a system of differential equations \citereference{CL15}:
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\item We can express the solution curve as a system of
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differential equations \citereference{CL15}:
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\begin{align*}
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DH(\bm{y}(s))\cdot \dot{\bm{y}}(s) &= 0 \\
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\text{det}\left(\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)\end{array}\right) &= \sigma_0 \\
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\text{det}\left(
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\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)
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\end{array}\right) &= \sigma_0 \\
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\lVert \dot{\bm{y}}(s) \rVert &= 1 \\
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\bm{y}(0) &= (\bm{x}_0, 0)
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,%
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@ -290,21 +288,23 @@
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predictor-corrector scheme, e.g., Euler's predictor and Newton's
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corrector \citereference{CL15}:
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\begin{align*}
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\hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot {\bm{y}}(\bm{s})\\
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\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}})
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\hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot
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{\bm{y}}(\bm{s})\\
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\bm{y} &= \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm}
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\mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} -
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(DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}})
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.%
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\end{align*}
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\end{itemize}
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\addreferences
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{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
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{CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy
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continuation method for solving systems of nonlinear and
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polynomial equations}. Communications in Information and
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Systems 15.2 (2015): 119-307.}
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polynomial equations}. Communications in Information and
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Systems 15.2 (2015): 119-307.}
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\stopreferences
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\end{frame}
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\begin{frame}
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\frametitle{Channel Decoding and Polynomial Equations}
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@ -314,36 +314,49 @@
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\item To describe the decoding problem we can use the code constraint
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polynomial \citereference{WT22}
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\begin{align*}
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%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}}
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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
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%h(\bm{x}) =
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% \underbrace{\sum_{i=1}^{n}\left(1-x_i^2\right)^2}_{\text{Bipolar
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% constraint}} + \underbrace{\sum_{j=1}^{m}\left(1 -
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% \left(\prod_{i\in
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% A(j)}x_i\right)\right)^2}_{\text{Parity constraint}}
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h(\bm{x}) = \sum_{i=1}^{n}\left(1-x_i^2\right)^2 +
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\sum_{j=1}^{m}\left(1 - \left(\prod_{i\in
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A(j)}x_i\right)\right)^2
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.%
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\end{align*}
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where $A(j) = \left\{i \in [1:n]: \bm{H}_{j,i} = 1\right\}, \hspace{3mm} j\in [1:m]$
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where $A(j) = \left\{i \in [1:n]: \bm{H}_{j,i} = 1\right\},
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\hspace{3mm} j\in [1:m]$
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represents the set of variables involved in parity check $j$.
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\item In a similar vein, we can define a polynomial system whose zeros
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correspond to codewords as
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\begin{align*}
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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}
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F(\bm{x}) = \left[
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\begin{array}{c}1 - x_1^2 \\ \vdots\\ 1 - x_n^2 \\ 1 -
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\prod_{i \in A(1)}x_i \\ \vdots\\ 1 -
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\prod_{i \in A(m)}x_i
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\end{array}\right] \overset{!}{=} \bm{0}
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.%
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\end{align*}
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\end{itemize}
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\addreferences
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{WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC Codes. IEICE Transactions on Fundamentals of Electronics, Communi-
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cations and Computer Sciences advpub (2022), 2022TAP0002.}
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{WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC
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Codes. IEICE Transactions on Fundamentals of Electronics, Communi-
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cations and Computer Sciences advpub (2022), 2022TAP0002.}
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\stopreferences
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Defining Homotopies for Channel Codes}
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\begin{itemize}
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\item Problem: Homotopy continuation algorithms / existing frameworks
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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
|
||||
\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
|
||||
\item Limited tests indicate that, for the systems we are interested in,
|
||||
finding a Gröbner basis yields a square system
|
||||
@ -352,8 +365,11 @@
|
||||
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{align*}
|
||||
\overbrace{\bm{H}}^{\text{Parity chek matrix}} &= \left[ \begin{array}{cc} 1 & 1 \end{array}\right] \\
|
||||
F(\bm{x}) &= \left[ \begin{array}{c}
|
||||
\overbrace{\bm{H}}^{\text{Parity check matrix}} &= \left[
|
||||
\begin{array}{cc} 1 & 1
|
||||
\end{array}\right] \\
|
||||
F(\bm{x}) &= \left[
|
||||
\begin{array}{c}
|
||||
1 - x_1^2 \\
|
||||
1 - x_2^2\\
|
||||
1 - x_1x_2
|
||||
@ -367,15 +383,17 @@
|
||||
\end{minipage}%
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{align*}
|
||||
\tilde{F}(\bm{x}) &= \left[ \begin{array}{c}
|
||||
\tilde{F}(\bm{x}) &= \left[
|
||||
\begin{array}{c}
|
||||
x_1 - x_2 \\
|
||||
x_2^2 - 1
|
||||
\end{array}\right] \\
|
||||
G(\bm{x}) &= \left[\begin{array}{c}
|
||||
x_1\\
|
||||
x_2
|
||||
G(\bm{x}) &= \left[
|
||||
\begin{array}{c}
|
||||
x_1 - y_1\\
|
||||
x_2 - y_2
|
||||
\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{minipage} \\
|
||||
|
||||
@ -384,11 +402,10 @@
|
||||
\addreferences
|
||||
{B01}{Buchberger, Bruno. "Gröbner bases: A short introduction for systems
|
||||
theorists." International Conference on Computer Aided Systems Theory.
|
||||
Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.}
|
||||
Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.}
|
||||
\stopreferences
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Path Tracker Implementation (Pseudo Code)}
|
||||
|
||||
@ -409,17 +426,17 @@
|
||||
baselinestretch=1, % Line spacing
|
||||
breaklines, % Allow line breaking
|
||||
tabsize=4, % Tab width
|
||||
]{python}
|
||||
func perform_prediction_step(y) {...}
|
||||
func perform_correction_step(y, step_size) {...}
|
||||
]{c}
|
||||
func perform_prediction_step(y, step_size) {...}
|
||||
func perform_correction_step(y) {...}
|
||||
func perform_step(y0) {
|
||||
for i in range(max_retries):
|
||||
step_size = step_size / 2
|
||||
|
||||
y = perform_prediction_step(y0)
|
||||
y = perform_prediction_step(y0, step_size)
|
||||
|
||||
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
|
||||
|
||||
@ -430,14 +447,13 @@ func perform_step(y0) {
|
||||
\vspace{1mm}
|
||||
|
||||
\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
|
||||
polynomial equations}. Communications in Information and
|
||||
Systems 15.2 (2015): 119-307.}
|
||||
polynomial equations}. Communications in Information and
|
||||
Systems 15.2 (2015): 119-307.}
|
||||
\stopreferences
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Decoding Algorithm Implementation (Pseudo Code)}
|
||||
|
||||
@ -453,7 +469,7 @@ func perform_step(y0) {
|
||||
baselinestretch=1, % Line spacing
|
||||
breaklines, % Allow line breaking
|
||||
tabsize=4, % Tab width
|
||||
]{python}
|
||||
]{c}
|
||||
func decode(y) {
|
||||
for i in range(max_iterations):
|
||||
y = perform_step(y)
|
||||
@ -466,87 +482,123 @@ func decode(y) {
|
||||
\end{minted}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\begin{frame}[fragile]
|
||||
\frametitle{Simulation results}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\begin{subfigure}[c]{0.48\textwidth}
|
||||
\centering
|
||||
|
||||
\vskip 0pt
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
domain=-5:5,
|
||||
width=\textwidth,
|
||||
height=0.75\textwidth,
|
||||
]
|
||||
\addplot+[mark=none, line width=1pt]
|
||||
{x^2};
|
||||
\addplot+[mark=none, line width=1pt]
|
||||
{x^3};
|
||||
\addplot+[mark=none, line width=1pt]
|
||||
{x^4};
|
||||
width=\textwidth,
|
||||
height=0.75\textwidth,
|
||||
ylabel={FER (-\,-\,-), BER (---)},
|
||||
ymode=log,
|
||||
legend,
|
||||
xlabel = {$Eb/N0$ (dB)},
|
||||
legend pos= south west,
|
||||
xmax=6
|
||||
]
|
||||
% \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{tikzpicture}
|
||||
|
||||
\caption{BCH(31,26) Code}
|
||||
\end{subfigure}%
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\begin{subfigure}[c]{0.48\textwidth}
|
||||
\centering
|
||||
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
domain=-5:5,
|
||||
width=\textwidth,
|
||||
height=0.75\textwidth,
|
||||
]
|
||||
\addplot+[mark=none, line width=1pt]
|
||||
{x};
|
||||
\addplot+[mark=none, line width=1pt]
|
||||
{2*x};
|
||||
\addplot+[mark=none, line width=1pt]
|
||||
{3*x};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\vspace*{-20mm}
|
||||
|
||||
\begin{tabular}{rl|ccccc}
|
||||
Parameter & Value \\ \hline
|
||||
$n_\text{iter}$ & for homotopy continuation & 20 \\
|
||||
$n_\text{iter}$ & for Newton corrector & 5 \\
|
||||
$\delta_\text{max}$ & for Newton corrector & 0.01 \\
|
||||
$\Delta s$ & for Euler predictor & 0.05 \\
|
||||
$n_\text{retries}$ & for Euler predictor & 5
|
||||
\end{tabular}
|
||||
|
||||
\bigskip
|
||||
|
||||
\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{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}
|
||||
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Next steps}
|
||||
|
||||
\begin{itemize}
|
||||
\item Simulations for other codes
|
||||
\item Thorough investigation into parameter choice
|
||||
\item Find more mathematical background / guarantees \begin{itemize}
|
||||
\item How do we have to choose $\sigma_0$?
|
||||
\item Guarantees for convergence? (i.e., what is the cause for
|
||||
decoding failures?)
|
||||
\item When do we actually get square systems using the Gröbner basis?
|
||||
\end{itemize}
|
||||
\item Other ideas: \begin{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}
|
||||
\item Find more mathematical background / guarantees
|
||||
\begin{itemize}
|
||||
\item How do we have to choose $\sigma_0$?
|
||||
\item Guarantees for convergence? (i.e., what is the cause for
|
||||
decoding failures?)
|
||||
\item When do we actually get square systems using
|
||||
the Gröbner basis?
|
||||
\end{itemize}
|
||||
\item Other ideas:
|
||||
\begin{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{frame}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user