Fix typo

Makefile

@@ -3,5 +3,8 @@ all:
 
 	latexmk src/2024-12-03/presentation.tex
 	mv build/presentation.pdf build/presentation_2024-12-03.pdf
+
+	latexmk src/2025-03-28/presentation.tex
+	mv build/presentation.pdf build/presentation_2025-03-28.pdf
 clean:
 	rm -rf build

lib/lib

@@ -0,0 +1 @@
+/home/andreas/workspace/work/hiwi/hiwi-update-presentations/lib

src/2025-03-28/presentation.tex

@@ -0,0 +1,589 @@
+\documentclass[10pt, aspectratio=169, usenames, dvipsnames]{beamer}
+
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+\usetikzlibrary{spy, external, intersections}
+%\tikzexternalize[prefix=build/]
+
+\usepackage{pgfplots}
+\pgfplotsset{compat=newest}
+\usepgfplotslibrary{fillbetween}
+
+\usepackage{listings}
+\usepackage{subcaption}
+\usepackage{bbm}
+
+\usepackage{xcolor}
+%\usepackage[outputdir=build/]{minted}
+\usepackage{minted}
+\usemintedstyle{gruvbox-light}
+
+%\definecolor{gruvbox-bg}{HTML}{282828}
+\definecolor{gruvbox-bg}{HTML}{f2e5bc}
+
+%
+%
+% Custom commands
+%
+%
+
+\input{lib/latex-common/common.tex}
+\pgfplotsset{colorscheme/rocket}
+
+\newcommand{\res}{src/2025-03-28/res}
+
+%
+%
+% CEL Template
+%
+%
+
+\newcommand{\templates}{lib/cel-template}
+
+\input{\templates/packages.tex}
+\input{\templates/modifications.tex}
+\input{\templates/makros_own.tex}
+
+% % Change the way the overview is displayed
+% \AtBeginSection[]
+% {
+%     \begin{frame}[t]
+%         \frametitle{Overview}
+%     \tableofcontents[sectionstyle=show/shaded,
+%                      subsectionstyle=show/show/shaded,
+%                      subsubsectionstyle=hide]
+%     \end{frame}
+% }
+% \AtBeginSubsubsection[]{}
+% \AtBeginSubsection[]{}
+
+%
+%
+% Set up document
+%
+%
+
+\title{HiWi Notes: Minimization of the Code Constraint Polynomial using
+Homotopy Continuation Methods}
+\subtitle{\small 28.03.2025}
+\author{\vspace{1.5mm} Andreas Tsouchlos}
+\date{ }
+
+\institute{Karlsruhe Institute of Technology (KIT),
+\\ Communications Engineering Lab (CEL) }
+
+\tikzstyle{every node}=[font=\small]
+\captionsetup[sub]{font=small}
+
+%
+%
+% Document body
+%
+%
+
+\begin{document}
+
+\begin{frame}[plain]
+    \maketitle
+\end{frame}
+
+\newcommand{\largecitereference}[1]
+{\textcolor{kit-green100}{ \large \textbf{{[#1]}} }}
+
+%
+% - The decoding problem as the search for a solution of polynomial system
+% - Homotopy continuation
+% - The problem with overdefined systems and the Groebner basis
+% - Implementation
+%
+
+% TODO: Check example homotopy
+% TODO: Add the name of the homotopy construction (t-1)G + tF and note that it
+% is not the only possibility
+\begin{frame}
+    \frametitle{Basic Idea of Homotopy Continuation \largecitereference{CL15}}
+
+    \begin{minipage}[c]{0.65\textwidth}
+        \begin{itemize}
+            \item Goal: Solve system of equations $F(\bm{x}) = \bm{0},
+                \hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$
+            \item Problem: Depending on $F$, solving this directly
+                may be difficult
+            \item Solution: Define \emph{homotopy function} $H(\bm{x}, t)$ with
+                \begin{gather*}
+                    H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x},
+                    1) = 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.,
+                \begin{gather*}
+                    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{itemize}
+
+        \vspace{5mm}
+
+        \addreferences
+        {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.}
+        \stopreferences
+    \end{minipage}%
+    \hfill
+    \begin{minipage}[c]{0.3\textwidth}
+        \begin{figure}
+            \centering
+
+            \newcommand{\figlength}{0.8\textwidth}
+
+            \vspace*{-2mm}
+
+            \begin{tikzpicture}
+                \begin{axis}[
+                        xmin=-1,xmax=1,
+                        ymin=-1,ymax=1,
+                        width=\figlength,
+                        height=\figlength,
+                        ticks=none,
+                        view={0}{90},
+                        title={$t=0$},
+                        title style={yshift=-1mm},
+                        % xlabel={$x_1$},
+                        % ylabel={$x_2$},
+                    ]
+                    \addplot3[point meta=\thisrow{Hmag},
+                        point meta min=0,
+                        point meta max=2.5,
+                        quiver={u=\thisrow{H1},
+                            v=\thisrow{H2},
+                            scale arrows=.25,
+                            every arrow/.append style={%
+                                line width=.5
+                                +\pgfplotspointmetatransformed/1000,
+                                -{Latex[length=0pt 5,width=0pt 3]}
+                            },
+                        },
+                        quiver/colored = {mapped color},
+                        -stealth,
+                    ]
+                    table[col sep=comma, discard if not={t}{0.0}]
+                    {\res/H.csv};
+
+                    \addplot[mark=*] coordinates {(0,0)} node[above]
+                    {$\bm{x}_0$};
+
+                \end{axis}
+            \end{tikzpicture}
+            \begin{tikzpicture}
+                \begin{axis}[
+                        xmin=-1,xmax=1,
+                        ymin=-1,ymax=1,
+                        width=\figlength,
+                        height=\figlength,
+                        ticks=none,
+                        view={0}{90},
+                        title={$t=0.5$},
+                        title style={yshift=-1mm},
+                        % xlabel={$x_1$},
+                        % ylabel={$x_2$},
+                    ]
+                    \addplot3[point meta=\thisrow{Hmag},
+                        point meta min=0,
+                        point meta max=2.5,
+                        quiver={u=\thisrow{H1},
+                            v=\thisrow{H2},
+                            scale arrows=.25,
+                            every arrow/.append style={%
+                                line width=.5
+                                +\pgfplotspointmetatransformed/1000,
+                                -{Latex[length=0pt 5,width=0pt 3]}
+                            },
+                        },
+                        quiver/colored = {mapped color},
+                        -stealth,
+                    ]
+                    table[col sep=comma, discard if not={t}{0.5}]
+                    {\res/H.csv};
+
+                    \draw[line width=1pt] (0,0) -- (0.25,-0.25);
+                    \addplot[mark=*] coordinates {(0.25, -0.25)};
+                \end{axis}
+            \end{tikzpicture}
+
+            \vspace{2mm}
+
+            \begin{tikzpicture}
+                \begin{axis}[
+                        xmin=-1,xmax=1,
+                        ymin=-1,ymax=1,
+                        width=\figlength,
+                        height=\figlength,
+                        ticks=none,
+                        view={0}{90},
+                        title={$t=1$},
+                        title style={yshift=-1mm},
+                        % xlabel={$x_1$},
+                        % ylabel={$x_2$},
+                    ]
+                    \addplot3[point meta=\thisrow{Hmag},
+                        point meta min=0,
+                        point meta max=2.5,
+                        quiver={u=\thisrow{H1},
+                            v=\thisrow{H2},
+                            scale arrows=.25,
+                            every arrow/.append style={%
+                                line width=.5
+                                +\pgfplotspointmetatransformed/1000,
+                                -{Latex[length=0pt 5,width=0pt 3]}
+                            },
+                        },
+                        quiver/colored = {mapped color},
+                        -stealth,
+                    ]
+                    table[col sep=comma, discard if not={t}{1.0}]
+                    {\res/H.csv};
+
+                    \draw[line width=1pt] (0,0) -- (0.5,-0.5);
+                    \addplot[mark=*] coordinates {(0.5,-0.5)} node[below right]
+                    {$\bm{x}_1$};
+                \end{axis}
+            \end{tikzpicture}
+
+            \caption{Visualization of ``snapshots'' of $H$ (e.g., $F, G$) as
+            vector fields}
+        \end{figure}
+    \end{minipage}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Path Tracing}
+
+    \vspace{-4mm}
+
+    \begin{itemize}
+        \item Reminder: We are trying to trace the solution curve
+            $H(\bm{x}, t) = \bm{0}$ from $t = 0$ to $t = 1$
+        \item We can express the solution curve as a system of
+            differential equations \citereference{CL15}:
+            \begin{align*}
+                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 \\
+                \lVert \dot{\bm{y}}(s) \rVert &= 1 \\
+                \bm{y}(0) &= (\bm{x}_0, 0)
+                ,%
+            \end{align*}
+            where $DH(\bm{y})$ is the Jacobian of $H(\bm{y})$ and
+            $\sigma_0 \in \left\{\pm 1\right\}$ defines the direction along
+            which we move on the curve.
+        \item For numerical stability, it is beneficial to solve this using a
+            predictor-corrector scheme, e.g., Euler's predictor and Newton's
+            corrector \citereference{CL15}:
+            \begin{align*}
+                \hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot
+                {\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{itemize}
+
+    \addreferences
+    {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.}
+    \stopreferences
+\end{frame}
+
+\begin{frame}
+    \frametitle{Channel Decoding and Polynomial Equations}
+
+    \vspace{-5mm}
+
+    \begin{itemize}
+        \item To describe the decoding problem we can use the code constraint
+            polynomial \citereference{WT22}
+            \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}) = \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*}
+            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 $j$.
+        \item In a similar vein, we can define a polynomial system whose zeros
+            correspond to codewords as
+            \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}
+                .%
+            \end{align*}
+    \end{itemize}
+
+    \addreferences
+    {WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC
+        Codes. IEICE Transactions on Fundamentals of Electronics, Communi-
+    cations and Computer Sciences advpub (2022), 2022TAP0002.}
+    \stopreferences
+\end{frame}
+
+\begin{frame}[fragile]
+    \frametitle{Defining Homotopies for Channel Codes}
+
+    \begin{itemize}
+        \item Problem: Homotopy continuation algorithms / existing frameworks
+            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''
+            \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
+        \item Example:
+    \end{itemize}
+
+    \begin{minipage}{0.45\textwidth}
+        \begin{align*}
+            \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
+            \end{array}\right]
+        \end{align*}
+    \end{minipage}%
+    \begin{minipage}{0.1\textwidth}
+        \begin{tikzpicture}
+            \draw[-{Latex}] (0,0) -- (1,0);
+        \end{tikzpicture}
+    \end{minipage}%
+    \begin{minipage}{0.45\textwidth}
+        \begin{align*}
+            \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
+            \end{array}\right]\\
+            H(\bm{x}, t) &= (1-t)G(\bm{x}) + tF(\bm{x})
+        \end{align*}
+    \end{minipage} \\
+
+    \vspace{5mm}
+
+    \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.}
+    \stopreferences
+\end{frame}
+
+\begin{frame}[fragile]
+    \frametitle{Path Tracker Implementation (Pseudo Code)}
+
+    \vspace{-6mm}
+
+    \begin{itemize}
+        \item Perform a predictor step followed by multiple corrector steps
+        \item If the corrector fails to converge, adjust the predictor step
+            size and try again \citereference{CL15}
+    \end{itemize}
+
+    \vspace{1mm}
+
+    \begin{minted}[
+        bgcolor=gruvbox-bg,  % Background color
+        fontsize=\small,     % Adjust font size
+        framesep=5mm,        % Frame separation
+        baselinestretch=1,   % Line spacing
+        breaklines,          % Allow line breaking
+        tabsize=4,           % Tab width
+    ]{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, step_size)
+
+        for k in range(max_corrector_iterations):
+            y = perform_correction_step(y)
+            if (corrector converged) break
+        if (corrector converged) break
+
+    return y
+}
+    \end{minted}
+
+    \vspace{1mm}
+
+    \addreferences
+    {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.}
+    \stopreferences
+\end{frame}
+
+\begin{frame}[fragile]
+    \frametitle{Decoding Algorithm Implementation (Pseudo Code)}
+
+    \begin{itemize}
+        \item If the algorithm doesn't converge, we still return the last
+            estimate in the hopes that this will limit the BER
+    \end{itemize}
+
+    \begin{minted}[
+        bgcolor=gruvbox-bg,  % Background color
+        fontsize=\small,     % Adjust font size
+        framesep=5mm,        % Frame separation
+        baselinestretch=1,   % Line spacing
+        breaklines,          % Allow line breaking
+        tabsize=4,           % Tab width
+    ]{c}
+func decode(y) {
+    for i in range(max_iterations):
+        y = perform_step(y)
+
+        x_hat = hard_decision(y)
+        if (H @ x_hat == 0) return x_hat
+
+    return x_hat
+}
+    \end{minted}
+\end{frame}
+
+\begin{frame}[fragile]
+    \frametitle{Simulation results}
+
+    \vspace*{-5mm}
+
+    \begin{itemize}
+        \item Simulation using the all-zeros codeword
+        \item Newton homotopy:
+            \begin{gather*}
+                G(\bm{x}) = F(x) - F(\bm{y}) \hspace{5mm} \Rightarrow
+                \hspace{5mm} H(\bm{x}) = F(\bm{x}) - (1 - t) F(\bm{y})
+            \end{gather*}
+    \end{itemize}
+
+    \begin{figure}[H]
+        \centering
+
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+
+            \vskip 0pt
+            \begin{tikzpicture}
+                \begin{axis}[
+                        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+[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}
+
+                \end{axis}
+            \end{tikzpicture}
+
+            \caption{BCH(31,26) Code}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+
+            \vspace*{-17mm}
+
+            \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
+            \smallskip
+
+            (No comprehensive investigation into choice of parameters
+            completed yet)
+        \end{subfigure}
+    \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}
+    \end{itemize}
+\end{frame}
+
+\end{document}

src/2025-03-28/res/check_regularity.py

@@ -0,0 +1,13 @@
+import sympy as sp
+
+
+def main():
+    x1,x2,x3 = sp.symbols("x_1 x_2 x_3")
+
+    x = sp.Matrix([x1, x2, x3])
+    F = sp.Matrix([1 - x1**2, 1 - x2**2, 1 - x3**2, 1 - x1*x2, 1 - x1*x3])
+    print(F.jacobian(x))
+    print(F.jacobian(x).rank())
+
+if __name__=="__main__":
+    main()

src/2025-03-28/res/gen_vec_fields.py

@@ -0,0 +1,46 @@
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+
+def main():
+    def F(x1, x2):
+        return np.array([x1 + x2, x2 + 0.5])
+
+    def G(x1, x2):
+        return np.array([x1, x2])
+
+    def H(x1, x2, t):
+        return (1 - t) * G(x1, x2) + t * F(x1, x2)
+
+    # x = np.linspace(-1, 1, 10)
+    # y = np.linspace(-1, 1, 10)
+    # X1, X2 = np.meshgrid(x, y)
+    #
+    # fig, axes = plt.subplots(1, 4, figsize=(20, 4))
+    #
+    # for i, t in enumerate(np.linspace(0, 1, 4)):
+    #     H_ = H(X1, X2, t)
+    #     axes[i].quiver(X1, X2, *H_, color='r')
+    #     axes[i].set_title(f't = {t}')
+    #
+    # plt.show()
+
+    df_dict = {"x1": [], "x2": [], "t": [], "H1": [], "H2": [], "Hmag": []}
+
+    for x1 in np.linspace(-1, 1, 10):
+        for x2 in np.linspace(-1, 1, 10):
+            for t in [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]:
+                df_dict["x1"].append(x1)
+                df_dict["x2"].append(x2)
+                df_dict["t"].append(t)
+                H_ = H(x1, x2, t)
+                df_dict["H1"].append(H_[0])
+                df_dict["H2"].append(H_[1])
+                df_dict["Hmag"].append(np.sqrt(H_[0]**2 + H_[1]**2))
+
+    df = pd.DataFrame(df_dict).sort_values(by=["t", "x2", "x1"])
+
+    df.to_csv("H.csv", index=False)
+
+if __name__=="__main__":
+    main()

src/2025-05-08/presentation.tex

@@ -0,0 +1,392 @@
+\documentclass[10pt, aspectratio=169, usenames, dvipsnames]{beamer}
+
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+\usetikzlibrary{spy, external, intersections}
+%\tikzexternalize[prefix=build/]
+
+\usepackage{pgfplots}
+\pgfplotsset{compat=newest}
+\usepgfplotslibrary{fillbetween}
+
+\usepackage{listings}
+\usepackage{subcaption}
+\usepackage{bbm}
+\usepackage{multirow}
+
+\usepackage{xcolor}
+%\usepackage[outputdir=build/]{minted}
+\usepackage{minted}
+\usemintedstyle{gruvbox-light}
+
+%\definecolor{gruvbox-bg}{HTML}{282828}
+\definecolor{gruvbox-bg}{HTML}{f2e5bc}
+
+%
+%
+% Custom commands
+%
+%
+
+\input{lib/latex-common/common.tex}
+\pgfplotsset{colorscheme/rocket}
+
+\newcommand{\res}{src/2025-05-08/res}
+
+%
+%
+% CEL Template
+%
+%
+
+\newcommand{\templates}{lib/cel-template}
+
+\input{\templates/packages.tex}
+\input{\templates/modifications.tex}
+\input{\templates/makros_own.tex}
+
+% % Change the way the overview is displayed
+% \AtBeginSection[]
+% {
+%     \begin{frame}[t]
+%         \frametitle{Overview}
+%     \tableofcontents[sectionstyle=show/shaded,
+%                      subsectionstyle=show/show/shaded,
+%                      subsubsectionstyle=hide]
+%     \end{frame}
+% }
+% \AtBeginSubsubsection[]{}
+% \AtBeginSubsection[]{}
+
+%
+%
+% Set up document
+%
+%
+
+\title{HiWi Notes: Minimization of the Code Constraint Polynomial using
+Homotopy Continuation Methods}
+\subtitle{\small 08.05.2025}
+\author{\vspace{1.5mm} Andreas Tsouchlos}
+\date{ }
+
+\institute{Karlsruhe Institute of Technology (KIT),
+\\ Communications Engineering Lab (CEL) }
+
+\tikzstyle{every node}=[font=\small]
+\captionsetup[sub]{font=small}
+
+%
+%
+% Document body
+%
+%
+
+\begin{document}
+
+\begin{frame}[plain]
+    \maketitle
+\end{frame}
+
+\newcommand{\largecitereference}[1]
+{\textcolor{kit-green100}{ \large \textbf{{[#1]}} }}
+
+\begin{frame}
+    \frametitle{The All-Zeros Assumption}
+
+    \vspace*{-5mm}
+
+    \begin{itemize}
+        \item Previous results were generated using the all-zeros
+            assumption
+        \item To cancel out any numerical effects,
+            subsequent results have been generated using randomly
+            generated codewords
+        \item Using randomly generated codewords paints a very different picture
+    \end{itemize}
+
+    \begin{figure}[H]
+        \centering
+
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+
+            \vskip 0pt
+            \begin{tikzpicture}
+                \begin{axis}[
+                        width=\textwidth,
+                        height=0.75\textwidth,
+                        ylabel={FER (-\,-\,-), BER (---)},
+                        ymode=log,
+                        legend,
+                        xlabel = {$Eb/N0$ (dB)},
+                        legend pos= south west,
+                        xmax=6
+                    ]
+
+                    \addplot+[black, 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+[black, 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+[scol3, densely dashed, mark=none, line
+                    width=1pt, forget plot]
+                    table[col sep=comma, x=SNR, y=FER]
+                    {\res/bch_31_26_hc_all_zeros.csv};
+                    \addplot+[scol3, mark=none, line width=1pt]
+                    table[col sep=comma, x=SNR, y=BER]
+                    {\res/bch_31_26_hc_all_zeros.csv};
+                    \addlegendentry{Hom. cont. - all zeros}
+
+                    \addplot+[scol1, densely dashed, mark=none, line
+                    width=1pt, forget plot]
+                    table[col sep=comma, x=SNR, y=FER]
+                    {\res/bch_31_26_hc_random.csv};
+                    \addplot+[scol1, mark=none, line width=1pt]
+                    table[col sep=comma, x=SNR, y=BER]
+                    {\res/bch_31_26_hc_random.csv};
+                    \addlegendentry{Hom. cont. - rand. cw.}
+
+                \end{axis}
+            \end{tikzpicture}
+
+            \caption{BCH(31,26) Code}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+
+            \vspace*{-17mm}
+
+            \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
+            \smallskip
+        \end{subfigure}
+    \end{figure}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Fixed Point Homotopy}
+
+    \vspace*{-6mm}
+
+    \begin{itemize}
+        \item Previous results were generated using the Newton homotopy
+            \begin{gather*}
+                G(\bm{x}) = F(x) - F(\bm{y}) \hspace{5mm} \Rightarrow
+                \hspace{5mm} H(\bm{x}) = F(\bm{x}) - (1 - t) F(\bm{y})
+            \end{gather*}
+        \item We could instead try the fixed point homotopy
+            \begin{gather*}
+                G(\bm{x}) = (\bm{x} - \bm{y}) \hspace{5mm}
+                \Rightarrow \hspace{5mm} H(\bm{x}) = (1-t)(\bm{x} -
+                \bm{y}) + t F(\bm{x})
+            \end{gather*}
+    \end{itemize}
+
+    \vspace*{-2mm}
+
+    \begin{figure}[H]
+        \centering
+
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+
+            \vskip 0pt
+            \begin{tikzpicture}
+                \begin{axis}[
+                        width=\textwidth,
+                        height=0.72\textwidth,
+                        ylabel={FER (-\,-\,-), BER (---)},
+                        ymode=log,
+                        legend,
+                        xlabel = {$Eb/N0$ (dB)},
+                        legend pos= south west,
+                        xmax=6,
+                        ymin = 0.1
+                    ]
+
+                    \addplot+[scol1, densely dashed, mark=none, line
+                    width=1pt, forget plot]
+                    table[col sep=comma, x=SNR, y=FER]
+                    {\res/bch_31_26_hc_fixed_point_random.csv};
+                    \addplot+[scol1, mark=none, line width=1pt]
+                    table[col sep=comma, x=SNR, y=BER]
+                    {\res/bch_31_26_hc_fixed_point_random.csv};
+                    \addlegendentry{Hom. cont. - fixed point, rand. cw.}
+
+                \end{axis}
+            \end{tikzpicture}
+
+            \vspace*{-2mm}
+            \caption{BCH(31,26) Code}
+        \end{subfigure}%
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+
+            \vspace*{-17mm}
+
+            \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
+            \smallskip
+        \end{subfigure}
+    \end{figure}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Parameter Exploration}
+
+    \vspace*{-7mm}
+
+    \begin{figure}
+        \centering
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+            \begin{itemize}
+                \item Fixed parameters:
+                    \begin{tabular}{rl|c}
+                        & Parameter & Value \\ \hline
+                        $E_\text{b} / N_0$ &
+                        & $\SI{6}{dB}$ \\
+                        $n_\text{iter}$    & for homotopy continuation & 200 \\
+                        $n_\text{iter}$    & for Newton corrector      & 20 \\
+                        $n_\text{retries}$ & for Euler predictor       & 20
+                    \end{tabular}
+                \item Random codewords, Newton homotopy, BCH(31, 26) code
+                \item Low decoding failure rate (DFR) $\rightarrow$
+                    We do often reach codewords, just not the correct ones
+            \end{itemize}
+        \end{subfigure} \hfill %
+        \begin{subfigure}[c]{0.48\textwidth}
+            \centering
+            \vskip 0pt
+            \scalebox{0.7}{
+                % tex-fmt: off
+                \begin{tabular}{c|c|ccccccc}
+                    DFR && \multicolumn{7}{c}{Euler step size}\\\hline
+                        && 0.001 & 0.005 & 0.01 & 0.05 & 0.1 & 0.5 & 1.0 \\ \hline
+                    \parbox[t]{2mm}{\multirow{7}{*}{\rotatebox[origin=c]{90}{Newtown threshold}}}
+                    & 0.001 & 0.070 & 0.055 & 0.085 & 0.070 & 0.075 & 0.050 & 0.060 \\
+                    & 0.005 & 0.055 & 0.075 & 0.060 & 0.070 & 0.100 & 0.070 & 0.050 \\
+                    & 0.01  & 0.065 & 0.060 & 0.075 & 0.085 & 0.095 & 0.040 & 0.045 \\
+                    & 0.05  & 0.090 & 0.080 & 0.085 & 0.085 & 0.045 & 0.065 & 0.055 \\
+                    & 0.1   & 0.100 & 0.085 & 0.045 & 0.080 & 0.065 & 0.055 & 0.075 \\
+                    & 0.5   & 0.070 & 0.075 & 0.125 & 0.055 & 0.070 & 0.055 & 0.090 \\
+                    & 1.0   & 0.055 & 0.095 & 0.060 & 0.050 & 0.055 & 0.075 & 0.055 \\
+                \end{tabular}
+                % tex-fmt: on
+            }
+        \end{subfigure}\\[2mm]
+        \begin{subfigure}[t]{0.48\textwidth}
+            \centering
+            \scalebox{0.7}{
+                % tex-fmt: off
+                \begin{tabular}{c|c|ccccccc}
+                    FER && \multicolumn{7}{c}{Euler step size}\\\hline
+                        && 0.001 & 0.005 & 0.01 & 0.05 & 0.1 & 0.5 & 1.0 \\ \hline
+                    \parbox[t]{2mm}{\multirow{7}{*}{\rotatebox[origin=c]{90}{Newtown threshold\footnotemark}}}
+                    & 0.001 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                    & 0.005 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                    & 0.01  & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                    & 0.05  & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                    & 0.1   & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                    & 0.5   & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                    & 1.0   & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
+                \end{tabular}
+                % tex-fmt: on
+            }
+        \end{subfigure} \hfill %
+        \begin{subfigure}[t]{0.48\textwidth}
+            \centering
+            \scalebox{0.7}{
+                % tex-fmt: off
+                \begin{tabular}{c|c|ccccccc}
+                    BER && \multicolumn{7}{c}{Euler step size}\\\hline
+                        && 0.001 & 0.005 & 0.01 & 0.05 & 0.1 & 0.5 & 1.0 \\ \hline
+                    \parbox[t]{2mm}{\multirow{7}{*}{\rotatebox[origin=c]{90}{Newtown threshold}}}
+                    & 0.001 & 0.497 & 0.496 & 0.509 &  0.495 & 0.508 & 0.495 & 0.503 \\
+                    & 0.005 & 0.509 & 0.503 & 0.491 &  0.502 & 0.506 & 0.502 & 0.502 \\
+                    & 0.01  & 0.498 & 0.492 & 0.520 &  0.509 & 0.500 & 0.502 & 0.504 \\
+                    & 0.05  & 0.503 & 0.499 & 0.507 &  0.494 & 0.492 & 0.500 & 0.506 \\
+                    & 0.1   & 0.499 & 0.499 & 0.500 &  0.508 & 0.502 & 0.488 & 0.507 \\
+                    & 0.5   & 0.510 & 0.503 & 0.510 &  0.495 & 0.507 & 0.515 & 0.501 \\
+                    & 1.0   & 0.502 & 0.506 & 0.506 &  0.502 & 0.500 & 0.505 & 0.500 \\
+                \end{tabular}
+                % tex-fmt: on
+            }
+        \end{subfigure}
+    \end{figure}
+
+    \footnotetext{``Newton threshold'' refers to the threshold used
+    for the convergence criterion of the newton corrector}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Tentative Conclusion}
+
+    \vspace*{-5mm}
+
+    \begin{itemize}
+        \item Original motivation
+            \begin{itemize}
+                \item Novel decoding technique
+                \item Possible decoding guarantees due to homotopy
+                    continuation behavior
+            \end{itemize}
+        \item Simulation results
+            \begin{itemize}
+                \item Implementing a working decoder has proven
+                    difficult, irrespective of the parameters / code chosen
+                \item The homotopy continuation implementation itself
+                    seems to be working, judging by the DFR
+                \item There are two alternatives that both describe
+                    the observed results:
+                    \begin{itemize}
+                        \item We always converge to the same
+                            codeword - error rates are due to
+                            Tx codewords being random
+                        \item We randomly converge to codewords that
+                            have nothing to do with what was sent
+                    \end{itemize}
+            \end{itemize}
+        \item Problems
+            \begin{itemize}
+                \item It is not obvious how we can apply the
+                    theory from homotopy continuation to the
+                    decoder
+                \item Decoding algorithm seems too comput. expensive (matrix
+                    inversion necessary for each iteration)
+            \end{itemize}
+        \item Conclusion
+            \begin{itemize}
+                \item There is no hard evidence yet that a decoder
+                    based on homotopy continuation can't work
+                \item However, pursuing this line of research doesn't seem
+                    particularly promising, due to missing motivation
+                    and missing evidence it can work at all
+            \end{itemize}
+    \end{itemize}
+\end{frame}
+
+\end{document}

src/2025-05-08/res/bch_31_11.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+2.0,  0.550964, 0.250955, 0.550964, 200
+4.0,  0.487805, 0.215342, 0.487805, 200
+6.0,  0.255102, 0.103152, 0.255102, 200
+8.0,  0.057339, 0.018756, 0.057339, 200
+10.0, 0.003046, 0.000718, 0.003046, 200
+12.0, 0.000022, 0.000003, 0.000022, 22

src/2025-05-08/res/bch_31_26_hc_all_zeros.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+1.0, 0.869565, 0.073072, 0.826087, 200
+2.0, 0.653595, 0.045435, 0.630719, 200
+3.0, 0.546448, 0.030672, 0.535519, 200
+4.0, 0.377358, 0.017712, 0.371698, 200
+5.0, 0.174825, 0.007219, 0.173077, 200
+6.0, 0.068989, 0.002648, 0.065885, 200

src/2025-05-08/res/bch_31_26_hc_fixed_point_random.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+1.0, 1.0, 0.532419, 0.635, 200
+2.0, 1.0, 0.523710, 0.470, 200
+3.0, 1.0, 0.520806, 0.425, 200
+4.0, 1.0, 0.518871, 0.220, 200
+5.0, 1.0, 0.512419, 0.125, 200
+6.0, 1.0, 0.504839, 0.085, 200

src/2025-05-08/res/bch_31_26_hc_newton.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
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+4.0,1.000000,0.302097,0.990000,200
+6.0,0.995025,0.185684,0.930348,200
+8.0,0.970874,0.106326,0.941748,200
+10.0,0.749064,0.043132,0.734082,200
+12.0,0.251889,0.009547,0.251889,200

src/2025-05-08/res/bch_31_26_hc_newton_random.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
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+4.0,1.000000,0.298548,0.960000,200
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+10.0,0.716846,0.040351,0.709677,200
+12.0,0.238379,0.009112,0.235995,200

src/2025-05-08/res/bch_31_26_hc_no_newton_random.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+2.0,1.000000,0.471774,0.250000,200
+4.0,1.000000,0.461613,0.260000,200
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+8.0,0.970874,0.454588,0.087379,200
+10.0,0.738007,0.340198,0.110701,200
+12.0,0.264201,0.119657,0.054161,200

src/2025-05-08/res/bch_31_26_hc_no_newton_zeros.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+2.0,0.506329,0.209637,0.470886,200
+4.0,0.111794,0.039002,0.081610,200
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+10.0,0.008497,0.000683,0.000000,200
+12.0,0.001114,0.000073,0.000000,200

src/2025-05-08/res/bch_31_26_hc_random.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+1.0, 1.0, 0.524194, 0.700, 200
+2.0, 1.0, 0.520161, 0.495, 200
+3.0, 1.0, 0.509516, 0.455, 200
+4.0, 1.0, 0.515161, 0.295, 200
+5.0, 1.0, 0.506290, 0.145, 200
+6.0, 1.0, 0.504516, 0.065, 200

src/2025-05-08/res/bch_31_26_hc_rx_fixed_point.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+2.0,1.000000,0.410484,0.860000,200
+4.0,1.000000,0.308710,0.895000,200
+6.0,1.000000,0.207581,0.900000,200
+8.0,0.909091,0.100000,0.868182,200
+10.0,0.602410,0.040420,0.575301,200
+12.0,0.156128,0.006421,0.156128,200

src/2025-05-08/res/bch_31_26_ml.csv

@@ -0,0 +1,9 @@
+SNR,FER,num_errors,num_iterations
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+1.00,4.566e-01,100,219
+2.00,2.695e-01,100,371
+3.00,9.990e-02,100,1001
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+5.00,5.692e-03,100,17568
+6.00,5.506e-04,100,181625
+7.00,4.400e-05,44,1000000

src/2025-05-08/res/bch_31_26_proximal.csv

@@ -0,0 +1,121 @@
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src/2025-05-08/res/bch_32_26_rx_fixed_point_random.csv

@@ -0,0 +1,7 @@
+SNR,FER,BER,DFR,frame_errors
+2.0,1.0,0.499032,0.895,200
+4.0,1.0,0.480000,0.865,200
+6.0,1.0,0.483710,0.885,200
+8.0,1.0,0.478548,0.800,200
+10.0,1.0,0.504516,0.575,200
+12.0,1.0,0.500323,0.135,200

src/2025-05-08/res/bch_7_3_hc.csv

@@ -0,0 +1,7 @@
+SNR,FER,num_errors,num_iterations
+ 2.0, 0.341880, 0.187546, 0.186325, 200
+ 4.0, 0.238949, 0.121010, 0.102748, 200
+ 6.0, 0.120555, 0.051580, 0.035564, 200
+ 8.0, 0.048614, 0.015973, 0.005348, 200
+10.0, 0.015249, 0.004466, 0.000610, 200
+12.0, 0.002916, 0.000846, 0.000000, 200

src/2025-05-08/res/check_regularity.py

@@ -0,0 +1,13 @@
+import sympy as sp
+
+
+def main():
+    x1,x2,x3 = sp.symbols("x_1 x_2 x_3")
+
+    x = sp.Matrix([x1, x2, x3])
+    F = sp.Matrix([1 - x1**2, 1 - x2**2, 1 - x3**2, 1 - x1*x2, 1 - x1*x3])
+    print(F.jacobian(x))
+    print(F.jacobian(x).rank())
+
+if __name__=="__main__":
+    main()

src/2025-05-08/res/gen_vec_fields.py

@@ -0,0 +1,46 @@
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+
+def main():
+    def F(x1, x2):
+        return np.array([x1 + x2, x2 + 0.5])
+
+    def G(x1, x2):
+        return np.array([x1, x2])
+
+    def H(x1, x2, t):
+        return (1 - t) * G(x1, x2) + t * F(x1, x2)
+
+    # x = np.linspace(-1, 1, 10)
+    # y = np.linspace(-1, 1, 10)
+    # X1, X2 = np.meshgrid(x, y)
+    #
+    # fig, axes = plt.subplots(1, 4, figsize=(20, 4))
+    #
+    # for i, t in enumerate(np.linspace(0, 1, 4)):
+    #     H_ = H(X1, X2, t)
+    #     axes[i].quiver(X1, X2, *H_, color='r')
+    #     axes[i].set_title(f't = {t}')
+    #
+    # plt.show()
+
+    df_dict = {"x1": [], "x2": [], "t": [], "H1": [], "H2": [], "Hmag": []}
+
+    for x1 in np.linspace(-1, 1, 10):
+        for x2 in np.linspace(-1, 1, 10):
+            for t in [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]:
+                df_dict["x1"].append(x1)
+                df_dict["x2"].append(x2)
+                df_dict["t"].append(t)
+                H_ = H(x1, x2, t)
+                df_dict["H1"].append(H_[0])
+                df_dict["H2"].append(H_[1])
+                df_dict["Hmag"].append(np.sqrt(H_[0]**2 + H_[1]**2))
+
+    df = pd.DataFrame(df_dict).sort_values(by=["t", "x2", "x1"])
+
+    df.to_csv("H.csv", index=False)
+
+if __name__=="__main__":
+    main()