From 41ad5d091203fea7ae152cee796cef95d2155d82 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Tue, 13 May 2025 14:53:33 +0200 Subject: [PATCH] Add 2025-05-08 presentation --- src/2025-05-08/presentation.tex | 392 ++++++ src/2025-05-08/res/H.csv | 1101 +++++++++++++++++ src/2025-05-08/res/bch_31_11.csv | 7 + src/2025-05-08/res/bch_31_26_hc_all_zeros.csv | 7 + .../res/bch_31_26_hc_fixed_point_random.csv | 7 + src/2025-05-08/res/bch_31_26_hc_newton.csv | 7 + .../res/bch_31_26_hc_newton_random.csv | 7 + .../res/bch_31_26_hc_no_newton_random.csv | 7 + .../res/bch_31_26_hc_no_newton_zeros.csv | 7 + src/2025-05-08/res/bch_31_26_hc_random.csv | 7 + .../res/bch_31_26_hc_rx_fixed_point.csv | 7 + src/2025-05-08/res/bch_31_26_ml.csv | 9 + src/2025-05-08/res/bch_31_26_proximal.csv | 121 ++ .../res/bch_32_26_rx_fixed_point_random.csv | 7 + src/2025-05-08/res/bch_7_3_hc.csv | 7 + src/2025-05-08/res/check_regularity.py | 13 + src/2025-05-08/res/gen_vec_fields.py | 46 + 17 files changed, 1759 insertions(+) create mode 100644 src/2025-05-08/presentation.tex create mode 100644 src/2025-05-08/res/H.csv create mode 100644 src/2025-05-08/res/bch_31_11.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_all_zeros.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_fixed_point_random.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_newton.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_newton_random.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_no_newton_random.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_no_newton_zeros.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_random.csv create mode 100644 src/2025-05-08/res/bch_31_26_hc_rx_fixed_point.csv create mode 100644 src/2025-05-08/res/bch_31_26_ml.csv create mode 100644 src/2025-05-08/res/bch_31_26_proximal.csv create mode 100644 src/2025-05-08/res/bch_32_26_rx_fixed_point_random.csv create mode 100644 src/2025-05-08/res/bch_7_3_hc.csv create mode 100644 src/2025-05-08/res/check_regularity.py create mode 100644 src/2025-05-08/res/gen_vec_fields.py diff --git a/src/2025-05-08/presentation.tex b/src/2025-05-08/presentation.tex new file mode 100644 index 0000000..5fe6a77 --- /dev/null +++ b/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}etwas + \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} diff --git 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+10.0,1.0,0.504516,0.575,200 +12.0,1.0,0.500323,0.135,200 diff --git a/src/2025-05-08/res/bch_7_3_hc.csv b/src/2025-05-08/res/bch_7_3_hc.csv new file mode 100644 index 0000000..2fa06d7 --- /dev/null +++ b/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 diff --git a/src/2025-05-08/res/check_regularity.py b/src/2025-05-08/res/check_regularity.py new file mode 100644 index 0000000..34ab697 --- /dev/null +++ b/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() diff --git a/src/2025-05-08/res/gen_vec_fields.py b/src/2025-05-08/res/gen_vec_fields.py new file mode 100644 index 0000000..6b3841a --- /dev/null +++ b/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()