706 lines
33 KiB
TeX
706 lines
33 KiB
TeX
\appendix
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\chapter{A Comparison of the Behaviour of Various Codes}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b / N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=\textwidth,
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height=0.75\textwidth,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=10,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot3[RedOrange, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
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\cite[\text{96.3.965}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b / N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=\textwidth,
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height=0.75\textwidth,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=10,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot3[RedOrange, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{BCH code with $n=31, k=26$\\[2\baselineskip]}
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\end{subfigure}
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b/N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=\textwidth,
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height=0.75\textwidth,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=14,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot3[RedOrange, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b / N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=\textwidth,
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height=0.75\textwidth,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=10,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot3[RedOrange, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.55.187}]{mackay_enc}}
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\end{subfigure}%
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b / N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=\textwidth,
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height=0.75\textwidth,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=10,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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\addplot3[RedOrange, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_40833844.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=408, k=204$
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\cite[\text{408.33.844}]{mackay_enc}}
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\end{subfigure}%
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\hfill
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[view={75}{30},
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zmode=log,
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xlabel={$E_b / N_0$ (dB)},
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ylabel={$\gamma$},
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zlabel={BER},
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width=\textwidth,
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height=0.75\textwidth,]
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\addplot3[surf,
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mesh/rows=17, mesh/cols=10,
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colormap/viridis] table [col sep=comma,
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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\addplot3[RedOrange, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.05},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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\addplot3[NavyBlue, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.01},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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\addplot3[ForestGreen, line width=1.5] table[col sep=comma,
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discard if not={gamma}{0.15},
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x=SNR, y=gamma, z=BER]
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{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{LDPC code (Progressive Edge Growth Construction) with $n=504, k=252$
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\cite[\text{PEGReg252x504}]{mackay_enc}}
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\end{subfigure}%
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\vspace{1cm}
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\begin{subfigure}[c]{\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[hide axis,
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xmin=10, xmax=50,
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ymin=0, ymax=0.4,
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legend style={draw=white!15!black,legend cell align=left}]
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\addlegendimage{surf, colormap/viridis}
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\addlegendentry{$\gamma = \left[ 0\text{ : }0.01\text{ : }0.16 \right] $};
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\addlegendimage{NavyBlue, line width=1.5pt}
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\addlegendentry{$\gamma = 0.01$};
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\addlegendimage{RedOrange, line width=1.5pt}
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\addlegendentry{$\gamma = 0.05$};
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\addlegendimage{ForestGreen, line width=1.5pt}
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\addlegendentry{$\gamma = 0.15$};
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}
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\caption{BER for $\omega = 0.05, K=100$ (different codes)}
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\label{fig:prox:results_3d_multiple}
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\end{figure}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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legend columns=1,
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legend pos=outer north east,
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[ForestGreen, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot[Emerald, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
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{res/hybrid/2d_ber_fer_dfr_963965.csv};
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\addplot[NavyBlue, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot[RoyalPurple, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
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{res/hybrid/2d_ber_fer_dfr_963965.csv};
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\addplot[RedOrange, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_963965.csv};
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\addplot[red, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/hybrid/2d_ber_fer_dfr_963965.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=96, k=48$
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\cite[\text{96.3.965}]{mackay_enc}}
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\end{subfigure}%
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\hfill%
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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legend columns=1,
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legend pos=outer north east,
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%legend columns=2,
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%legend style={at={(0.5,-0.45)},anchor=south},
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[ForestGreen, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[Emerald, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
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{res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[NavyBlue, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[RoyalPurple, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
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{res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[RedOrange, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_bch_31_26.csv};
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\addplot[red, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/hybrid/2d_ber_fer_dfr_bch_31_26.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{BCH code with $n=31, k=26$\\[\baselineskip]}
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\end{subfigure}%
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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legend columns=1,
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legend pos=outer north east,
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xmin=0.5, xmax=6, xtick={1, ..., 5},
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[ForestGreen, mark=*, solid,]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.15},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[Emerald, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.15},
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discard if gt={SNR}{5.5},]
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{res/hybrid/2d_ber_fer_dfr_20433484.csv};
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\addplot[NavyBlue, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.01},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[RoyalPurple, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.01},
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discard if gt={SNR}{5.5},]
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{res/hybrid/2d_ber_fer_dfr_20433484.csv};
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\addplot[RedOrange, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5},]
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{res/proximal/2d_ber_fer_dfr_20433484.csv};
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\addplot[red, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma,
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discard if not={gamma}{0.05},
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discard if gt={SNR}{5.5},]
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{res/hybrid/2d_ber_fer_dfr_20433484.csv};
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\end{axis}
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\end{tikzpicture}
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\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=204, k=102$
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\cite[\text{204.33.484}]{mackay_enc}}
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\end{subfigure}%
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\hfill%
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\begin{subfigure}[c]{0.48\textwidth}
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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grid=both,
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xlabel={$E_b / N_0$}, ylabel={FER},
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ymode=log,
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legend columns=1,
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legend pos=outer north east,
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%legend columns=2,
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%legend style={at={(0.5,-0.45)},anchor=south},
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ymax=1.5, ymin=8e-5,
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width=\textwidth,
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height=0.75\textwidth,
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]
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\addplot[ForestGreen, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot[Emerald, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
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{res/hybrid/2d_ber_fer_dfr_20455187.csv};
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\addplot[NavyBlue, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
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\addplot[RoyalPurple, mark=triangle, densely dashed]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
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{res/hybrid/2d_ber_fer_dfr_20455187.csv};
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\addplot[RedOrange, mark=*, solid]
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table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
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{res/proximal/2d_ber_fer_dfr_20455187.csv};
|
|
\addplot[red, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_20455187.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{$\left( 5, 10 \right)$-regular \ac{LDPC} code with $n=204, k=102$
|
|
\cite[\text{204.55.187}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$}, ylabel={FER},
|
|
ymode=log,
|
|
legend columns=1,
|
|
legend pos=outer north east,
|
|
%legend columns=2,
|
|
%legend style={at={(0.5,-0.45)},anchor=south},
|
|
ymax=1.5, ymin=8e-5,
|
|
width=\textwidth,
|
|
height=0.75\textwidth,
|
|
]
|
|
|
|
\addplot[ForestGreen, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\addplot[Emerald, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/hybrid/2d_ber_fer_dfr_40833844.csv};
|
|
|
|
\addplot[NavyBlue, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/hybrid/2d_ber_fer_dfr_40833844.csv};
|
|
|
|
\addplot[RedOrange, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_40833844.csv};
|
|
\addplot[red, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_40833844.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
|
|
\caption{$\left( 3, 6 \right)$-regular \ac{LDPC} code with $n=408, k=204$
|
|
\cite[\text{408.33.844}]{mackay_enc}}
|
|
\end{subfigure}%
|
|
\hfill%
|
|
\begin{subfigure}[c]{0.48\textwidth}
|
|
\centering
|
|
\begin{tikzpicture}
|
|
\begin{axis}[
|
|
grid=both,
|
|
xlabel={$E_b / N_0$}, ylabel={FER},
|
|
ymode=log,
|
|
legend columns=1,
|
|
legend pos=outer north east,
|
|
ymax=1.5, ymin=8e-5,
|
|
width=\textwidth,
|
|
height=0.75\textwidth,
|
|
]
|
|
|
|
\addplot[ForestGreen, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot[Emerald, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.15}]
|
|
{res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
|
|
\addplot[NavyBlue, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot[RoyalPurple, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.01}]
|
|
{res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
|
|
\addplot[RedOrange, mark=*, solid]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/proximal/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\addplot[red, mark=triangle, densely dashed]
|
|
table [x=SNR, y=FER, col sep=comma, discard if not={gamma}{0.05}]
|
|
{res/hybrid/2d_ber_fer_dfr_pegreg252x504.csv};
|
|
\end{axis}
|
|
\end{tikzpicture}\\
|
|
|
|
\caption{\ac{LDPC} code (progressive edge growth construction) with $n=504, k=252$
|
|
\cite[\text{PEGReg252x504}]{mackay_enc}}
|
|
\label{fig:prox:improved:comp:504}
|
|
\end{subfigure}%
|
|
|
|
\vspace{1cm}
|
|
|
|
\begin{subfigure}[c]{\textwidth}
|
|
\centering
|
|
|
|
\begin{tikzpicture}
|
|
\begin{axis}[hide axis,
|
|
xmin=10, xmax=50,
|
|
ymin=0, ymax=0.4,
|
|
legend columns=3,
|
|
legend style={draw=white!15!black,legend cell align=left}]
|
|
\addlegendimage{ForestGreen, mark=*, solid}
|
|
\addlegendentry{proximal, $\gamma = 0.15$}
|
|
|
|
\addlegendimage{NavyBlue, mark=*, solid}
|
|
\addlegendentry{proximal, $\gamma = 0.01$}
|
|
|
|
\addlegendimage{RedOrange, mark=*, solid}
|
|
\addlegendentry{proximal, $\gamma = 0.05$}
|
|
|
|
\addlegendimage{Emerald, mark=triangle, densely dashed}
|
|
\addlegendentry{improved, $\gamma = 0.15$}
|
|
|
|
\addlegendimage{RoyalPurple, mark=triangle, densely dashed}
|
|
\addlegendentry{improved, $\gamma = 0.01$}
|
|
|
|
\addlegendimage{red, mark=triangle, densely dashed}
|
|
\addlegendentry{improved, $\gamma = 0.05$}
|
|
\end{axis}
|
|
\end{tikzpicture}
|
|
\end{subfigure}
|
|
|
|
\caption{Comparison of improvement in decoding performance for various
|
|
codes}
|
|
\label{fig:prox:improved:comp}
|
|
\end{figure}
|
|
|
|
\chapter{\acs{LP} Decoding using \acs{ADMM} as a Proximal Algorithm}%
|
|
\label{chapter:LD Decoding using ADMM as a Proximal Algorithm}
|
|
|
|
\todo{Find out how to properly title and section appendix}
|
|
|
|
%For problems of the form%
|
|
%\begin{align*}
|
|
% \text{minimize}\hspace{2mm} & f\left( \boldsymbol{x} \right)
|
|
% + g\left( \boldsymbol{A}\boldsymbol{x} \right) \\
|
|
% \text{subject to}\hspace{2mm} & \boldsymbol{x} \in \mathbb{R}^n
|
|
%,\end{align*}%
|
|
%%
|
|
%a version of \ac{ADMM}, \textit{linearized \ac{ADMM}}, can be expressed
|
|
%as a proximal algorithm \cite[Sec. 4.4.2]{proximal_algorithms}:
|
|
%%
|
|
%\begin{align*}
|
|
% \boldsymbol{x} &\leftarrow \textbf{prox}_{\mu f}\left( \boldsymbol{x}
|
|
% - \frac{\mu}{\lambda}\boldsymbol{A}^\text{T}\left( \boldsymbol{A}\boldsymbol{x}
|
|
% - \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
|
|
% \boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left( \boldsymbol{A}\boldsymbol{x}
|
|
% + \boldsymbol{u} \right) \\
|
|
% \boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{A} \boldsymbol{x} - \boldsymbol{z}
|
|
%.\end{align*}
|
|
|
|
In order to express \ac{LP} decoding using \ac{ADMM} through proximal operators,
|
|
it can be rewritten to fit the template for \textit{linearized \ac{ADMM}} given
|
|
in \cite[Sec. 4.4.2]{proximal_algorithms}.
|
|
We start with the general formulation of the \ac{LP} decoding problem:%
|
|
%
|
|
\begin{align*}
|
|
\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}} \\
|
|
\text{subject to}\hspace{2mm} & \boldsymbol{T}_j \tilde{\boldsymbol{c}} \in \mathcal{P}_{d_j}
|
|
\hspace{5mm} \forall j \in \mathcal{J}
|
|
.\end{align*}
|
|
%
|
|
The constraints can be moved into the objective function: %
|
|
%
|
|
\begin{align}
|
|
\begin{aligned}
|
|
\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
|
|
+ \sum_{j\in \mathcal{J}} I_{P_{d_j}}\left(
|
|
\boldsymbol{T}_j\tilde{\boldsymbol{c}} \right) \\
|
|
\text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n,
|
|
\end{aligned}
|
|
\label{eq:app:sum_reformulated}
|
|
\end{align}%
|
|
%
|
|
using the \textit{indicator functions}
|
|
$I_{\mathcal{P}_{d_j}} : \mathbb{R}^{d_j} \rightarrow \left\{ 0, +\infty \right\},
|
|
\hspace{3mm} j\in \mathcal{J}$, defined as%
|
|
%
|
|
\begin{align*}
|
|
I_{\mathcal{P}_{d_j}}\left( \boldsymbol{t} \right) :=
|
|
\begin{cases}
|
|
0 & \boldsymbol{t} \in \mathcal{P}_{d_j} \\
|
|
+\infty & \boldsymbol{t} \not\in \mathcal{P}_{d_j}
|
|
\end{cases}
|
|
.\end{align*}%
|
|
%
|
|
Further defining
|
|
%
|
|
\begin{align*}
|
|
\boldsymbol{T} := \begin{bmatrix}
|
|
\boldsymbol{T}_1 \\
|
|
\boldsymbol{T}_2 \\
|
|
\vdots \\
|
|
\boldsymbol{T}_m
|
|
\end{bmatrix}
|
|
\hspace{5mm}\text{and}\hspace{5mm}
|
|
g\left( \boldsymbol{t} \right) = \sum_{j\in\mathcal{J}} I_{\mathcal{P}_{d_j}}\left(
|
|
\boldsymbol{B}_j \boldsymbol{t} \right)
|
|
,\end{align*}%
|
|
%
|
|
\todo{Define $\boldsymbol{B}_j$}%
|
|
problem (\ref{eq:app:sum_reformulated}) becomes%
|
|
%
|
|
\begin{align}
|
|
\begin{aligned}
|
|
\text{minimize}\hspace{2mm} & \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
|
|
+ g\left( \boldsymbol{T}\tilde{\boldsymbol{c}} \right) \\
|
|
\text{subject to}\hspace{2mm} & \tilde{\boldsymbol{c}} \in \mathbb{R}^n.
|
|
\end{aligned}
|
|
\label{eq:app:func_reformulated}
|
|
\end{align}
|
|
%
|
|
\todo{Fix $\mu f$ and $\lambda g$ in the steps below}%
|
|
In this form, it fits the template for linearized \ac{ADMM}.
|
|
The iterative algorithm can then be expressed as%
|
|
%
|
|
\begin{align}
|
|
\begin{aligned}
|
|
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
|
|
- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
|
|
\boldsymbol{z} &\leftarrow \textbf{prox}_{\lambda g}\left(\boldsymbol{T}\tilde{\boldsymbol{c}}
|
|
+ \boldsymbol{u} \right) \\
|
|
\boldsymbol{u} &\leftarrow \boldsymbol{u} + \boldsymbol{T} \tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z}.
|
|
\end{aligned}
|
|
\label{eq:app:admm_prox}
|
|
\end{align}
|
|
%
|
|
|
|
Using the definition of the proximal operator, the $\tilde{\boldsymbol{c}}$ update step
|
|
can be rewritten to match the definition given in section \ref{sec:lp:Decoding Algorithm}:%
|
|
%
|
|
\begin{align*}
|
|
\tilde{\boldsymbol{c}} &\leftarrow \textbf{prox}_{\mu f}\left( \tilde{\boldsymbol{c}}
|
|
- \frac{\mu}{\lambda}\boldsymbol{T}^\text{T}\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z} + \boldsymbol{u} \right) \right) \\
|
|
&= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
|
|
- \frac{\mu}{2} \left\Vert \boldsymbol{T}^\text{T}
|
|
\left( \boldsymbol{T}\tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z} + \boldsymbol{u} \right) \right\Vert_2^2 \right) \\
|
|
&\overset{\text{(a)}}{=} \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}
|
|
\tilde{\boldsymbol{c}}
|
|
- \frac{\mu}{2} \left\Vert \boldsymbol{T}\tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z} + \boldsymbol{u} \right\Vert_2^2 \right) \\
|
|
&= \argmin_{\tilde{\boldsymbol{c}}}\left( \boldsymbol{\gamma}^\text{T}\tilde{\boldsymbol{c}}
|
|
- \frac{\mu}{2} \left\Vert \begin{bmatrix}
|
|
\boldsymbol{T}_1 \\
|
|
\boldsymbol{T}_2 \\
|
|
\vdots \\
|
|
\boldsymbol{T}_m
|
|
\end{bmatrix}
|
|
\tilde{\boldsymbol{c}}
|
|
- \begin{bmatrix}
|
|
\boldsymbol{z}_1 \\
|
|
\boldsymbol{z}_2 \\
|
|
\vdots \\
|
|
\boldsymbol{z}_m
|
|
\end{bmatrix}
|
|
+ \begin{bmatrix}
|
|
\boldsymbol{u}_1 \\
|
|
\boldsymbol{u}_2 \\
|
|
\vdots \\
|
|
\boldsymbol{u}_m
|
|
\end{bmatrix} \right\Vert_2^2 \right),
|
|
\hspace{5mm}\boldsymbol{z}_j,\boldsymbol{u}_j \in \mathbb{F}_2^{d_j},
|
|
\hspace{2mm} j\in\mathcal{J}\\
|
|
&= \argmin_{\tilde{\boldsymbol{c}}} \left( \boldsymbol{\gamma}^\text{T} \tilde{\boldsymbol{c}}
|
|
- \frac{\mu}{2} \sum_{j \in J} \left\Vert \boldsymbol{T}_j \tilde{\boldsymbol{c}}
|
|
- \boldsymbol{z}_j + \boldsymbol{u}_j \right\Vert_2^2 \right)
|
|
.\end{align*}
|
|
%
|
|
Step (a) can be justified by observing that multiplication with $\boldsymbol{T}^\text{T}$
|
|
only reorders components, leaving their values unchanged.
|
|
Similarly to the $\boldsymbol{c}$ update, the $\boldsymbol{z}$ update step can be rewritten.
|
|
Since $g\left( \cdot \right)$ is separable, so is its proximal operator
|
|
\cite[Sec. 2.1]{proximal_algorithms}. The $\boldsymbol{z}$ update step can then
|
|
be expressed as a number of smaller steps:%
|
|
%
|
|
\begin{gather*}
|
|
\boldsymbol{z} \leftarrow \textbf{prox}_{\lambda g} \left(\boldsymbol{T}\tilde{\boldsymbol{c}}
|
|
+ \boldsymbol{u} \right) \\[0.5em]
|
|
\iff \\[0.5em]
|
|
\begin{alignedat}{3}
|
|
\boldsymbol{z}_j &\leftarrow \textbf{prox}_{\lambda I_{\mathcal{P}_{d_j}}}\left(
|
|
\boldsymbol{T}_j \tilde{\boldsymbol{c}} + \boldsymbol{u}_j \right),
|
|
\hspace{5mm} && \forall j\in\mathcal{J} \\
|
|
& \overset{\text{(b)}}{=} \Pi_{\mathcal{P}_{d_j}}\left( \boldsymbol{T}_j
|
|
\tilde{\boldsymbol{c}}
|
|
+ \boldsymbol{u}_j \right), \hspace{5mm} && \forall j\in\mathcal{J}
|
|
,\end{alignedat}
|
|
\end{gather*}
|
|
%
|
|
where (b) results from the fact that appying the proximal operator on the
|
|
indicator function of a convex set amounts to a projection onto the set
|
|
\cite[Sec. 1.2]{proximal_algorithms}.
|