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Added footnote about discrete -> continuous; Added quotation marks; minor other changes

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Andreas Tsouchlos 2023-03-12 20:23:22 +01:00
parent 6513fd2297
commit 52ba8c67ee
2 changed files with 18 additions and 9 deletions
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5
latex/thesis/abbreviations.tex
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@ -97,6 +97,11 @@
long = probability density function long = probability density function
} }
\DeclareAcronym{PMF}{
short = PMF,
long = probability mass function
}
% %
% V % V
% %

22
latex/thesis/chapters/decoding_techniques.tex
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@ -25,14 +25,14 @@ available optimization algorithms.
Generally, the original decoding problem considered is either the \ac{MAP} or Generally, the original decoding problem considered is either the \ac{MAP} or
the \ac{ML} decoding problem:% the \ac{ML} decoding problem:%
% %
\begin{align*} \begin{align}
\hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}}_{\text{\ac{MAP}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y} p_{\boldsymbol{C} \mid \boldsymbol{Y}} \left(\boldsymbol{c} \mid \boldsymbol{y}
\right)\\ \right) \label{eq:dec:map}\\
\hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}} \hat{\boldsymbol{c}}_{\text{\ac{ML}}} &= \argmax_{\boldsymbol{c} \in \mathcal{C}}
f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c} f_{\boldsymbol{Y} \mid \boldsymbol{C}} \left( \boldsymbol{y} \mid \boldsymbol{c}
\right) \right) \label{eq:dec:ml}
.\end{align*}% .\end{align}%
% %
The goal is to arrive at a formulation, where a certain objective function The goal is to arrive at a formulation, where a certain objective function
$g : \mathbb{R}^n \rightarrow \mathbb{R}^n $ must be minimized under certain constraints:% $g : \mathbb{R}^n \rightarrow \mathbb{R}^n $ must be minimized under certain constraints:%
@ -707,7 +707,11 @@ non-convex optimization formulation of the \ac{MAP} decoding problem.
In order to derive the objective function, the authors begin with the In order to derive the objective function, the authors begin with the
\ac{MAP} decoding rule, expressed as a continuous maximization problem% \ac{MAP} decoding rule, expressed as a continuous maximization problem%
\footnote{The }% \footnote{The expansion of the domain to be continuous doesn't constitute a
material difference.
The only change is that what previously were \acp{PMF} now have to be expressed
in terms of \acp{PDF}}
over $\boldsymbol{x}$
:% :%
% %
\begin{align} \begin{align}
@ -726,7 +730,7 @@ The likelihood $f_{\boldsymbol{Y} \mid \tilde{\boldsymbol{X}}}
determined by the channel model. determined by the channel model.
The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also The prior \ac{PDF} $f_{\tilde{\boldsymbol{X}}}\left( \tilde{\boldsymbol{x}} \right)$ is also
known, as the equal probability assumption is made on known, as the equal probability assumption is made on
$\mathcal{C}\left( \boldsymbol{H} \right)$. $\mathcal{C}$.
However, since the considered domain is continuous, However, since the considered domain is continuous,
the prior \ac{PDF} cannot be ignored as a constant during the minimization the prior \ac{PDF} cannot be ignored as a constant during the minimization
as is often done, and has a rather unwieldy representation:% as is often done, and has a rather unwieldy representation:%
@ -843,14 +847,14 @@ The second step thus becomes%
\hspace{5mm}\gamma > 0,\text{ small} \hspace{5mm}\gamma > 0,\text{ small}
.\end{align*} .\end{align*}
% %
While the approximation of the prior \ac{PDF} made in \ref{eq:prox:prior_pdf_approx} While the approximation of the prior \ac{PDF} made in equation (\ref{eq:prox:prior_pdf_approx})
theoretically becomes better theoretically becomes better
with larger $\gamma$, the constraint that $\gamma$ be small is important, with larger $\gamma$, the constraint that $\gamma$ be small is important,
as it keeps the effect of $h\left( \boldsymbol{x} \right) $ on the landscape as it keeps the effect of $h\left( \boldsymbol{x} \right) $ on the landscape
of the objective function small. of the objective function small.
Otherwise, unwanted stationary points, including local minima, are introduced. Otherwise, unwanted stationary points, including local minima, are introduced.
The authors say that in practice, the value of $\gamma$ should be adjusted The authors say that ``in practice, the value of $\gamma$ should be adjusted
according to the decoding performance \cite[Sec. 3.1]{proximal_paper}. according to the decoding performance.'' \cite[Sec. 3.1]{proximal_paper}.
%The components of the gradient of the code-constraint polynomial can be computed as follows:% %The components of the gradient of the code-constraint polynomial can be computed as follows:%
%% %%