40 lines
2.1 KiB
TeX
40 lines
2.1 KiB
TeX
\chapter{Discussion}%
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\label{chapter:discussion}
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While the modified proximal decoding algorithm presented in section
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\ref{sec:prox:Improved Implementation} shows some promising results, further
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investigation is required to determine how different choices of parameters
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affect the decoding performance.
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Additionally, a more mathematically rigorous foundation for determining the
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potentially wrong components of the estimate is desirable.
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As mentioned in section \ref{subsec:prox:conv_properties}, the alternating
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minimization of the two gradients in the proximal decoding algorithm leads to
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an oscillation after a number of iterations.
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One approach to alleviate this problem might be to use \ac{ADMM} instead of
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the proximal gradient method to solve the optimization problem.
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This is because due to the introduction of the dual variable, the minimization
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of each part of the objective function would no longer take place with regard
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to the same exact variable.
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Additionally, ``\ac{ADMM} will converge even when the x- and z-minimization
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steps are not carried out exactly [\ldots]''
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\cite[Sec. 3.4.4]{distr_opt_book}, which is advantageous, as the
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constraints are never truly satisfied; not even after the minimization step
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dealing with the constraint part of the objective function.
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Despite this, an initial examination by Yanxia Lu in
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\cite[Sec. 4.2.4.]{yanxia_lu_thesis} shows only limited success.
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Another interesting approach might be the combination of proximal and \ac{LP}
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decoding.
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Performing an initial number of iterations using proximal decoding to obtain
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a rough first estimate and subsequently using \ac{LP} decoding with only the
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violated constraints may be a way to achieve a shorter running time, because
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of the low-complexity nature of proximal decoding.
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This could be usefull, for example, to mitigate the slow convergence of
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\ac{ADMM} \cite[3.2.2]{distr_opt_book}.
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Subsequently introducing additional parity checks might be a way of combining
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the best properties of proximal decoding, \ac{LP} decoding using \ac{ADMM} and
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\textit{adaptive \ac{LP} decoding} \cite{alp} to obtain a decoder relatively
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efficiently approximating \ac{ML} performance.
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