59 lines
3.2 KiB
TeX
59 lines
3.2 KiB
TeX
\chapter{Discussion}%
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\label{chapter:discussion}
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A modification of the implementation to reduce the memory requirements, even
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at some cost with regard to the running time, would allow for the examination
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of longer codes.
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This in turn would make possible studying the behavior of the decoding
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algorithms covered here in error-rate regions where traditional approaches
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exhibit an error floor.
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The decoding algorithms could then be assessed for use in very
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high reliability applications, where traditional methods like \ac{BP} or the
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min-sum-algorithm fall short.
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\todo{Doesn't make sense}
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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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It is also important to note that while in this thesis proximal decoding was
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examined with respect to its performance in \ac{AWGN} channels, in
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\cite{proximal_paper} it is presented as a method applicable to non-trivial
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channel models such as \ac{LDPC}-coded massive \ac{MIMO} channels, perhaps
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broadening its usefulness beyond what is shown here.
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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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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 efficiently
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approximating \ac{ML} performance.
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\todo{It turns out that ADMM is more compuationally efficient than proximal
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decoding.
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Find a way to combine them that still makes sense (maybe exploiting the
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fact that the BER is so much better than the FER, in constrasto to ADMM)}
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