In this work, a novel data-driven methodology for designing polar codes for
channels with and without memory is proposed. The methodology is suitable for
the case where the channel is given as a "black-box" and the designer has
access to the channel for generating observations of its inputs and outputs,
but does not have access to the explicit channel model. The proposed method
leverages the structure of the successive cancellation (SC) decoder to devise a
neural SC (NSC) decoder. The NSC decoder uses neural networks (NNs) to replace
the core elements of the original SC decoder, the check-node, the bit-node and
the soft decision. Along with the NSC, we devise additional NN that embeds the
channel outputs into the input space of the SC decoder. The proposed method is
supported by theoretical guarantees that include the consistency of the NSC.
Also, the NSC has computational complexity that does not grow with the channel
memory size. This sets its main advantage over successive cancellation trellis
(SCT) decoder for finite state channels (FSCs) that has complexity of
$O(|\mathcal{S}|^3 N\log N)$, where $|\mathcal{S}|$ denotes the number of
channel states. We demonstrate the performance of the proposed algorithms on
memoryless channels and on channels with memory. The empirical results are
compared with the optimal polar decoder, given by the SC and SCT decoders. We
further show that our algorithms are applicable for the case where there SC and
SCT decoders are not applicable.

通过新颖的数据驱动方法，提出了一种针对具有和不具有记忆的信道设计极化码的方法。该方法利用连续取消 (SC) 译码器的结构来设计神经 SC (NSC) 译码器，使用神经网络 (NNs) 替代原始 SC 译码器的核心元素，以及在 SC 译码器的输入空间中嵌入信道输出的附加 NN。该方法不仅具备理论保证，还具有计算复杂性随信道记忆大小而不增长的优势。该算法在无记忆信道和具有记忆信道上展示了其性能，并与最优极化译码器进行了比较，同时还适用于 SC 和 SCT 译码器不适用的情况。

基于数据驱动的神经极化码研究未知无记忆信道及带有记忆信道

Data-Driven Neural Polar Codes for Unknown Channels With and Without  Memory

We propose an iterative method for approximately computing the capacity of
discrete memoryless channels, possibly under additional constraints on the
input distribution. Based on duality of convex programming, we derive explicit
upper and lower bounds for the capacity. The presented method requires $O(M^2 N
\sqrt{\log N}/\varepsilon)$ to provide an estimate of the capacity to within
$\varepsilon$, where $N$ and $M$ denote the input and output alphabet size; a
single iteration has a complexity $O(M N)$. We also show how to approximately
compute the capacity of memoryless channels having a bounded continuous input
alphabet and a countable output alphabet under some mild assumptions on the
decay rate of the channel's tail. It is shown that discrete-time Poisson
channels fall into this problem class. As an example, we compute sharp upper
and lower bounds for the capacity of a discrete-time Poisson channel with a
peak-power input constraint.

提出一种迭代计算分散无记忆信道容量的方法，包括对输入分布的附加约束；利用凸规划的对偶性，获得了容量的显式上下界。该方法的复杂度为 O（M ^ 2 N√（log N）/ε），其中 N 和 M 分别表示输入和输出字母表的大小；单次迭代的复杂度为 O（MN）。同时，针对有限连续输入和可数输出字母表的无记忆信道提出了近似计算容量的方法，在一些关于信道尾部的渐变速率的温和假设下可以实现（离散时间泊松信道属于该问题类），并给出了其在峰值功率输入约束下的上下界估计路段作为案例研究

信道容量的高效逼近

Efficient Approximation of Channel Capacities

In this paper we introduce a fundamental principle for optimal communication
over general memoryless channels in the presence of noiseless feedback, termed
posterior matching. Using this principle, we devise a (simple, sequential)
generic feedback transmission scheme suitable for a large class of memoryless
channels and input distributions, achieving any rate below the corresponding
mutual information. This provides a unified framework for optimal feedback
communication in which the Horstein scheme (BSC) and the Schalkwijk-Kailath
scheme (AWGN channel) are special cases. Thus, as a corollary, we prove that
the Horstein scheme indeed attains the BSC capacity, settling a longstanding
conjecture. We further provide closed form expressions for the error
probability of the scheme over a range of rates, and derive the achievable
rates in a mismatch setting where the scheme is designed according to the wrong
channel model. Several illustrative examples of the posterior matching scheme
for specific channels are given, and the corresponding error probability
expressions are evaluated. The proof techniques employed utilize novel
relations between information rates and contraction properties of iterated
function systems.

本文提出了一种基本原则，即后验匹配，用于通讯过程中的反馈传输方案，在一定条件下，在广泛类别的无记忆信道和输入分布情况下，实现相对应的互信息以下的任何传输速率，该方案提供了一种优化反馈通讯的统一框架，并证明了 Horstein 方案确实实现了 BSC 容量，同时提供了反馈方案的错误概率表达式和可实现的速率范围。