Zhang Shuhao, Liu Nan, Kang Wei, Permuter Haim
School of Information Science and Engineering, Southeast University, Nanjing 211189, China.
National Mobile Communications Research Laboratory, Southeast University, Nanjing 211189, China.
Entropy (Basel). 2024 Aug 21;26(8):711. doi: 10.3390/e26080711.
The complete characterization of the almost-entropic region yields rate regions for network coding problems. However, this characterization is difficult and open. In this paper, we propose a novel algorithm to determine whether an arbitrary vector in the entropy space is entropic or not, by parameterizing and generating probability mass functions by neural networks. Given a target vector, the algorithm minimizes the normalized distance between the target vector and the generated entropic vector by training the neural network. The algorithm reveals the entropic nature of the target vector, and obtains the underlying distribution, accordingly. The proposed algorithm was further implemented with convolutional neural networks, which naturally fit the structure of joint probability mass functions, and accelerate the algorithm with GPUs. Empirical results demonstrate improved normalized distances and convergence performances compared with prior works. We also conducted optimizations of the Ingleton score and Ingleton violation index, where a new lower bound of the Ingleton violation index was obtained. An inner bound of the almost-entropic region with four random variables was constructed with the proposed method, presenting the current best inner bound measured by the volume ratio. The potential of a computer-aided approach to construct achievable schemes for network coding problems using the proposed method is discussed.
几乎熵区域的完整表征为网络编码问题提供了速率区域。然而，这种表征既困难又尚未解决。在本文中，我们提出了一种新颖的算法，通过神经网络对概率质量函数进行参数化和生成，来确定熵空间中的任意向量是否为熵向量。给定一个目标向量，该算法通过训练神经网络来最小化目标向量与生成的熵向量之间的归一化距离。该算法揭示了目标向量的熵性质，并据此获得潜在分布。所提出的算法进一步使用卷积神经网络实现，卷积神经网络自然地适配联合概率质量函数的结构，并通过图形处理器（GPU）加速算法。实证结果表明，与先前的工作相比，归一化距离和收敛性能均有所改善。我们还对英格尔顿分数和英格尔顿违反指数进行了优化，得到了英格尔顿违反指数的一个新的下界。使用所提出的方法构建了具有四个随机变量的几乎熵区域的内界，呈现了以体积比衡量的当前最佳内界。讨论了使用所提出的方法通过计算机辅助方法构建网络编码问题可实现方案的潜力。
