Nomura Ryo, Yagi Hideki
Center for Data Science, Waseda University, Tokyo 169-8050, Japan.
Department of Computer and Network Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan.
Entropy (Basel). 2024 Sep 6;26(9):766. doi: 10.3390/e26090766.
Two typical fixed-length random number generation problems in information theory are considered for sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to -divergences have been characterized using the information spectrum quantity. The -divergence is a general non-negative measure between two probability distributions on the basis of a convex function . The class of -divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to -divergences. However, optimum achievable rates with respect to -divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of -divergence. To do so, we first derive general formulas of the optimum achievable rates with respect to -divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on -divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions . As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. optimum achievable rates and optimistic achievable rates are also investigated.
针对信源，考虑了信息论中两个典型的固定长度随机数生成问题。一个是信源可分解性问题，另一个是内在随机性问题。在这些问题中的每一个中，相对于给定近似度量的最优可达速率是我们主要关注的问题之一，并且已经使用两种不同的信息量进行了刻画：信息谱和平滑Rényi熵。最近，相对于散度的最优可达速率已经使用信息谱量进行了刻画。散度是基于凸函数的两个概率分布之间的一般非负度量。散度类包括几个重要的度量，如变分距离、KL散度、Hellinger距离等。因此，考虑相对于散度的随机数生成问题是有意义的。然而，在这两个问题中，使用平滑Rényi熵相对于散度的最优可达速率尚未明确。在本文中，我们尝试使用平滑Rényi熵分析最优可达速率，并扩展散度类。为此，我们首先在与先前研究相同的条件下，推导这两个问题中相对于散度的最优可达速率的一般公式。接下来，我们放宽对散度的条件并推广得到的一般公式。然后，我们将我们的一般公式特殊化为几个指定的函数。结果，我们发现从我们的一般公式很容易推导出几个重要度量的最优可达速率。此外，从平滑Rényi熵的角度揭示了可分解性和内在随机性之间的一种关系。还研究了最优可达速率和乐观可达速率。
