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## 1. chinaXiv:202108.00116 [pdf]

Subjects: Mathematics >> Mathematics （General）

 In this paper, we discuss the properties of functions generated using Hausdorff metric.

## 2. chinaXiv:202112.00129 [pdf]

Subjects: Mathematics >> Mathematics （General）

 最近，一种全新的集合被提出，名为随机排列集(Random Permutation Set RPS)。随机排列集考虑了某一集合的所有可能的排列，并可以看作是证据理论的一种推广。不确定性是随机排列集的一个重要特征。一个简单的问题是如何测量随机排列集的不确定度。为了解决这一问题，本文提出了随机排列集的信息熵(Entropy of Random Permutation Set)。当忽略排列事件(Permutation Event)中元素的顺序时，随机排列集的信息熵退化为证据理论中的邓熵。当每个排列事件只包含一个元素时，随机排列集的信息熵退化为概率论中的香农熵。因此，随机排列集的信息熵可以看作是邓熵和香农熵的推广。 本文用数值算例说明了随机排列集的信息熵的有效性。

## 3. chinaXiv:202112.00122 [pdf]

Subjects: Mathematics >> Statistics and Probability

 The paper considers Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform one on the d-dimensional ball and give the asymptotic estimation of their expectation as $n \to \infty$. Further We considers the above problem on a mixed process, i.e., n discrete random variables are produced by the Poisson process.

## 4. chinaXiv:202112.00055 [pdf]

Subjects: Mathematics >> Applied Mathematics

 We analyze properties of degree and clustering of a hyperbolic geometric model of complex networks in small parameter case $\tau<1, 2\sigma<1$. We find that the probability of k-degree goes to 0 and the global clustering coefficient goes to 0 in probability too as the number of nodes $N\to\infty$ for some specific growth $R(N)$ of the region radius. Here the scale-free degree is failed and the connection between neighbors are very weak. The transition of properties of the model with the parameter $\sigma$ changes seems to show that the mobility is important to keep society full and stable communication, otherwise a silence society. Some analysis technique and method are first applied for such model.

## 5. chinaXiv:202112.00085 [pdf]

Subjects: Computer Science >> Integration Theory of Computer Science

 最近，一种名为随机排列集(Random Permutation Set RPS)的集合被提出，它考虑了一个集合中所有元素的排列。 为了度量随机排列集的不确定度，随机排列集的熵被提出。 然而，随机排列集的最大熵原理却没有详细地论证。为了解决这一问题，本文提出了随机排列集的最大熵。随机排列集的最大熵的解析解及其对应的PMF条件得到了证明与讨论。我们用数值例子说明了随机排列集的最大熵的性质。实验结果表明，随机排列集的最大熵和最大邓熵、最大香农熵相互兼容。 当忽略排列事件(Permutation Event)中元素的顺序时，随机排列集的最大熵将退化为最大邓熵。当每个排列事件只包含一个元素时，随机排列集的最大熵将退化为最大香农熵。

## 6. chinaXiv:202112.00002 [pdf]

Subjects: Mathematics >> Control and Optimization.

 在机器学习和数学优化研究领域, 深度学习优化问题易优性的数学解释极具挑战性. 损失函数存在高维、非凸、不光滑等特质性, 然而也能通过梯度下降法搜索到全局最优值. 损失函数地貌分析成为揭示深度学习优化问题易优性本质的重要研究方向. 为促进可解释、可信的深度学 习在更关键领域的应用, 本文回顾了损失函数地貌特征（局部极小点的数量和空间分布、最优点之间的连通性、临界点的最优性）、梯度下降法收敛性、以及损失函数地貌可视化等方面的研究进展和挑战.

## 7. chinaXiv:202107.00011 [pdf]

Subjects: Mathematics >> Mathematics （General）

 This paper discusses the properties the spaces of fuzzy sets in a metric s- pace equipped with the endograph metric and the sendograph metric, re- spectively. We fist discuss the level characterizations of the Γ-convergence and the endograph metric, and point out the elementary relationships among Γ-convergence, endograph metric and the sendograph metric. On the basis of these results, we present the characterizations of total boundedness, rel- ative compactness and compactness in the space of compact positive α-cuts fuzzy sets equipped with the endograph metric, and in the space of com- pact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these two kinds of spaces, respectively.

## 8. chinaXiv:202105.00070 [pdf]

Subjects: Statistics >> Mathematical Statistics
Subjects: Computer Science >> Computer Application Technology
Subjects: Information Science and Systems Science >> Basic Disciplines of Information Science and Systems Science

 统计独立性是统计学和机器学习领域的基础性概念，如何表示和度量统计独立性是该领域的基本问题。Copula理论提供了统计相关性表示的理论工具，而Copula熵理论则给出了度量统计独立性的概念工具。本文综述了Copula熵的理论和应用，概述了其基本概念定义、定理和性质，以及估计方法。介绍了Copula熵研究的最新进展，包括其在统计学四个基本问题（结构学习、关联发现、变量选择和时序因果发现等）上的理论应用。讨论了四个理论应用之间的关系，以及其对应的深层次的相关性和因果性概念之间的联系，并将Copula熵的（条件）独立性度量框架与基于核函数和距离的相关性度量框架进行了对比。简述了Copula熵在水文学、环境气象学、生态学、化学信息学、认知神经学、计算神经学、系统生物学、生物信息学、临床诊断学、老年医学、公共卫生学、经济政策学、政治学，以及能源工程、制造工程、可靠性工程、航空工程、通信工程、测绘工程和金融工程等领域的实际应用。

## 9. chinaXiv:202110.00083 [pdf]

Subjects: Mathematics >> Mathematics （General）

 In this paper, we give characterizations of compactness in fuzzy set space with Lpmetric. Based on these results, we present completions of fuzzy set spaces with Lpmetric. The fuzzy sets discussed in this paper are fuzzy sets in general metric space. We also point out that the results in this paper generalize the results given in our previous paper.