尺度同调与拓扑熵

Variably Scaled Persistence Kernels (VSPKs) for persistent homology applications

In recent years, various kernels have been proposed in the context of persistent homology to deal with persistence diagrams in supervised learning approaches. In this paper, we consider the idea of variably scaled kernels, for approximating functions and data, and we interpret it in the framework of persistent homology. We call them Variably Scaled Persistence Kernels (VSPKs). These new kernels are then tested in different classification experiments. The obtained results show that they can improve the performance and the efficiency of existing standard kernels.

Integration in Čech theories and a bound on entropy

The evaluation of Alexander-Spanier cochains over formal simplices in a topological space leads to a notion of integration of Alexander-Spanier cohomology classes over Čech homology classes. The integral defines an explicit and non-degenerate pairing between the Alexander-Spanier cohomology and the Čech homology. Instead of working on the limits that define both groups, most of the discussion is carried out "at scale $\mathcal U$", for an open covering $\mathcal U$. As an application, we generalize a result of Manning to arbitrary compact spaces $X$: we prove that the topological entropy of $f…

拓扑熵猜想

Topological Entropy Conjecture

1974年，M. Shub提出了拓扑熵猜想，即不等式$\logρ\leq ent(f)$是否成立，其中$f$是紧致流形$M$上的连续自映射，$ent(f)$是$f$的拓扑熵，$ρ$是$f$在同调群$H_{*}(M;\mathbb{Z})=\bigoplus\limits_{i=0}^n{H_{i}(M;\mathbb{Z})}$上诱导的线性变换$f_*$的最大绝对特征值. 1986年，A. B. Katok给出了一个反例，使得不等式$\logρ\leq ent(f)$是无效的.本文在紧致Hausdorff空间$X$上定义了$f$ -Čech同调群$\check{H}_{i}(X,f;\mathbb{Z})$和拓扑纤维熵$ent(f_L)$，其中$n=n(J)$使得$\check{H}_*(X;\mathbb{Z})$存在，其中$f\in C^0(X)$和$J$是所有覆盖的集合.然后证明了$\logρ\leq ent(f_L)$是有效的。

Shapley同调：神经网络样本影响的拓扑分析

Shapley Homology: Topological Analysis of Sample Influence for Neural Networks

为训练机器学习模型而收集的数据样本通常被假设为独立同分布（iid）。最近的研究表明，这种假设可能是有问题的，因为它简化了结构化数据的流形。这激发了不同的研究领域，如数据投毒，模型改进和机器学习模型的解释。在这项工作中，我们研究了样本的影响，确定其基础流形的内在拓扑特征。我们提出了Shapley同调框架，它提供了一个定量度量的影响，一个简单的复杂的同源性的样本。通过将影响解释为概率度量，我们进一步定义了反映数据流形复杂性的熵。我们的实证研究表明，当使用0维同源性时，在相邻图上，具有较高影响分数的样本对神经网络确定图连通性的准确性有更大的影响，并且对一些规则文法，其较高的熵值意味着更难学习。

字符串拓扑数据分析

Topological Data Analysis for the String Landscape

持久同源通过使用离散复合体的序列来计算数据集的多尺度拓扑。在本文中，我们提出持久同调可能是一个有用的工具，研究弦真空景观的结构。作为该程序的缩小版本，我们使用持久同调来表征模空间上的IIB型通量真空的分布：刚性Calabi-Yau，加权射影空间中的超曲面和对称六环面$T^6=(T^2)^3$。这些例子表明，持久性配对和多参数持久性包含有用的信息表征的景观，除了通常的信息包含在标准的持久性同源。我们还研究了如何限制到特殊真空现象有趣的低能量特性影响的拓扑分布。

Summary and Distance between Sets of Texts based on Topological Data Analysis

In this paper, we use topological data analysis (TDA) tools such as persistent homology, persistent entropy and bottleneck distance, to provide a {\it TDA-based summary} of any given set of texts and a general method for computing a distance between any two literary styles, authors or periods. To this aim, deep-learning word-embedding techniques are combined with these tools in order to study the topological properties of texts embedded in a metric space. As a case of study, we use the written texts of three poets of the Spanish Golden Age: Francisco de Quevedo, Luis de Góngora and Lope de Veg…