论文标题
持久性和契约功能对应关系
Persistence and the Sheaf-Function Correspondence
论文作者
论文摘要
捆函数通信将与$ M $上的Grothendieck组一起在真实的分析歧管$ M $上确定了构造功能组。当$ m $是有限的尺寸实际矢量空间时，Kashiwara-Schapira最近在$ M $上引入了$ k $ - 矢量空间之间的卷积距离。在本文中，我们表征了真实有限维矢量空间上的构造函数组的距离，该函数可以通过通过横扫功能对应的卷积距离来控制。我们的主要结果断言，这种距离几乎是微不足道的：只要两个可构造功能具有相同的Euler积分，它们就会消失。我们在拓扑数据分析中制定了结果的后果：在相互交织的距离中，持续不断的持续模块的非平凡添加剂不变。
The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold $M$ with the Grothendieck group of constructible sheaves on $M$. When $M$ is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $k$-vector spaces on $M$. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exists non-trivial additive invariants of persistence modules that are continuous for the interleaving distance.
