Persistence in discrete Morse theory
Persistence in discrete Morse theory
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坚持离散莫尔斯理论
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Ulrich Bauer
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文献类型：
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作者：
Ulrich Bauer
本论文的目标是将两种不同的理论结合起来，它们是关于标函数临界点的理论，以及它们与拓扑学的关系：离散莫尔斯理论和持续同调理论。虽然目标和基本技术不同，但两种理论中出现的某些主题非常相似。离散莫尔斯理论提供了经典莫尔斯理论的几个核心概念的组合等价物，如离散莫尔斯函数，离散梯度向量场，临界点，以及消除向量场临界点的消去定理。由于它的简单性，它不仅保持了经典理论的直观性，而且通过提供在光滑环境下会变得相当复杂的明确和规范的构造，在某种意义上允许超越经典理论。持久同调量化了函数的拓扑特征。它定义了同调类在临界点的生与死，确定了这些同调类的对（持久性对），并提供了它们稳定性（持久性）的定量概念。（离散）莫尔斯理论对函数的子级集的同伦类型进行了陈述，而持久性则与它们的同调有关。虽然同调是同伦等价的一个不变量，但匡威并不正确：不是每一个在同调中产生同构的映射都是同伦等价。在这篇论文中，我们建立了两个理论之间的联系，并使用这种组合来解决问题，这是不容易接近的任何单一的理论。特别是，我们解决了在一定的公差范围内，从一个给定的输入函数的表面上的函数的临界点的数量最小化的问题。
The goal of this thesis is to bring together two different theories about critical points of a scalar function and their relation to topology: Discrete Morse theory and Persistent homology. While the goals and fundamental techniques are different, there are certain themes appearing in both theories that closely resemble each other. In certain cases, the two threads can be joined, leading to new insights beyond the classical realm of one particular theory.Discrete Morse theory provides combinatorial equivalents of several core concepts of classical Morse theory, such as discrete Morse functions, discrete gradient vector fields, critical points, and a cancelation theorem for the elimination of critical points of a vector field. Because of its simplicity, it not only maintains the intuition of the classical theory but allows to surpass it in a certain sense by providing explicit and canonical constructions that would become quite complicated in the smooth setting.Persistent homology quantifies topological features of a function. It defines the birth and death of homology classes at critical points, identifies pairs of these (persistence pairs), and provides a quantitative notion of their stability (persistence).Whereas (discrete) Morse theory makes statements about the homotopy type of the sublevel sets of a function, persistence is concerned with their homology. While homology is an invariant of homotopy equivalences, the converse is not true: not every map inducing an isomorphism in homology is a homotopy equivalence. In this thesis we establish a connection between both theories and use this combination to solve problems that are not easily accessibly by any single theory alone. In particular, we solve the problem of minimizing the number of critical points of a function on a surface within a certain tolerance from a given input function.