随机映射图的概率收敛与稳定性
Probabilistic Convergence and Stability of Random Mapper Graphs
AI总结：
研究随机映射图与拓扑空间Reeb图的概率收敛性，提出增强映射图并证明其在概率密度下近似Reeb图，结合可构造余sheaf理论与核密度估计方法。
摘要：
我们研究拓扑空间X配备连续函数f: X→R的随机映射图与Reeb图之间的概率收敛性。我们首先通过将映射图和Reeb图解释为实数轴的余sheaf和分层覆盖的分类化方法。然后引入一种改进的映射图，证明其在随机采样概率密度函数下近似Reeb图。我们的技术基于可构造余sheaf的交织距离和拓扑估计。通过Munch和Wang(2018)的方法，我们证明映射图近似Reeb图。然后构造映射图的同构关系。最后基于Bobrowski等(2017)的方法，证明在足够大的样本下可以恢复超水平集的映射图。本文首次将余sheaf理论应用于概率设置中的映射构造，是结合sheaf理论、概率与统计的持续努力的一部分，以支持随机数据的拓扑数据分析。
英文摘要：
We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.
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