稀疏区域持久图的收敛性
Convergence of persistence diagram in the sparse regime
摘要：
本文的目的是研究与Čech过滤相关的持久性图的渐近行为。持久性图是几何对象的拓扑和代数结构的图形描述符。我们考虑在缩放随机样本$r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$上的Čech过滤，使得$r_n\to 0$为$n\to\infty$。我们把持久性图作为一个点过程，并建立其极限定理在稀疏制度：$nr_n^d\to0$，$n\to\infty$。在这种情况下，我们表明，$k$ th持久性图的渐近依赖于序列$n^{k+2}r_n^{d(k+1)}$的极限值。如果$n^{k+2}r_n^{d(k+1)} \to \infty$，在Vague度量下，定标持久图几乎必然收敛于确定性Radon测度.如果$r_n$的衰减速度快于$n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$，则持久性图弱收敛于一个极限点过程，而无需归一化。最后，如果是$n^{k+2}r_n^{d(k+1)} \to 0$，则持久图的概率分布序列应该被归一化，并且由此产生的收敛将根据$\mathcal M_0$ -拓扑来处理。
英文摘要：
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.
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