一族密度标度的滤过复形
A Family of Density-Scaled Filtered Complexes
摘要：
我们开发了新的方法，使用持久的同源性来推断一个未知的黎曼流形$(M, g)$的同源性，从一个任意光滑的概率密度函数采样的点云。标准的基于距离的过滤复合体，如切赫复合体，通常很难区分噪声和很小的特征。我们解决这个问题，通过定义一个家庭的“密度缩放过滤复合物”，其中包括一个密度缩放的Čech复杂和一个密度缩放的Vietoris-Rips复杂。我们证明了密度标度的Čech复形对于一个区间内的过滤值是同伦等价于$M$的，该区间的起始点以概率收敛于$0$作为点的数目$N \to \infty$，其终点以$N \to \infty$的形式趋近于无穷大。相比之下，标准的Čech复形可能只在非常小的过滤值范围内同伦等价于$M$。密度缩放的过滤复合物还具有在保形变换（例如缩放）下不变的性质。我们实现了一个过滤复杂的$\widehat{DVR}$，近似密度缩放的Vietoris-撕裂复杂，我们凭经验测试我们的实现的性能。作为示例，我们使用$\widehat{DVR}$来识别具有不同密度的簇，并且我们将$\widehat{DVR}$应用于Lorenz动力系统的时间延迟嵌入。我们的实现是稳定的（在几乎肯定满足的条件下），并设计用于处理点云中不位于$M$上的离群值。
英文摘要：
We develop novel methods for using persistent homology to infer the homology of an unknown Riemannian manifold $(M, g)$ from a point cloud sampled from an arbitrary smooth probability density function. Standard distance-based filtered complexes, such as the Čech complex, often have trouble distinguishing noise from features that are simply small. We address this problem by defining a family of "density-scaled filtered complexes" that includes a density-scaled Čech complex and a density-scaled Vietoris--Rips complex. We show that the density-scaled Čech complex is homotopy-equivalent to $M$ for filtration values in an interval whose starting point converges to $0$ in probability as the number of points $N \to \infty$ and whose ending point approaches infinity as $N \to \infty$. By contrast, the standard Čech complex may only be homotopy-equivalent to $M$ for a very small range of filtration values. The density-scaled filtered complexes also have the property that they are invariant under conformal transformations, such as scaling. We implement a filtered complex $\widehat{DVR}$ that approximates the density-scaled Vietoris--Rips complex, and we empirically test the performance of our implementation. As examples, we use $\widehat{DVR}$ to identify clusters that have different densities, and we apply $\widehat{DVR}$ to a time-delay embedding of the Lorenz dynamical system. Our implementation is stable (under conditions that are almost surely satisfied) and designed to handle outliers in the point cloud that do not lie on $M$.
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