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 \v{C}ech 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 \v{C}ech complex and a density-scaled Vietoris--Rips complex. We show that the density-scaled \v{C}ech 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 \v{C}ech 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 $M$.
翻译:我们开发了使用持久性同质学的新方法, 以从任意的平滑概率密度函数的点云样样本中推断出一个未知的瑞曼元( M, g) 的同质 。 标准的远程过滤综合体, 如 & v{ C} 技术综合体, 往往难以区分噪音和简单小的特征。 我们通过定义一个“ 密度大小的过滤综合体” 的组合来解决这个问题, 包括一个密度缩放的 { {C} 复杂和密度尺度的越越野- 里普- 复合体。 我们显示, 密度尺度的内基值相当于 美元, 在一个间隔期间, 标准级的过滤综合体值等于 $ 。 开始点的概率等于 0 美元, 到 分级的过滤综合体, 我们的递增量的系统, 也能够确定我们系统内部的比值 。