We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain intrinsic persistence diagrams, which are less sensitive to the particular embedding of the manifold in the Euclidean space. We show that this approach is robust to outliers and deduce a method for pattern recognition in signals, with applications in real data.

翻译：我们从有限的样本中用多种方式估计内在距离的问题。我们证明,样品所定义的具有可计算度量的测量空间(称为样本Fermat距离)在Gromov-Hausdorf的意义上与 a.s.相融合。限制的物体是配有人群Fermat距离的多元体,它既能测量该金属的几何,又能测量产生样本的密度。这一结果用于获取内在持久性图,而该图对将该金属具体嵌入Euclidean空间不那么敏感。我们表明,这一方法对外星非常有力,并推断出一种在信号中识别模式的方法,在真实数据中应用。