Intrinsic persistent homology via density-based metric learning

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.