Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. theČech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.
We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual Čech filtration of such a subset can be endowed with a vector bundle structure, that we call a Čech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its first persistent Stiefel-Whitney class. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis.
We describe an algorithm that takes as an input a CW complex and returns a simplicial complex of the same homotopy type. This algorithm, although well-known in the literature, requires some work to make it computationally tractable. We pay close attention to weak simplicial approximation, which we implement for generalized barycentric and edgewise subdivisions. We also propose a new subdivision process, based on Delaunay complexes. In order to facilitate the computation of a simplicial approximation, we introduce a simplification step, based on edge contractions. We define a new version of simplicial mapping cone, which requires less simplices. Last, we illustrate the algorithm with the real projective spaces, the 3-dimensional lens spaces and the Grassmannian of 2-planes in R 4 .
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