Rigorous cubical approximation and persistent homology of continuous functions

65. Pawel Dlotko, Thomas Wanner:
    Rigorous cubical approximation and persistent homology of continuous functions
    Computers & Mathematics with Applications 75(5), pp. 1648-1666, 2018.

Abstract

The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.

Links

The preprint version of the paper can be downloaded from https://arxiv.org/abs/1610.03833, while the published version of the paper can be found at https://doi.org/10.1016/j.camwa.2017.11.027.

Bibtex

@article{dlotko:wanner:18a,
   author = {Pawe{\l} D{\l}otko and Thomas Wanner},
   title = {Rigorous cubical approximation and persistent homology of
            continuous functions},
   journal = {Computers \& Mathematics with Applications},
   volume = {75},
   number = {5},
   year = {2018},
   pages = {1648--1666},
   doi = {10.1016/j.camwa.2017.11.027}
   }