Quantitative homotopy theory in topological data analysis
AJ Blumberg, MA Mandell - Foundations of Computational Mathematics, 2013 - Springer
Foundations of Computational Mathematics, 2013•Springer
This paper lays the foundations of an approach to applying Gromov's ideas on quantitative
topology to topological data analysis. We introduce the “contiguity complex”, a simplicial
complex of maps between simplicial complexes defined in terms of the combinatorial notion
of contiguity. We generalize the Simplicial Approximation Theorem to show that the
contiguity complex approximates the homotopy type of the mapping space as we subdivide
the domain. We describe algorithms for approximating the rate of growth of the components …
topology to topological data analysis. We introduce the “contiguity complex”, a simplicial
complex of maps between simplicial complexes defined in terms of the combinatorial notion
of contiguity. We generalize the Simplicial Approximation Theorem to show that the
contiguity complex approximates the homotopy type of the mapping space as we subdivide
the domain. We describe algorithms for approximating the rate of growth of the components …
Abstract
This paper lays the foundations of an approach to applying Gromov’s ideas on quantitative topology to topological data analysis. We introduce the “contiguity complex”, a simplicial complex of maps between simplicial complexes defined in terms of the combinatorial notion of contiguity. We generalize the Simplicial Approximation Theorem to show that the contiguity complex approximates the homotopy type of the mapping space as we subdivide the domain. We describe algorithms for approximating the rate of growth of the components of the contiguity complex under subdivision of the domain; this procedure allows us to computationally distinguish spaces with isomorphic homology but different homotopy types.
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