Approximating persistent homology in Euclidean space through collapses

The Čech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the Čech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at sma...

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Bibliographic Details
Published inApplicable algebra in engineering, communication and computing Vol. 26; no. 1-2; pp. 73 - 101
Main Authors Botnan, Magnus Bakke, Spreemann, Gard
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2015
Subjects
Artificial Intelligence
Computer Hardware
Computer Science
Original Paper
Symbolic and Algebraic Manipulation
Theory of Computation
Computational topology
Persistent homology
Approximation
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Summary:The Čech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the Čech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at smaller scales than what the application calls for. In this paper we propose two methods to approximate the Čech persistence module. Both are constructed on the level of spaces, i.e. as sequences of simplicial complexes induced by nerves. We also show how the bottleneck distance between such persistence modules can be understood by how tightly they are sandwiched on the level of spaces. In turn, this implies the correctness of our approximation methods. Finally, we implement our methods and apply them to some example point clouds in Euclidean space.
ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-014-0247-y