# Generalized persistence modules and some of their invariants

## Abstract/Contents

- Abstract
- To analyze a data set topologically, the first step is to construct its topology. There are several methods to construct a topology, most of which depend on some parameters that must be chosen manually for the resulting topology to be useful. A general method of persistent homology proceeds as follows: 1) Construct a sequence of topologies from a sequence of parameter values. 2) Construct a persistence module by applying a homology functor to the sequence of topologies. 3) Extract persistence module invariants from the persistence module. The invariants serve as topological descriptors of the original data set. If the topology construction method has only one parameter, the constructed persistence module is called ordinary or one-dimensional. In this case, the classification theorem of finitely generated graded modules over a polynomial ring can be used to define the persistence barcode, which is a complete module invariant. If the topology construction method has more than one parameter, the constructed persistence module is called multidimensional. The classification theorem used in the case of ordinary persistence modules does not extend naturally, and neither does the definition of the persistence barcode. In fact, it is known that no complete invariant exists for multidimensional persistence modules, and as such, any incomplete invariant is considered useful. This thesis generalizes the theory of multidimensional persistence modules and introduces some new concepts in hopes of facilitating the discovery of new discrete invariants. The main contributions are 1) the basic study of generalized persistence modules - we view persistence modules as functors from a small category to the category of modules; 2) exterior critical series - a new invariant based on module presentation that is complete for the class of finitely presented one-dimensional per- sistence modules; 3) persistence reparametrization - a functorial process that can be used to reduce the persistence dimension and detect "persistent" features; 4) region encoding - a conservative generalization of the persistence barcode that can be de- fined for general persistence modules; 5) zigzag rank invariant - a generalization of the rank invariant that can be defined for zigzag persistence modules; and 6) de- tailed descriptions of algorithms to compute all aforementioned invariants and their complexity analysis.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2015 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Vongmasa, Pawin | |
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Associated with | Stanford University, Institute for Computational and Mathematical Engineering. | |

Primary advisor | Carlsson, Gunnar | |

Thesis advisor | Carlsson, Gunnar | |

Thesis advisor | Guibas, Leonidas J | |

Thesis advisor | Mazzeo, Rafe | |

Advisor | Guibas, Leonidas J | |

Advisor | Mazzeo, Rafe |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Pawin Vongmasa. |
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Note | Submitted to the Institute for Computational and Mathematical Engineering. |

Thesis | Thesis (Ph.D.)--Stanford University, 2015. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2015 by Pawin Vongmasa
- License
- This work is licensed under a Creative Commons Attribution Share Alike 3.0 Unported license (CC BY-SA).

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