## Homological Illusions of Persistence and Stability (2008)

Citations: | 12 - 3 self |

### BibTeX

@TECHREPORT{Morozov08homologicalillusions,

author = {Dmitriy Morozov},

title = {Homological Illusions of Persistence and Stability},

institution = {},

year = {2008}

}

### OpenURL

### Abstract

In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram. We start with the question of ridding the function of topological noise as suggested by its persistence diagram. We give an algorithm for hierarchically finding such ε-simplifications on 2-manifolds as well as answer the question of when it is impossible to simplify a function in higher dimensions. We continue by examining time-varying functions. The original algorithm computes the persistence pairing from an ordering of the simplices in a triangulation and takes worstcase time cubic in the number of simplices. We describe how to maintain the pairing in linear time per transposition of consecutive simplices. A side effect of the update algorithm is an elementary proof of the stability of persistence diagrams. We introduce a parametrized family of persistence diagrams called persistence vineyards and illustrate the concept with a vineyard describing a folding of a small peptide. We also base a simple algorithm to compute the rank invariant of a collection of functions on the update procedure.