Year: 2008
Type: dissertation
Abstract: In this thesis we explore and extend the theory of persistent homology, which\ncaptures topological features of a function by pairing its critical values. The\nresult is represented by a collection of points in the extended plane called\npersistence diagram. We start with the question of ridding the function of topological noise as\nsuggested by its persistence diagram. We give an algorithm for hierarchically\nfinding such epsilon-simplifications on 2-manifolds as well as answer the question\nof when it is impossible to simplify a function in higher dimensions. We continue by examining time-varying functions. The original algorithm computes\nthe persistence pairing from an ordering of the simplices in a triangulation and\ntakes worst-case time cubic in the number of simplices. We describe how to\nmaintain the pairing in linear time per transposition of consecutive simplices.\nA side effect of the update algorithm is an elementary proof of the stability of\npersistence diagrams. We introduce a parametrized family of persistence diagrams\ncalled persistence vineyards and illustrate the concept with a vineyard\ndescribing a folding of a small peptide. We also base a simple algorithm to\ncompute the rank invariant of a collection of functions on the update procedure. Guided by the desire to reconstruct stratified spaces from noisy samples, we use\nthe vineyard of the distance function restricted to a 1-parameter family of\nneighborhoods of a point to assess the local homology of a sampled stratified space at\nthat point.\nWe prove the correctness of\nthis assessment under the assumption of a sufficiently dense sample. We also\ngive an algorithm that constructs the vineyard and makes the local assessment in\ntime at most cubic in the size of the Delaunay triangulation of the point\nsample. Finally, to refine the measurement of local homology the thesis extends the\nnotion of persistent homology\nto sequences of kernels, images, and cokernels of maps induced\nby inclusions in a filtration of pairs of spaces.\nSpecifically,\nwe note that persistence in this context is well defined,\nwe prove that the persistence diagrams are stable,\nand we explain how to compute them.\nAdditionally, we use image persistence to cope with functions on noisy domains.
Source: DukeSpace (Duke University)
Authors: Herbert Edelsbrunner, Dmitriy Morozov
Institution: Duke University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2115896258