Abstract:
<p>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.</p><p>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 epsilon-simplifications on 2-manifolds as well as answer the question
of when it is impossible to simplify a function in higher dimensions.</p><p>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 worst-case 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.</p><p>Guided by the desire to reconstruct stratified spaces from noisy samples, we use
the vineyard of the distance function restricted to a 1-parameter family of
neighborhoods of a point to assess the local homology of a sampled stratified space at
that point.
We prove the correctness of
this assessment under the assumption of a sufficiently dense sample. We also
give an algorithm that constructs the vineyard and makes the local assessment in
time at most cubic in the size of the Delaunay triangulation of the point
sample.</p><p>Finally, to refine the measurement of local homology the thesis extends the
notion of persistent homology
to sequences of kernels, images, and cokernels of maps induced
by inclusions in a filtration of pairs of spaces.
Specifically,
we note that persistence in this context is well defined,
we prove that the persistence diagrams are stable,
and we explain how to compute them.
Additionally, we use image persistence to cope with functions on noisy domains.</p>
