18 research outputs found
The persistence landscape and some of its properties
Persistence landscapes map persistence diagrams into a function space, which
may often be taken to be a Banach space or even a Hilbert space. In the latter
case, it is a feature map and there is an associated kernel. The main advantage
of this summary is that it allows one to apply tools from statistics and
machine learning. Furthermore, the mapping from persistence diagrams to
persistence landscapes is stable and invertible. We introduce a weighted
version of the persistence landscape and define a one-parameter family of
Poisson-weighted persistence landscape kernels that may be useful for learning.
We also demonstrate some additional properties of the persistence landscape.
First, the persistence landscape may be viewed as a tropical rational function.
Second, in many cases it is possible to exactly reconstruct all of the
component persistence diagrams from an average persistence landscape. It
follows that the persistence landscape kernel is characteristic for certain
generic empirical measures. Finally, the persistence landscape distance may be
arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu
Hierarchies and Ranks for Persistence Pairs
We develop a novel hierarchy for zero-dimensional persistence pairs, i.e.,
connected components, which is capable of capturing more fine-grained spatial
relations between persistence pairs. Our work is motivated by a lack of spatial
relationships between features in persistence diagrams, leading to a limited
expressive power. We build upon a recently-introduced hierarchy of pairs in
persistence diagrams that augments the pairing stored in persistence diagrams
with information about which components merge. Our proposed hierarchy captures
differences in branching structure. Moreover, we show how to use our hierarchy
to measure the spatial stability of a pairing and we define a rank function for
persistence pairs and demonstrate different applications.Comment: Topology-based Methods in Visualization 201