78,032 research outputs found
The structure and stability of persistence modules
We give a self-contained treatment of the theory of persistence modules
indexed over the real line. We give new proofs of the standard results.
Persistence diagrams are constructed using measure theory. Linear algebra
lemmas are simplified using a new notation for calculations on quiver
representations. We show that the stringent finiteness conditions required by
traditional methods are not necessary to prove the existence and stability of
the persistence diagram. We introduce weaker hypotheses for taming persistence
modules, which are met in practice and are strong enough for the theory still
to work. The constructions and proofs enabled by our framework are, we claim,
cleaner and simpler.Comment: New version. We discuss in greater depth the interpolation lemma for
persistence module
Parametrized Homology via Zigzag Persistence
This paper develops the idea of homology for 1-parameter families of
topological spaces. We express parametrized homology as a collection of real
intervals with each corresponding to a homological feature supported over that
interval or, equivalently, as a persistence diagram. By defining persistence in
terms of finite rectangle measures, we classify barcode intervals into four
classes. Each of these conveys how the homological features perish at both ends
of the interval over which they are defined
Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are
common objects in the computer graphics or topological data analysis
literature. Defining good metrics between these objects has become an important
question for applications, where it matters to quantify the extent by which two
given Reeb graphs differ. Recent contributions emphasize this aspect, proposing
novel distances such as {\em functional distortion} or {\em interleaving} that
are provably more discriminative than the so-called {\em bottleneck distance},
being true metrics whereas the latter is only a pseudo-metric. Their main
drawback compared to the bottleneck distance is to be comparatively hard (if at
all possible) to evaluate. Here we take the opposite view on the problem and
show that the bottleneck distance is in fact good enough {\em locally}, in the
sense that it is able to discriminate a Reeb graph from any other Reeb graph in
a small enough neighborhood, as efficiently as the other metrics do. This
suggests considering the {\em intrinsic metrics} induced by these distances,
which turn out to be all {\em globally} equivalent. This novel viewpoint on the
study of Reeb graphs has a potential impact on applications, where one may not
only be interested in discriminating between data but also in interpolating
between them
Quantifying Transversality by Measuring the Robustness of Intersections
By definition, transverse intersections are stable under infinitesimal
perturbations. Using persistent homology, we extend this notion to a measure.
Given a space of perturbations, we assign to each homology class of the
intersection its robustness, the magnitude of a perturbations in this space
necessary to kill it, and prove that robustness is stable. Among the
applications of this result is a stable notion of robustness for fixed points
of continuous mappings and a statement of stability for contours of smooth
mappings
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
- …