13,355 research outputs found
Persistence modules with operators in Morse and Floer theory
We introduce a new notion of persistence modules endowed with operators. It
encapsulates the additional structure on Floer-type persistence modules coming
from the intersection product with classes in the ambient (quantum) homology,
along with a few other geometric situations. We provide sample applications to
the -geometry of Morse functions and to Hofer's geometry of Hamiltonian
diffeomorphisms, that go beyond spectral invariants and traditional persistent
homology.Comment: 31 pages, 4 figure
A Comparison Framework for Interleaved Persistence Modules
We present a generalization of the induced matching theorem and use it to
prove a generalization of the algebraic stability theorem for
-indexed pointwise finite-dimensional persistence modules. Via
numerous examples, we show how the generalized algebraic stability theorem
enables the computation of rigorous error bounds in the space of persistence
diagrams that go beyond the typical formulation in terms of bottleneck (or log
bottleneck) distance
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
In 2009, Chazal et al. introduced -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and generalize readily to multidimensional
persistence modules. In this paper, we develop the theory of multidimensional
interleavings, with a view towards applications to topological data analysis.
We present four main results. First, we show that on 1-D persistence modules,
is equal to the bottleneck distance . This result, which first
appeared in an earlier preprint of this paper, has since appeared in several
other places, and is now known as the isometry theorem. Second, we present a
characterization of the -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that restricts to a metric on isomorphism
classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in
Foundations of Computational Mathematics. 36 page
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