34,647 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
Induced Matchings and the Algebraic Stability of Persistence Barcodes
We define a simple, explicit map sending a morphism of
pointwise finite dimensional persistence modules to a matching between the
barcodes of and . Our main result is that, in a precise sense, the
quality of this matching is tightly controlled by the lengths of the longest
intervals in the barcodes of and . As an
immediate corollary, we obtain a new proof of the algebraic stability of
persistence, a fundamental result in the theory of persistent homology. In
contrast to previous proofs, ours shows explicitly how a -interleaving
morphism between two persistence modules induces a -matching between
the barcodes of the two modules. Our main result also specializes to a
structure theorem for submodules and quotients of persistence modules, and
yields a novel "single-morphism" characterization of the interleaving relation
on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational
Geometry. Includes a proof that no definition of induced matching can be
fully functorial (Proposition 5.10), and an extension of our single-morphism
characterization of the interleaving relation to multidimensional persistence
modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde
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
Metrics for generalized persistence modules
We consider the question of defining interleaving metrics on generalized
persistence modules over arbitrary preordered sets. Our constructions are
functorial, which implies a form of stability for these metrics. We describe a
large class of examples, inverse-image persistence modules, which occur
whenever a topological space is mapped to a metric space. Several standard
theories of persistence and their stability can be described in this framework.
This includes the classical case of sublevelset persistent homology. We
introduce a distinction between `soft' and `hard' stability theorems. While our
treatment is direct and elementary, the approach can be explained abstractly in
terms of monoidal functors.Comment: Final version; no changes from previous version. Published online Oct
2014 in Foundations of Computational Mathematics. Print version to appea
The observable structure of persistence modules
In persistent topology, q-tame modules appear as a natural and large class of
persistence modules indexed over the real line for which a persistence diagram
is definable. However, unlike persistence modules indexed over a totally
ordered finite set or the natural numbers, such diagrams do not provide a
complete invariant of q-tame modules. The purpose of this paper is to show that
the category of persistence modules can be adjusted to overcome this issue. We
introduce the observable category of persistence modules: a localization of the
usual category, in which the classical properties of q-tame modules still hold
but where the persistence diagram is a complete isomorphism invariant and all
q-tame modules admit an interval decomposition
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