7,893 research outputs found
Persistence modules, shape description, and completeness
Persistence modules are algebraic constructs that can be used to describe the shape of an object starting from a geometric representation of it. As shape descriptors, persistence modules are not complete, that is they may not distinguish non-equivalent shapes.
In this paper we show that one reason for this is that homomorphisms between persistence modules forget the geometric nature of the problem. Therefore we introduce geometric homomorphisms between persistence modules, and show that in some cases they perform better. A combinatorial structure, the -tree, is shown to be an invariant for geometric isomorphism classes in the case of persistence modules obtained through the 0th persistent homology functor
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
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
Early aspects: aspect-oriented requirements engineering and architecture design
This paper reports on the third Early Aspects: Aspect-Oriented Requirements Engineering and Architecture Design Workshop, which has been held in Lancaster, UK, on March 21, 2004. The workshop included a presentation session and working sessions in which the particular topics on early aspects were discussed. The primary goal of the workshop was to focus on challenges to defining methodical software development processes for aspects from early on in the software life cycle and explore the potential of proposed methods and techniques to scale up to industrial applications
Phase extraction in disordered isospectral shapes
The phase of the electronic wave function is not directly measurable but,
quite remarkably, it becomes accessible in pairs of isospectral shapes, as
recently proposed in the experiment of Christopher R. Moon {\it et al.},
Science {\bf 319}, 782 (2008). The method is based on a special property,
called transplantation, which relates the eigenfunctions of the isospectral
pairs, and allows to extract the phase distributions, if the amplitude
distributions are known. We numerically simulate such a phase extraction
procedure in the presence of disorder, which is introduced both as Anderson
disorder and as roughness at edges. With disorder, the transplantation can no
longer lead to a perfect fit of the wave functions, however we show that a
phase can still be extracted - defined as the phase that minimizes the misfit.
Interestingly, this extracted phase coincides with (or differs negligibly from)
the phase of the disorder-free system, up to a certain disorder amplitude, and
a misfit of the wave functions as high as , proving a robustness of
the phase extraction method against disorder. However, if the disorder is
increased further, the extracted phase shows a puzzle structure, no longer
correlated with the phase of the disorder-free system. A discrete model is
used, which is the natural approach for disorder analysis. We provide a proof
that discretization preserves isospectrality and the transplantation can be
adapted to the discrete systems.Comment: Accepted for Phys.Rev.
Wasserstein Stability for Persistence Diagrams
The stability of persistence diagrams is among the most important results in
applied and computational topology. Most results in the literature phrase
stability in terms of the bottleneck distance between diagrams and the
-norm of perturbations. This has two main implications: it makes the
space of persistence diagrams rather pathological and it is often provides very
pessimistic bounds with respect to outliers. In this paper, we provide new
stability results with respect to the -Wasserstein distance between
persistence diagrams. This includes an elementary proof for the setting of
functions on sufficiently finite spaces in terms of the -norm of the
perturbations, along with an algebraic framework for -Wasserstein distance
which extends the results to wider class of modules. We also provide apply the
results to a wide range of applications in topological data analysis (TDA)
including topological summaries, persistence transforms and the special but
important case of Vietoris-Rips complexes
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