366 research outputs found
Dualities in persistent (co)homology
We consider sequences of absolute and relative homology and cohomology groups
that arise naturally for a filtered cell complex. We establish algebraic
relationships between their persistence modules, and show that they contain
equivalent information. We explain how one can use the existing algorithm for
persistent homology to process any of the four modules, and relate it to a
recently introduced persistent cohomology algorithm. We present experimental
evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue
on Topological Data Analysi
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
Positive Alexander Duality for Pursuit and Evasion
Considered is a class of pursuit-evasion games, in which an evader tries to
avoid detection. Such games can be formulated as the search for sections to the
complement of a coverage region in a Euclidean space over a timeline. Prior
results give homological criteria for evasion in the general case that are not
necessary and sufficient. This paper provides a necessary and sufficient
positive cohomological criterion for evasion in a general case. The principal
tools are (1) a refinement of the Cech cohomology of a coverage region with a
positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore
homology of the coverage gaps with a positive cone encoding time orientation,
and (3) a positive variant of Alexander Duality. Positive cohomology decomposes
as the global sections of a sheaf of local positive cohomology over the time
axis; we show how this decomposition makes positive cohomology computable as a
linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive
(co)homology generalized to arbitrary degrees; Positive Alexander Duality
generalized from homological degrees 0,1; Morse and smoothness conditions
generalized; illustrations of positive homology added. minor corrections in
proofs, notation, organization, and language made throughout. variant of
Borel-Moore homology now use
Topics in Persistent Homology: From Morse Theory for Minimal Surfaces to Efficient Computation of Image Persistence
We study some problems and develop some theory related to persistent homology, separated into two lines of investigation.
In the first part, we introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent
(co)homology, generalizing previous results in terms of barcodes by de Silva, Morozov, and Vejdemo-Johansson. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for an efficient algorithm to compute barcodes of images and induced matchings of such morphisms, which performs computations in terms of relative cohomology and then translates to absolute homology via the aforementioned dualities. Our method is based on a previous algorithm by Cohen-Steiner, Edelsbrunner, Harer, and Morozov that did not make use of relative cohomology. Using it is crucial, however, because our algorithm applies the clearing optimization introduced by Chen and Kerber, which works particularly well in the context of relative cohomology. We provide an implementation of our algorithm for inclusions of filtrations of Vietoris–Rips complexes in the framework of the software Ripser by Ulrich Bauer.
In the second part, we introduce local connectedness conditions on a broad class of functionals that ensure that the persistent homology of their associated sublevel set filtration is q-tame, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the unstable minimal surface theorem given by Morse and Tompkins in terms of persistent Čech homology in a modern and rigorous framework. Moreover, we show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a filtration consisting of paracompact Hausdorff spaces is 0 if it satisfies a similar local connectedness condition to the one used to ensure q-tameness, generalizing a result by Mardešić for locally connected spaces to the setting of filtrations. In contrast to singular homology, the persistent Čech homology of a compact filtration is always upper semi-continuous, which has structural implications in the q-tame case: using a result by Chazal, Crawley-Boevey, and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper semi-continuous q-tame persistence module can be decomposed as a product of interval modules
Clear and Compress: Computing Persistent Homology in Chunks
We present a parallelizable algorithm for computing the persistent homology
of a filtered chain complex. Our approach differs from the commonly used
reduction algorithm by first computing persistence pairs within local chunks,
then simplifying the unpaired columns, and finally applying standard reduction
on the simplified matrix. The approach generalizes a technique by G\"unther et
al., which uses discrete Morse Theory to compute persistence; we derive the
same worst-case complexity bound in a more general context. The algorithm
employs several practical optimization techniques which are of independent
interest. Our sequential implementation of the algorithm is competitive with
state-of-the-art methods, and we improve the performance through parallelized
computation.Comment: This result was presented at TopoInVis 2013
(http://www.sci.utah.edu/topoinvis13.html
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