199 research outputs found
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
Persistence for Circle Valued Maps
We study circle valued maps and consider the persistence of the homology of
their fibers. The outcome is a finite collection of computable invariants which
answer the basic questions on persistence and in addition encode the topology
of the source space and its relevant subspaces. Unlike persistence of real
valued maps, circle valued maps enjoy a different class of invariants called
Jordan cells in addition to bar codes. We establish a relation between the
homology of the source space and of its relevant subspaces with these
invariants and provide a new algorithm to compute these invariants from an
input matrix that encodes a circle valued map on an input simplicial complex.Comment: A complete algorithm to compute barcodes and Jordan cells is provided
in this version. The paper is accepted in in the journal Discrete &
Computational Geometry. arXiv admin note: text overlap with arXiv:1210.3092
by other author
Computing Topological Persistence for Simplicial Maps
Algorithms for persistent homology and zigzag persistent homology are
well-studied for persistence modules where homomorphisms are induced by
inclusion maps. In this paper, we propose a practical algorithm for computing
persistence under coefficients for a sequence of general
simplicial maps and show how these maps arise naturally in some applications of
topological data analysis.
First, we observe that it is not hard to simulate simplicial maps by
inclusion maps but not necessarily in a monotone direction. This, combined with
the known algorithms for zigzag persistence, provides an algorithm for
computing the persistence induced by simplicial maps.
Our main result is that the above simple minded approach can be improved for
a sequence of simplicial maps given in a monotone direction. A simplicial map
can be decomposed into a set of elementary inclusions and vertex collapses--two
atomic operations that can be supported efficiently with the notion of simplex
annotations for computing persistent homology. A consistent annotation through
these atomic operations implies the maintenance of a consistent cohomology
basis, hence a homology basis by duality. While the idea of maintaining a
cohomology basis through an inclusion is not new, maintaining them through a
vertex collapse is new, which constitutes an important atomic operation for
simulating simplicial maps. Annotations support the vertex collapse in addition
to the usual inclusion quite naturally.
Finally, we exhibit an application of this new tool in which we approximate
the persistence diagram of a filtration of Rips complexes where vertex
collapses are used to tame the blow-up in size.Comment: This is the revised and full version of the paper that is going to
appear in the Proceedings of 30th Annual Symposium on Computational Geometr
A Geometric Perspective on Sparse Filtrations
We present a geometric perspective on sparse filtrations used in topological
data analysis. This new perspective leads to much simpler proofs, while also
being more general, applying equally to Rips filtrations and Cech filtrations
for any convex metric. We also give an algorithm for finding the simplices in
such a filtration and prove that the vertex removal can be implemented as a
sequence of elementary edge collapses
Computational Complexity of the Interleaving Distance
The interleaving distance is arguably the most prominent distance measure in
topological data analysis. In this paper, we provide bounds on the
computational complexity of determining the interleaving distance in several
settings. We show that the interleaving distance is NP-hard to compute for
persistence modules valued in the category of vector spaces. In the specific
setting of multidimensional persistent homology we show that the problem is at
least as hard as a matrix invertibility problem. Furthermore, this allows us to
conclude that the interleaving distance of interval decomposable modules
depends on the characteristic of the field. Persistence modules valued in the
category of sets are also studied. As a corollary, we obtain that the
isomorphism problem for Reeb graphs is graph isomorphism complete.Comment: Discussion related to the characteristic of the field added. Paper
accepted to the 34th International Symposium on Computational Geometr
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
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