25,461 research outputs found
Computing Multidimensional Persistence
The theory of multidimensional persistence captures the topology of a
multifiltration -- a multiparameter family of increasing spaces.
Multifiltrations arise naturally in the topological analysis of scientific
data. In this paper, we give a polynomial time algorithm for computing
multidimensional persistence. We recast this computation as a problem within
computational algebraic geometry and utilize algorithms from this area to solve
it. While the resulting problem is Expspace-complete and the standard
algorithms take doubly-exponential time, we exploit the structure inherent
withing multifiltrations to yield practical algorithms. We implement all
algorithms in the paper and provide statistical experiments to demonstrate
their feasibility.Comment: This paper has been withdrawn by the authors. Journal of
Computational Geometry, 1(1) 2010, pages 72-100.
http://jocg.org/index.php/jocg/article/view/1
A new approximation Algorithm for the Matching Distance in Multidimensional Persistence
Topological Persistence has proven to be a promising framework for dealing with problems concerning shape analysis and comparison. In this contexts, it was originally introduced by taking into account 1-dimensional properties of shapes, modeled by real-valued functions. More recently, Topological Persistence has been generalized to consider multidimensional properties of shapes, coded by vector-valued functions. This extension has led to introduce suitable shape descriptors, named the multidimensional persistence Betti numbers functions, and a distance to compare them, the so-called multidimensional matching distance. In this paper we propose a new computational framework to deal with the multidimensional matching distance. We start by proving some new theoretical results, and then we use them to formulate an algorithm for computing such a distance up to an arbitrary threshold error
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
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
Stable comparison of multidimensional persistent homology groups with torsion
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made
Stable comparison of multidimensional persistent homology groups with torsion
The present lack of a stable method to compare persistent homology groups
with torsion is a relevant problem in current research about Persistent
Homology and its applications in Pattern Recognition. In this paper we
introduce a pseudo-distance d_T that represents a possible solution to this
problem. Indeed, d_T is a pseudo-distance between multidimensional persistent
homology groups with coefficients in an Abelian group, hence possibly having
torsion. Our main theorem proves the stability of the new pseudo-distance with
respect to the change of the filtering function, expressed both with respect to
the max-norm and to the natural pseudo-distance between topological spaces
endowed with vector-valued filtering functions. Furthermore, we prove a result
showing the relationship between d_T and the matching distance in the
1-dimensional case, when the homology coefficients are taken in a field and
hence the comparison can be made.Comment: 10 pages, 3 figure
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