Corpus ID: 119677946

# Interleaving Distance as a Limit

@article{Meehan2017InterleavingDA,
title={Interleaving Distance as a Limit},
author={Killian Meehan and David C. Meyer},
journal={arXiv: Algebraic Topology},
year={2017}
}
• Published 30 October 2017
• Mathematics
• arXiv: Algebraic Topology
Persistent homology is a way of determining the topological properties of a data set. It is well known that each persistence module admits the structure of a representation of a finite totally ordered set. In previous work, the authors proved an analogue of the isometry theorem of Bauer and Lesnick for representations of a certain class of finite posets. The isometry was between the interleaving metric of Bubenik, de Silva and Scott and a bottleneck metric which incorporated algebraic… Expand
5 Citations
An Isometry Theorem for Generalized Persistence Modules
• Mathematics
• 2017
In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules canExpand
Tracking the variety of interleavings.
• Mathematics
• 2020
In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules featureExpand
A T ] 2 5 O ct 2 01 8 TOPOLOGICAL SPACES OF PERSISTENCE MODULES AND THEIR PROPERTIES
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. WeExpand
Topological spaces of persistence modules and their properties
• Mathematics, Computer Science
• J. Appl. Comput. Topol.
• 2018
This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties. Expand
Data Analysis Using Representation Theory and Clustering Algorithms
• Computer Science
• WSEAS TRANSACTIONS ON COMPUTERS
• 2021
This work aims to expand the knowledge of the area of data analysis through both persistence homology, as well as representations of directed graphs by finding the agglomerative hierarchical clustering method that will be the optimal clustering algorithm among these three; K-Means, PAM, and Random Forest methods. Expand

#### References

SHOWING 1-10 OF 20 REFERENCES
An Isometry Theorem for Generalized Persistence Modules
• Mathematics
• 2017
In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules canExpand
Induced matchings and the algebraic stability of persistence barcodes
• Mathematics, Computer Science
• J. Comput. Geom.
• 2015
This work shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $delta-matching between the barcodes of the two modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules. Expand Algebraic Stability of Zigzag Persistence Modules • Mathematics, Computer Science • Algebraic & Geometric Topology • 2018 This paper functorially extends each zigzag persistence module to a two-dimensional persistence module, and establishes an algebraic stability theorem for these extensions, which yields a stability result for free two- dimensional persistence modules. Expand Metrics for Generalized Persistence Modules • Computer Science, Mathematics • Found. Comput. Math. • 2015 This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems. Expand The theory of multidimensional persistence • Mathematics, Computer Science • SCG '07 • 2007 This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension. Expand Persistence Modules on Commutative Ladders of Finite Type • Mathematics, Computer Science • Discret. Comput. Geom. • 2016 It is proved that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander–Reiten quivers. Expand Persistence stability for geometric complexes • Mathematics, Computer Science • ArXiv • 2012 The properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces are studied. Expand The Theory of the Interleaving Distance on Multidimensional Persistence Modules • M. Lesnick • Mathematics, Computer Science • Found. Comput. Math. • 2015 The theory of multidimensional interleavings is developed, with a view toward applications to topological data analysis, and it is shown that when the authors define their persistence modules over a prime field, d_\mathrm{I}$\$dI satisfies a universality property. Expand
Stability of persistence diagrams
• Mathematics, Computer Science
• SCG
• 2005
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagramExpand
On the Local Behavior of Spaces of Natural Images
• Mathematics, Computer Science
• International Journal of Computer Vision
• 2007
A theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle and a polynomial representation is used to give coordinatization to various subspaces ofℳ. Expand