The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Lesnick, Michael

doi:10.1007/s10208-015-9255-y

Cited by 182 publications

(248 citation statements)

References 46 publications

Supporting

Mentioning

244

Contrasting

Order By: Relevance

“…3 Moreover, these diagrams admit a bottleneck distance d Bot and the following isometry theorem establishes that the assignment dgm which sends a (tame) persistence module to its corresponding diagram preserves distances. Theorem 2.6 (see [6,10,19]). The equality…”

Section: The Interpolation Lemmamentioning

confidence: 99%

Higher Interpolation and Extension for Persistence Modules

Bubenik¹,

Silva²,

Nanda³

2017

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

Abstract. The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of nonexpansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. Our main construction gives an isometric embedding of a metric space into the metric space of persistence modules with values in the spacetime of this metric space. As a consequence of such "higher interpolation," it becomes possible to compare Vietoris-Rips andČech complexes built within the space of persistence modules. [7,16,22]. The output of this process is a collection of persistent homology groups, which are typically represented via a barcode or a persistence diagram. Recent applications of persistence often confront dynamically evolving data [2,17], and in these cases one requires the ability to make inferences about the dynamics from collections of persistence diagrams. Substantial efforts have been devoted to this end; among the best-known outcomes are vineyards [12], Fréchet means [24], and persistence landscapes [4].In this work, we provide a new geometric lens with which to view the space of persistence diagrams. Our main result is in fact a statement about the space of (sufficiently tame) persistence modules-these consist of vector spaces and linear maps indexed by the real line R, and their representation theory produces persistence diagrams [23]. The class Mod of persistence modules admits an interleaving metric, and the interpolation lemma from [8] establishes that Mod is a path metric space-two modules which are e-interleaved for 0 ≤ e < ∞ can always be connected by a path in Mod of length e. This lemma plays a key role in

show abstract

Section: The Interpolation Lemmamentioning

confidence: 99%

Higher Interpolation and Extension for Persistence Modules

Bubenik¹,

Silva²,

Nanda³

2017

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

show abstract

“…Questions of stability for multi-dimensional persistence modules have been studied, both in the size function community ( [14,15]) and in the context of persistent homology by Lesnick [53]. 4.2.3.…”

Section: 22mentioning

confidence: 99%

“…Stability. Based on this machinery, the authors are able to prove a number of stability-related theorems, that all lead to the fundamental isometry theorem, occuring in [18, Theorem 4.11], and also proven independently by Lesnick [53]:…”

Section: Order Module View Of Interleavingmentioning

confidence: 99%

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

Vejdemo-Johansson¹

2014

Algebraic Topology: Applications and New Directions

View full text Add to dashboard Cite

Abstract. The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results.In this survey paper, we shall examine the various choices in use, and what they allow us to prove. We shall also discuss the inherent differences between the choices people use, and speculate on potential directions of research to resolve these differences.Johnstone [51] named his book on topos theory "Sketches of an elephant", referencing a joke: three blind wise men encounter an elephant. They each try to describe it to each other. The wise man who caught hold of the elephant's trunk says "An elephant is like a snake."; the wise man holding the ear says "An elephant is like a palm leaf."; and the wise man holding its leg says "An elephant is like a tree.".The joke is highly relevant to topos theory; which has its roots in logic, in geometry, and in topology, with the three perspectives being fundamentally different and enriching each other in surprising ways.The title of this paper is similar, but different. The platypus is well-known to be a hybrid of an animal: sharing traits both with the phylum of birds and with the phylum of mammals. The field of persistent homology is in a similar situation to the platypus: there are at least two different viewpoints of what persistent homology should be, and they interact in sometimes unexpected ways.

show abstract

“…The stability theorem of [7] asserts (under some restrictions on the modules) that the map [13] it was shown to be an isometry. For proofs in the case of q-tame modules, see [6].…”

Section: Diagramsmentioning

confidence: 99%

“…The isometry theorem of [7,13,2,6] amounts to the following specific assertions about q-tame modules and their diagrams:…”

Section: Interleavings and Matchingsmentioning

confidence: 99%

Geometry in the space of persistence modules

Silva

Nanda

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

Topological persistence is, by now, an established paradigm for constructing robust topological invariants from pointcloud data: the data are converted into a filtered simplicial complex, the complex gives rise to a persistence module, and the module is described by a persistence diagram. In this paper, we study the geometry of the spaces of persistence modules and diagrams, with special attention toČech and Rips complexes. The metric structures are determined in terms of interleaving maps between modules and matchings between diagrams. We show that the relationship between theČech and Rips complexes is governed by certain 'coherence' conditions on the corresponding families of interleavings or matchings.

show abstract

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Cited by 182 publications

References 46 publications

Higher Interpolation and Extension for Persistence Modules

Higher Interpolation and Extension for Persistence Modules

Sketches of a platypus: a survey of persistent homology and its algebraic foundations

Geometry in the space of persistence modules

Contact Info

Product

Resources

About