Metrics for Generalized Persistence Modules

Bubenik, Peter; Silva, Vin; Scott, Jonathan

doi:10.1007/s10208-014-9229-5

Cited by 112 publications

(163 citation statements)

References 26 publications

Supporting

Mentioning

162

Contrasting

Order By: Relevance

“…(1) is a direct consequence of the definitions; (2), (3), (4), (5), and (6) follow from (1), and (7) from (6).…”

Section: The Shift Is Linearmentioning

confidence: 99%

Metrics and Stabilization in One Parameter Persistence

Chachólski¹,

Riihimäki²

2020

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

We propose the use of persistent homology in a supervised way. We believe homological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to a model. Fitting this model is what we believe supervised homological persistence is. We develop theory behind constructing such models and we give evidence of the usefulness of this approach in concrete data analysis tasks.

show abstract

“…(1) is a direct consequence of the definitions; (2), (3), (4), (5), and (6) follow from (1), and (7) from (6).…”

Section: The Shift Is Linearmentioning

confidence: 99%

Metrics and Stabilization in One Parameter Persistence

Chachólski¹,

Riihimäki²

2020

SIAM J. Appl. Algebra Geometry

View full text Add to dashboard Cite

show abstract

“…5 Given a 1-Lipschitz map f ∈ Met(M, N ), define f R : M R → N R via the mapping (x, s) → (f (x), s). Proof.…”

Section: The Interpolation Lemmamentioning

confidence: 99%

“…For a small category C, we will denote the category of functors from C to D by D C . Although we will survey some relevant definitions and results here, the reader is invited to consult [5,6,8,10,23] for detailed background material on the categorical and metric aspects of persistence modules.…”

mentioning

confidence: 99%

Higher Interpolation and Extension for Persistence Modules

Bubenik¹,

Silva²,

Nanda³

2017

SIAM J. Appl. Algebra Geometry

Self Cite

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

“…1.1 Related work de Silva et al (2016) showed that any Reeb graph can be identified with a constructible Set-valued cosheaf on R [16]. Thus, Reeb graphs can be thought of as generalized persistence modules in the setting of Bubenik et al (2015) [8]. A generalized persistence module is any functor F : P → C from a poset P to a category C [8].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, Reeb graphs can be thought of as generalized persistence modules in the setting of Bubenik et al (2015) [8]. A generalized persistence module is any functor F : P → C from a poset P to a category C [8]. When P = (R, ≤) the poset of real numbers and C = vect k is the category of finite dimensional k-vector spaces, we obtain the notion of a pointwise finite dimensional (p.f.d.)…”

Section: Introductionmentioning

confidence: 99%

Tree decomposition of Reeb graphs, parametrized complexity, and applications to phylogenetics

Stefanou

2020

J Appl. and Comput. Topology

View full text Add to dashboard Cite

Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s, is equal to a coproduct of at most 2 s trees with (n + s) leaves. Reeb graphs are therefore classified up to isomorphism by their tree decomposition. An implication of this result, is that the isomorphism problem for Reeb graphs is fixed parameter tractable when the parameter is the first Betti number. We propose ordered Reeb graphs as a model for time consistent phylogenetic networks and propose a certain Hausdorff distance as a metric on these structures.

show abstract

Metrics for Generalized Persistence Modules

Cited by 112 publications

References 26 publications

Metrics and Stabilization in One Parameter Persistence

Metrics and Stabilization in One Parameter Persistence

Higher Interpolation and Extension for Persistence Modules

Tree decomposition of Reeb graphs, parametrized complexity, and applications to phylogenetics

Contact Info

Product

Resources

About