Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Bubenik, Peter; Elchesen, Alex

doi:10.1007/s41468-022-00091-9

Cited by 11 publications

(7 citation statements)

References 23 publications

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“…More relevant for us is the recent article [8] by Bubenik and Elchesen. In this work, the group completion of the monoid of persistence diagrams is described, i.e., K 0 (Diag) is defined (semi-)explicitly.…”

Section: K-theory and Persistencementioning

confidence: 99%

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Zig-zag modules: cosheaves and $k$-theory

Grady,

Schenfisch

2023

Homology, Homotopy and Applications

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Section: K-theory and Persistencementioning

confidence: 99%

“…In [8], Bubenik and Elchesen describe the group completion of a monoid of persistence diagrams. The resulting equivalence classes are called virtual persistence diagrams and can be realized by extending the diagrams to include arbitrary points in the (extended) first quadrant, i.e., not just points above the diagonal.…”

Section: Virtual Diagramsmentioning

confidence: 99%

Zig-zag modules: cosheaves and $k$-theory

Grady,

Schenfisch

2023

Homology, Homotopy and Applications

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“…We denote by the result of applying the resulting Möbius inversion formula to the invariant . Invariants obtained in this way are sometimes referred to as “generalized persistence diagrams.” See, e.g., [AENYb, BBE22, BE22, BOO KM21, MP22, Pat18].…”

Section: Motivation and Related Invariantsmentioning

confidence: 99%

Homological approximations in persistence theory

BLANCHETTE¹,

Brüstle²,

Hanson³

2022

Can. J. Math.-J. Can. Math.

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We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement. Contents 1. Introduction 1 2. Notation and Terminology 4 3. Motivation and related invariants 6 4. Relative homological algebra 9 5. Classes and morphisms of spread modules. 14 6. Homological invariants in persistence theory 17 7. Examples and comparison to other invariants 19 References 23

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“…It follows that for all p ∈ [1, ∞], d p metrizes the quotient topology on X/A. Lemma 3.6 ([8,9]). Let (X, d, A) be a metric pair and 1 ≤ q ≤ p ≤ ∞.…”

Section: Basic Properties Of Spaces Of Persistence Diagramsmentioning

confidence: 99%

“…We call (D(X, A), W p ), a space of persistence diagrams. These metrics canonically extend to countable formal sums [9].…”

Section: Introductionmentioning

confidence: 99%

Topological and metric properties of spaces of generalized persistence diagrams

Bubenik¹,

Hartsock²

2022

Preprint

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Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is σ-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.

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Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Cited by 11 publications

References 23 publications

Zig-zag modules: cosheaves and $k$-theory

Zig-zag modules: cosheaves and $k$-theory

Homological approximations in persistence theory

Topological and metric properties of spaces of generalized persistence diagrams

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