$$A_\infty $$ A ∞ -persistence

Belchí, Francisco; Murillo, Aniceto

doi:10.1007/s00200-014-0241-4

Cited by 9 publications

(4 citation statements)

References 17 publications

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“…The possible algebraic (resp., coalgebraic) structures on persistent cohomology (resp., homology) have not deserved much attention up to the present time in the literature as far we know. We want to remark that in the preprint article [2] by F. Belchí and A. Murillo, the authors seem to proceed in our opinion in a different manner as ours (regardless of the fact they consider A ∞ -coalgebras, as pointed out in the last paragraph of this section), since they produce barcodes by means of the higher structure maps on the cohomology of the entire complex in a similar fashion to those produced in plain persistent homology theory. Our point of depart is somehow similar, since, in the case the binary operation on the totally ordered set P is given by taking minimum, the A ∞ -algebra structure of the cohomology H • (D) of the Rees dg algebra is obtained in essentially the same (but dual) manner as the A ∞ -coalgebra they consider (using the morphisms f i,i+1 of that article).…”

Section: Introductionmentioning

confidence: 78%

“…However, we believe that our manner of proceeding is more systematic and it clarifies the underlying structures, for it also indicates that, in order to study the higher multiplications in complete generality we should consider instead an intermediate object (see the last paragraph in Subsection 3.1). Furthermore, if the filtration under consideration is better behaved from the point of view of algebra (as the skeletal filtration), our A ∞ -algebra has more richer structure than (the dual to) that considered in [2]. Finally, in our case, we do not produce any further barcodes but we use the A ∞ -algebraic structure on the complete persistent cohomology group to compare the classic ones.…”

Section: Introductionmentioning

confidence: 99%

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A higher homotopic extension of persistent (co)homology

Herscovich

2017

J. Homotopy Relat. Struct.

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Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in R n induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvili to get a unique (up to noncanonical equivalence) A∞-algebra structure on the complete persistent cohomology of the filtered simplicial (or topological) set. We then provide a construction of a (pseudo)metric on the set of all (generalized) barcodes (that is, of all cohomological degrees) enriched with the A∞-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A∞-algebra structure chosen (among those equivalent to each other). We think that this distance might deserve some attention for topological data analysis, for it in particular can recognize different linking or foldings patterns, as in the Borromean rings. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology. This result was observed in [9] under some restricted assumptions, which we do not suppose.Mathematics subject classification 2010: 16E45, 16W70, 18G55, 55U10, 68U05.

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

A higher homotopic extension of persistent (co)homology

Herscovich

2017

J. Homotopy Relat. Struct.

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“…In Belchí and Stefanou (2021), the authors study the structure and stability of a family of barcodes that absorb information from an A ∞ -coalgebra structure on persistent homology. See also Belchí and Murillo (2015).…”

Section: Introductionmentioning

confidence: 99%

Persistent cup product structures and related invariants

Mémoli,

Stefanou,

Zhou

2023

J Appl. and Comput. Topology

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One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length $$\ell \ge 0$$ ℓ ≥ 0 and the other is the filtration parameter. This new persistence structure, called the persistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter $$\ell $$ ℓ , we obtain a 1-dimensional persistence module, called the persistent $$\ell $$ ℓ -cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a persistent invariant, which extends both the rank invariant (also referred to as persistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant.

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“…In persistent homology one studies particular structures coming from exact couples associated to a filtration of a complex (see [1], and references therein). As noted by F. Belchí and A. Murillo in [2], one may gain some profit from the A ∞ -(co)algebra structure on cohomology. Their point of view is however completely different from ours (regardless of the fact they consider A ∞ -coalgebras, as pointed out in the last paragraph of this section), since they produce bars codes by means of the higher structure maps on the cohomology of the entire complex in a similar fashion to those produced in plain persistent cohomology theory.…”

Section: Introductionmentioning

confidence: 99%

A-infinity-algebras, spectral sequences and exact couples

Herscovich

2014

Preprint

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We study in this article a possible further structure of homotopic nature on multiplicative spectral sequences. More precisely, since Kadeishvili's theorem asserts that, given a dg (or A∞-)algebra, its cohomology has also a structure of A∞-algebra such that both become quasi-isomorphic, and in a multiplicative spectral sequence one considers the cohomology of dg algebras when moving from a term to the next one, a natural problem that arises is to study how this two possible structures intertwine. We give such a homotopic structure proposal, called A∞-enhancement of multiplicative spectral sequences, which could be of interest in our opinion. As far we know, this construction was studied only recently by S. Lapin, even though he did not state any definition. Seeing that the procedure considered by Lapin is rather complicated to handle, we propose an equivalent but in our opinion easier approach. In particular, from our definition we show that the canonical multiplicative spectral sequence obtained from a filtered dg (or A∞-)algebra, which could be viewed as the main example, has such an A∞-enhancement.

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$$A_\infty $$ A ∞ -persistence

Cited by 9 publications

References 17 publications

A higher homotopic extension of persistent (co)homology

A higher homotopic extension of persistent (co)homology

Persistent cup product structures and related invariants

A-infinity-algebras, spectral sequences and exact couples

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