On Vietoris–Rips complexes of ellipses

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On Vietoris–Rips complexes of ellipses. / Adamaszek, Michał; Adams, Henry; Reddy, Samadwara.

I: Journal of Topology and Analysis, Bind 11, Nr. 3, 2019, s. 661-690.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Adamaszek, M, Adams, H & Reddy, S 2019, 'On Vietoris–Rips complexes of ellipses', Journal of Topology and Analysis, bind 11, nr. 3, s. 661-690. https://doi.org/10.1142/S1793525319500274

APA

Adamaszek, M., Adams, H., & Reddy, S. (2019). On Vietoris–Rips complexes of ellipses. Journal of Topology and Analysis, 11(3), 661-690. https://doi.org/10.1142/S1793525319500274

Vancouver

Adamaszek M, Adams H, Reddy S. On Vietoris–Rips complexes of ellipses. Journal of Topology and Analysis. 2019;11(3):661-690. https://doi.org/10.1142/S1793525319500274

Author

Adamaszek, Michał ; Adams, Henry ; Reddy, Samadwara. / On Vietoris–Rips complexes of ellipses. I: Journal of Topology and Analysis. 2019 ; Bind 11, Nr. 3. s. 661-690.

Bibtex

@article{519da0c7273346b1ade47e1551c97159,
title = "On Vietoris–Rips complexes of ellipses",
abstract = "For (Formula presented.) a metric space and (Formula presented.) a scale parameter, the Vietoris–Rips simplicial complex (Formula presented.) (resp. (Formula presented.)) has (Formula presented.) as its vertex set, and a finite subset (Formula presented.) as a simplex whenever the diameter of (Formula presented.) is less than (Formula presented.) (resp. at most (Formula presented.)). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev 13, 16, they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses (Formula presented.) of small eccentricity, meaning (Formula presented.). Indeed, we show that there are constants (Formula presented.) such that for all (Formula presented.), we have (Formula presented.) and (Formula presented.), though only one of the two-spheres in (Formula presented.) is persistent. Furthermore, we show that for any scale parameter (Formula presented.), there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.",
keywords = "clique complex, ellipses, homotopy, persistent homology, Vietoris–Rips complex",
author = "Micha{\l} Adamaszek and Henry Adams and Samadwara Reddy",
year = "2019",
doi = "10.1142/S1793525319500274",
language = "English",
volume = "11",
pages = "661--690",
journal = "Journal of Topology and Analysis",
issn = "1793-5253",
publisher = "World Scientific Publishing Co. Pte. Ltd.",
number = "3",

}

RIS

TY - JOUR

T1 - On Vietoris–Rips complexes of ellipses

AU - Adamaszek, Michał

AU - Adams, Henry

AU - Reddy, Samadwara

PY - 2019

Y1 - 2019

N2 - For (Formula presented.) a metric space and (Formula presented.) a scale parameter, the Vietoris–Rips simplicial complex (Formula presented.) (resp. (Formula presented.)) has (Formula presented.) as its vertex set, and a finite subset (Formula presented.) as a simplex whenever the diameter of (Formula presented.) is less than (Formula presented.) (resp. at most (Formula presented.)). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev 13, 16, they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses (Formula presented.) of small eccentricity, meaning (Formula presented.). Indeed, we show that there are constants (Formula presented.) such that for all (Formula presented.), we have (Formula presented.) and (Formula presented.), though only one of the two-spheres in (Formula presented.) is persistent. Furthermore, we show that for any scale parameter (Formula presented.), there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.

AB - For (Formula presented.) a metric space and (Formula presented.) a scale parameter, the Vietoris–Rips simplicial complex (Formula presented.) (resp. (Formula presented.)) has (Formula presented.) as its vertex set, and a finite subset (Formula presented.) as a simplex whenever the diameter of (Formula presented.) is less than (Formula presented.) (resp. at most (Formula presented.)). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev 13, 16, they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses (Formula presented.) of small eccentricity, meaning (Formula presented.). Indeed, we show that there are constants (Formula presented.) such that for all (Formula presented.), we have (Formula presented.) and (Formula presented.), though only one of the two-spheres in (Formula presented.) is persistent. Furthermore, we show that for any scale parameter (Formula presented.), there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.

KW - clique complex

KW - ellipses

KW - homotopy

KW - persistent homology

KW - Vietoris–Rips complex

UR - http://www.scopus.com/inward/record.url?scp=85040084127&partnerID=8YFLogxK

U2 - 10.1142/S1793525319500274

DO - 10.1142/S1793525319500274

M3 - Journal article

AN - SCOPUS:85040084127

VL - 11

SP - 661

EP - 690

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

SN - 1793-5253

IS - 3

ER -

ID: 196738887