The Gromov–Hausdorff distance between
spheres

Lim, Sunhyuk; Mémoli, Facundo; Smith, Zane

doi:10.2140/gt.2023.27.3733

Cited by 6 publications

(4 citation statements)

References 25 publications

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“…The proof is identical for the Čech case. The stability theorem, Theorem B, gives from ( 5), where d GH .S n ; S m / is the Gromov-Hausdorff distance between spheres endowed with their geodesic distances for any 0 < m < n. A better lower bound is found in [53]:…”

Section: A Hausmann Type Theorem For 2-vietoris-rips and 2-čech Thick...mentioning

confidence: 99%

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The persistent topology of optimal transport based metric thickenings

Adams,

Mémoli,

Moy

et al. 2024

Algebr. Geom. Topol.

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The persistent topology of optimal transport based metric thickenings 395 complexes of the circle at all scales (See Adamaszek and Adams [1]) are not accompanied by a broader Morse theory (though some techniques feel Morse-theoretic).A second strategy is very recent. Lim, Mémoli and Okutan [52] and Okutan [62] show that the Vietoris-Rips simplicial complex filtration is equivalent to thickenings of the Kuratowski embedding into L 1 .M / or any other injective metric space. In particular, the two filtrations have the same persistent homology. This overcomes obstacle (i): the inclusion of a metric space into a thickening of its Kuratowski embedding is continuous, and indeed an isometry onto its image. This connection has created new opportunities, such as the Morse theoretic techniques employed by Katz [45;46;47;48]. This Morse theory allows one to prove the first new homotopy types that occur for Vietoris-Rips simplicial complexes of the circle and 2-sphere, but have not yet inspired progress for larger scales, or for spheres above dimension two.

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Section: A Hausmann Type Theorem For 2-vietoris-rips and 2-čech Thick...mentioning

confidence: 99%

“…d GH ..S n ; `2/; .S m ; `2// In a more detailed analysis, Proposition 9.13 of[53] provides the lower bound 1 2 for d GH ..S n ; `2/; .S m ; `2// when n ¤ m, which is larger than the lower bound p 2 4 given in(5). As for the geodesic distance case, one can use[53, Corollary 9.8(1)] to obtain d GH .S n ; S m / arcsin…”

mentioning

confidence: 98%

The persistent topology of optimal transport based metric thickenings

Adams,

Mémoli,

Moy

et al. 2024

Algebr. Geom. Topol.

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“…Now, by Proposition 9.7, the right-hand side is bounded above by [63] for improved lower bounds via considerations based on a certain version of the Borsuk-Ulam theorem. In fact, there the factor 1 2 is removed, leading, for example, to the bound d GH .S m ; S n / FillRad.S minfm;ng / for all 0 Ä m < n Ä 1.…”

Section: Sunhyuk Lim Facundo Mémoli and Osman Berat Okutanmentioning

confidence: 99%

“…What we know is that this is indeed not tight when X and Y are spheres of different dimension since, in Corollary 9.39, we show that `VR .S m ; S n / D 1 4 arccos . 1=.m C 1// for any 0 < m < n. However, in[63, Theorem B] it is proved that…”

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confidence: 99%

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

Lim,

Mémoli,

Okutan

2024

Algebr. Geom. Topol.

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The Gromov–Wasserstein Distance Between Spheres

Arya,

Auddy,

Clark

et al. 2024

Found Comput Math

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The Gromov–Wasserstein distance—a generalization of the usual Wasserstein distance—permits comparing probability measures defined on possibly different metric spaces. Recently, this notion of distance has found several applications in Data Science and in Machine Learning. With the goal of aiding both the interpretability of dissimilarity measures computed through the Gromov–Wasserstein distance and the assessment of the approximation quality of computational techniques designed to estimate the Gromov–Wasserstein distance, we determine the precise value of a certain variant of the Gromov–Wasserstein distance between unit spheres of different dimensions. Indeed, we consider a two-parameter family $$\{d_{{{\text {GW}}}p,q}\}_{p,q=1}^{\infty }$$ { d GW p , q } p , q = 1 ∞ of Gromov–Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we are able to determine the exact value of the distance $$d_{{{\text {GW}}}4,2}$$ d GW 4 , 2 between all pairs of unit spheres of different dimensions endowed with their Euclidean distance and their uniform measure.

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The Gromov–Hausdorff distance between spheres

Cited by 6 publications

References 25 publications

The persistent topology of optimal transport based metric thickenings

The persistent topology of optimal transport based metric thickenings

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

The Gromov–Wasserstein Distance Between Spheres

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