Estimating the gap of finite metric spaces of strict p-negative type

Wolf, Ronald de

doi:10.1016/j.laa.2018.07.005

Linear Algebra and its Applications

2018

DOI: 10.1016/j.laa.2018.07.005

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Estimating the gap of finite metric spaces of strict p-negative type

Ronald de Wolf

Abstract: Let (X, d) be a finite metric space. This paper first discusses the spectrum of the p-distance matrix of a finite metric space of pnegative type and then gives upper and lower bounds for the so called gap Γ(X, p) of a finite metric space of strict p-negative type. Furthermore estimations for Γ(X, p) under a certain glueing construction for finite metric spaces are given and finally be applied to finite ultrametric spaces.

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Cited by 4 publications

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“…The existence of b is shown in[17, Theorem 4.2]. The independence of the value of b, 1 was noted in[17, Remark 4.4]. Statement (2) is[17, Theorem 4.8].…”

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confidence: 96%

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Distance matrices of subsets of the Hamming cube

Doust

Robertson

Stoneham

et al. 2021

Indagationes Mathematicae

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Let D denote the distance matrix for an n + 1 point metric space (X, d). In the case that X is an unweighted metric tree, the sum of the entries in D −1 is always equal to 2/n. Such trees can be considered as affinely independent subsets of the Hamming cube H n , and it was conjectured that the value 2/n was minimal among all such subsets. In this paper we confirm this conjecture and give a geometric interpretation of our result which applies to any subset of H n .

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“…The existence of b is shown in[17, Theorem 4.2]. The independence of the value of b, 1 was noted in[17, Remark 4.4]. Statement (2) is[17, Theorem 4.8].…”

mentioning

confidence: 96%

“…The independence of the value of b, 1 was noted in[17, Remark 4.4]. Statement (2) is[17, Theorem 4.8]. Suppose that (X, d) is a finite metric space of strict 1-negative type with distance matrix D. Then M(X) < ∞ andM(X) = 1 D −11, 1 .…”

mentioning

confidence: 99%

Distance matrices of subsets of the Hamming cube

Doust

Robertson

Stoneham

et al. 2021

Indagationes Mathematicae

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Negative Type and Bi-lipschitz Embeddings into Hilbert Space

Robertson

2024

Bull. Malays. Math. Sci. Soc.

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The usual theory of negative type (and p-negative type) is heavily dependent on an embedding result of Schoenberg, which states that a metric space isometrically embeds in some Hilbert space if and only if it has 2-negative type. A generalisation of this embedding result to the setting of bi-lipschitz embeddings was given by Linial, London and Rabinovich. In this article we use this newer embedding result to define the concept of distorted p-negative type and extend much of the known theory of p-negative type to the setting of bi-lipschitz embeddings. In particular we show that a metric space $$(X,d_{X})$$ ( X , d X ) has p-negative type with distortion C ($$0\le p<\infty $$ 0 ≤ p < ∞ , $$1\le C<\infty $$ 1 ≤ C < ∞ ) if and only if $$(X,d_{X}^{p/2})$$ ( X , d X p / 2 ) admits a bi-lipschitz embedding into some Hilbert space with distortion at most C. Analogues of strict p-negative type and polygonal equalities in this new setting are given and systematically studied. Finally, we provide explicit examples of these concepts in the bi-lipschitz setting for the bipartite graphs $$K_{m,n}$$ K m , n .

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A problem on distance matrices of subsets of the Hamming cube

Doust

Wolf

2022

Proc. Amer. Math. Soc. Ser. B

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Let D D denote the distance matrix for an n + 1 n+1 point metric space ( X , d ) (X,d) . In the case that X X is an unweighted metric tree, the sum of the entries in D − 1 D^{-1} is always equal to 2 / n 2/n . Such trees can be considered as affinely independent subsets of the Hamming cube H n H_n , and it was conjectured that the value 2 / n 2/n was minimal among all such subsets. In this paper we confirm this conjecture and give a geometric interpretation of our result which applies to any subset of H n H_n .

show abstract

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Estimating the gap of finite metric spaces of strict p-negative type

Cited by 4 publications

References 22 publications

Distance matrices of subsets of the Hamming cube

Distance matrices of subsets of the Hamming cube

Negative Type and Bi-lipschitz Embeddings into Hilbert Space

A problem on distance matrices of subsets of the Hamming cube

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