An Optimization Parameter for Seriation of Noisy Data

Ghandehari, Mahya; Janssen, Jeannette

doi:10.1137/18m1174544

Cited by 7 publications

(3 citation statements)

References 20 publications

(28 reference statements)

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“…In [GJ19], the authors introduced a function Γ 1 on the space of matrices, which attains 0 exactly when it is applied to a Robinson matrix. While Γ 1 is easy to compute, it fails to be continuous in cut-norm (or equivalently the graph limit topology).…”

Section: Similar Graph Parametersmentioning

confidence: 99%

Graph sequences sampled from Robinson graphons

Ghandehari¹,

Janssen²

2020

Preprint

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The function Γ on the space of graphons, introduced in [CGH + 15], aims to measure the extent to which a graphon w exhibits the Robinson property: for all x < y < z, w(x, z) ≤ min{w(x, y), w(y, z)}.Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. Function Γ is compatible with the cut-norm • ✷ , in the sense that graphons close in cut-norm have similar Γ-values. Here we show the converse, by proving that every graphon w can be approximated by a Robinson graphon R w so that w − R w ✷ is bounded in terms of Γ(w). We then use classical techniques from functional analysis to show that a converging graph sequence {G n } converges to a Robinson graphon if and only if Γ(G n ) → 0. Finally, using probabilistic techniques we show that the rate of convergence of Γ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly w exhibits the Robinson property.

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Section: Similar Graph Parametersmentioning

confidence: 99%

Graph sequences sampled from Robinson graphons

Ghandehari¹,

Janssen²

2020

Preprint

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“…Chepoi and Seston [10] presented a factor 16 approximation for the ℓ ∞ -fitting problem. For a similarity matrix A, Ghandehari and Janssen [18] introduced a parameter Γ 1 (A) and showed that one can construct a Robinson similarity R (with the same order of lines and columns as A) such that ||A − R|| 1 ≤ 26Γ 1 (A) 1 3 . The result of Atkins et al [3] in the case of the Fiedler vectors with no repeated values was generalized by Fogel et al [17] to the case when the entries of A are subject to a uniform noise or some entries are not given.…”

Section: Introductionmentioning

confidence: 99%

Modules in Robinson Spaces

Carmona¹,

Chepoi²,

Naves³

et al. 2022

Preprint

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A Robinson space is a dissimilarity space (X, d) (i.e., a set X of size n and a dissimilarity d on X) for which there exists a total order < on X such that x < y < z implies that d(x, z) ≥ max{d(x, y), d(y, z)}. Recognizing if a dissimilarity space is Robinson has numerous applications in seriation and classification. An mmodule of (X, d) (generalizing the notion of a module in graph theory) is a subset M of X which is not distinguishable from the outside of M , i.e., the distance from any point of X \ M to all points of M is the same. If p is any point of X, then {p} and the maximal by inclusion mmodules of (X, d) not containing p define a partition of X, called the copoint partition. In this paper, we investigate the structure of mmodules in Robinson spaces and use it and the copoint partition to design a simple and practical divide-and-conquer algorithm for recognition of Robinson spaces in optimal O(n 2 ) time.

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“…One's first thought for this problem is to simply compare every entry in the matrix with its preceding neighbors and add up the "errors"; clearly, you will only be Robinson if you end up with 0. In [22], they defined a parameter Γ1 that does this, and while Γ1 is easy to compute, it fails to be continuous in cut norm (or equivalently the graph limit topology). Thus Γ1 is not a suitable Robinson measurement for growing networks as it does not respect limits of graph sequences.…”

Section: Introductionmentioning

confidence: 99%

Cut norm discontinuity of triangular truncation of graphons

Mishura¹

2021

Preprint

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The space of L p graphons, symmetric measurable functions w : [0, 1] 2 → R with finite p-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator Mχ acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as n → ∞, the norm of the triangular truncation operator Tn on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of L p graphons, the norm growth of Tn generalizes to the unboundedness of Mχ. We also show that the norm of Tn grows to infinity on symmetric matrices equipped with the operator norm.

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An Optimization Parameter for Seriation of Noisy Data

Cited by 7 publications

References 20 publications

Graph sequences sampled from Robinson graphons

Graph sequences sampled from Robinson graphons

Modules in Robinson Spaces

Cut norm discontinuity of triangular truncation of graphons

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