Sharp L 1-Poincaré inequalities correspond to optimal hypersurface cuts

Steinerberger, Stefan

doi:10.1007/s00013-015-0778-x

Cited by 13 publications

(8 citation statements)

References 21 publications

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“…Notably, almost all explicit examples of designs in [34] have the property of being at distance at most 1 from every other vertex in the graph, however only one, the one for the Sylvester graph, is a maximal independent set. A computer experiment in Sage on the Sylvester graph shows that there exist maximal independent sets of bigger size (namely, 11 vertices), however their order as of a graphical design seems to be higher.…”

Section: Resultsmentioning

confidence: 99%

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Graphical Designs and Extremal Combinatorics

Golubev

2019

Preprint

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A graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the Laplacian operator have mean value zero. In this paper, we show that extremal independent sets make extremal graphical designs, that is, a design on which the maximum possible number of eigenfunctions have mean value zero. We then provide examples of such graphs and sets, which arise naturally in extremal combinatorics. We also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs, and provide examples for them as well. We investigate the behavior of graphical designs under the operation of weak graph product. In addition, we present a family of extremal graphical designs for the hypercube graph.

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Section: Resultsmentioning

confidence: 99%

“…A similar modification to the isoperimetric constant is made for Euclidean domains in [34], and for simplicial complexes in [31]. We make use of the following theorem, which is a slight variation of the Cheeger inequality for graphs.…”

Section: Resultsmentioning

confidence: 99%

Graphical Designs and Extremal Combinatorics

Golubev

2019

Preprint

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“…be too small. A result of the third author [20] (refining a result of Dyer & Frieze [9]) states that for open and convex Ω ⊂ R n and all open subsets A ⊂ Ω…”

Section: The Continuous Case: Hot Spotsmentioning

confidence: 99%

On the diffusion geometry of graph Laplacians and applications

Cheng

Rachh

Steinerberger

2019

Applied and Computational Harmonic Analysis

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We study directed, weighted graphs G = (V, E) and consider the (not necessarily symmetric) averaging operatorwhere p ij are normalized edge weights. Given a vertex i ∈ V , we define the diffusion distance to a set B ⊂ V as the smallest number of steps d B (i) ∈ N required for half of all random walks started in i and moving randomly with respect to the weights p ij to visit B within d B (i) steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if u satisfies Lu = λu on V andis a remarkably good approximation of |u| in the sense of having very high correlation.The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.2010 Mathematics Subject Classification. 35R02 (primary) and 35J05, 94C15 (secondary).

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“…After this paper was completed we learned from Stefan Steinerberger of his interesting related work on Poincaré inequalities in the Euclidean setting (cf. [34]). Moreover, in [34] one can also find a large set of references that go all the way back to Yau [40].…”

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confidence: 99%

“…[34]). Moreover, in [34] one can also find a large set of references that go all the way back to Yau [40]. In [33], Steinerberger also studies uncertainty inequalities, but we should note that [34] and [34] are independent of each other.…”

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confidence: 99%

Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

Martín

Milman

2016

Journal of Functional Analysis

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We extend the recent L 1 uncertainty inequalities obtained in [13] to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets µ({w ≤ r}) can be controlled by rI(r), where I is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincaré inequalities*, and interpolation, to prove L p , 1 ≤ p < ∞, uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities of [20], and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.

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Sharp L 1-Poincaré inequalities correspond to optimal hypersurface cuts

Cited by 13 publications

References 21 publications

Graphical Designs and Extremal Combinatorics

Graphical Designs and Extremal Combinatorics

On the diffusion geometry of graph Laplacians and applications

Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

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