A Compactness Theorem for functions on Poisson point clouds

Caroccia, Marco

doi:10.48550/arxiv.2205.05465

Cited by 1 publication

(2 citation statements)

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“…The question whether and how percolation techniques can be applied to other graph PDEs like for instance the Laplace or p-Laplace equation is much harder. Recent results in two dimensions show that at least Dirichlet energies Gamma-converge for percolation length scales [10,24]. Combining quantitative versions of these arguments with the techniques from [16, Section 5.5] can potentially produce convergence rates.…”

Section: Discussionmentioning

confidence: 99%

“…While the fields of percolation theory and partial differential equations (PDEs) on graphs (including finite difference methods and semi-supervised learning) are very well developed, there are relatively few results that connect them, such as [10,24] which deals with Gamma-convergence of discrete Dirichlet energies on Poisson clouds or [36] on distance learning from a Poisson cloud on an unknown manifold. In the following we give a brief overview of first-passage percolation and graph PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

Bungert¹,

Calder²,

Tim³

2022

Preprint

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In this paper we prove the first quantitative convergence rates for the graph infinity Laplace equation for length scales at the connectivity threshold. In the graph-based semi-supervised learning community this equation is also known as Lipschitz learning. The graph infinity Laplace equation is characterized by the metric on the underlying space, and convergence rates follow from convergence rates for graph distances. At the connectivity threshold, this problem is related to Euclidean first passage percolation, which is concerned with the Euclidean distance function d h (x, y) on a homogeneous Poisson point process on R d , where admissible paths have step size at most h > 0. Using a suitable regularization of the distance function and subadditivity we prove that d hs (0, se1)/s → σ as s → ∞ almost surely where σ ≥ 1 is a dimensional constant and hs log(s) 1 d . A convergence rate is not available due to a lack of approximate superadditivity when hs → ∞. Instead, we prove convergence rates for the ratio d h (0,se 1 ) d h (0,2se 1 ) → 1 2 when h is frozen and does not depend on s. Combining this with the techniques that we developed in (Bungert, Calder, Roith, IMA Journal of Numerical Analysis, 2022), we show that this notion of ratio convergence is sufficient to establish uniform convergence rates for solutions of the graph infinity Laplace equation at percolation length scales.

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Section: Discussionmentioning

confidence: 99%