Related papers: Lipschitz regularity of graph Laplacians on random…

The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…

We study uniform Lipschitz regularity estimates for elliptic systems in divergence form with continuous coefficients, based on rapidly oscillating periodic coefficients derived from homogenization theory. We extend a result by Avellaneda…

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief…

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard…

This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this…

Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph…

Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph…

The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold $\mathcal M$. In this work, we propose the estimation of the manifold Helmholtzian from point…

In this paper local Lipschitz regularity of weak solutions to certain singular elliptic equations involving one-Laplacian is studied. Equations treated here also contains another well-behaving elliptic operator such as $p$-Laplacian with…

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some…

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…

Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…

$M$-Lipschitz mappings of graphs (or equivalently graph-indexed random walks) are a generalization of standard random walk on $\mathbb{Z}$. For $M \in \N$, an \emph{$M$-Lipschitz mapping} of a connected rooted graph $G = (V,E)$ is a mapping…

Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically…

This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from $N$ random samples on a $d$-dimensional manifold embedded in a possibly high dimensional…

We develop a general theory of random walks on hypergraphs which includes, as special cases, the different models that are found in literature. In particular, we introduce and analyze general random walk Laplacians for hypergraphs, and we…

Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points.…

Let $G$ be an $n$-vertex graph with Laplacian eigenvalues $0=\lambda_1(G)\le \lambda_2(G)\le\cdots\le \lambda_n(G)$. Motivated by the Alon-Boppana bound and the Ramanujan phenomenon for regular graphs, Spielman conjectured that, for every…

Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the…