Root Laplacian Eigenmaps help in spectral embedding of graphs.
problem Efficient spectral embedding of graphs.
method Square root of graph-Laplacian operator.
result Improved spectral embedding techniques.
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48 results for “Graph Laplacians”
Study shows SNN graph Laplacians converge to k-NN graph Laplacians under large scale asymptotics.
problem Understanding the convergence of SNN graph Laplacians to k-NN graph Laplacians.
method Analyzing the asymptotic behavior of SNN and k-NN graph Laplacians.
result The graph Laplacians of SNN and k-NN graphs converge to the same limit under large scale asymptotics.
Paper learns Cartesian product graphs with Laplacian constraints.
problem Learning Cartesian product graphs from Laplacian constraints.
method Penalized maximum likelihood estimation (MLE) and efficient algorithm.
result Statistical consistency for Cartesian product Laplacian estimation.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on …
Spectral sparsification improves Laplacian-constrained graph learning.
problem Improving accuracy of Laplacian-constrained graph learning.
method Spectral graph sparsification as a post-estimation operation.
result Improved accuracy of Laplacian-constrained graph learning.
We argue that the standard graph Laplacian is preferable for spectral partitioning of signed graphs compared to the signed Laplacian. Simple examples demonstrate that partitioning based on signs of components of the leading eigenvectors of the signed Laplacian may be meaningless, in contrast to partitioning based on th…
Paper proves convergence of bi-stochastically normalized graph Laplacian to manifold Laplacian and robustness to outlier noise.
problem Convergence of bi-stochastically normalized graph Laplacian to manifold Laplacian and robustness to outlier noise.
method Proves convergence of bi-stochastically normalized graph Laplacian to manifold Laplacian with rates, and proposes an approximate and constrained matrix scaling problem to achieve the same consistency rate.
result Graph Laplacian consistency rate matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors.
Paper describes eigenvalues of genus 3 surfaces graphs.
problem Understanding eigenvalues of genus 3 surfaces.
method Analyzes graphs derived from pair of pants decompositions.
result Complete description of eigenvalue sets for genus 3.
Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a well-behaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also introduce a kernel-fr…
Develops methods to analyze manifold singularities using graph Laplacian.
problem Analyzing geometric properties of singularities in datasets.
method Theory and methods using the graph Laplacian to provide explicit bounds on manifold singularities.
result Explicit bounds on the graph Laplacian for functions near manifold singularities.
Study spectral properties of graph Laplacian for manifold data.
problem Understanding spectral properties of graph Laplacian for manifold data.
method Non-asymptotic error bounds on spectral properties of empirical graph Laplacian.
result Eigenvalues and eigenspaces of empirical graph Laplacian are close to Laplace-Beltrami operator of manifold.
Novel Haar-Laplacian for directed graphs enhances spectral graph applications.
problem Lack of suitable Laplacian for directed graphs in spectral graph theory.
method Inspired by Haar-like transformation, introduces a Hermitian matrix preserving direction and weight.
result HaarNet outperforms in weight prediction and denoising on directed graphs.
Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistencystat.ML
Paper proves conditions for estimating precision matrices with Laplacian constraints.
problem Estimating high-dimensional precision matrices with Laplacian constraints.
method Minimizing Stein's loss with conditions on graph connectivity and Laplacian constraints.
result High-dimensional consistency achieved with Laplacian constraints, independent of graph structure.
We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians:…
Graph poly-Laplacian method improves regression accuracy.
problem Regression with noisy labels on graphs.
method Graph poly-Laplacian regularization for non-parametric regression.
result Rate of convergence matches known results for smoothing splines.
Propagation-regularization improves GNN performance by infusing extra graph information.
problem The effectiveness of graph Laplacian regularization in GNNs is questioned and improved upon.
method Introducing Propagation-regularization (P-reg) to enhance GNN performance.
result P-reg boosts GNN performance on various tasks across multiple datasets.
New method clusters evolving networks using spatio-temporal graph Laplacian.
problem Clustering communities in time-varying graphs.
method Extends spectral clustering to dynamic graphs using CCA and spatio-temporal graph Laplacian.
result The spatio-temporal graph Laplacian clearly interprets cluster evolution over time.
Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
The paper tackles sparse graph learning under Laplacian-related constraints, improving upon existing methods.
problem Learning a sparse undirected graph from multivariate data under Laplacian-related constraints.
method Modifications to penalized log-likelihood approaches to enforce total positivity and lasso/adaptive lasso penalties using ADMM.
result The proposed constrained adaptive lasso approach significantly outperforms existing Laplacian-based approaches.
A sign is introduced in the usual Laplacian on graphs and the corresponding analogue of the isoperimetric constant for this Laplacian is presented, i.e. a geometric quantity which enables to bound from above and below the first eigenvalue. The introduction of the sign in the Laplacian is motivated by the study of -l…
BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.
problem Comparing combinatorial and Hodge Laplacians for discrete data.
method Introducing Boundary-Induced Graph (BIG) Laplacians using DEC.
result BIG Laplacian eigenvalues converge to Hodge Laplacian for simple shapes.
Towards Better Laplacian Representation in Reinforcement Learning with Generalized Graph Drawingcs.LG
New method learns high-quality Laplacian representations for reinforcement learning.
problem Lack of accurate Laplacian representations in large or continuous state spaces.
method Reformulated spectral graph drawing objective to have eigenvectors as unique global minimizer.
result Learned Laplacian representations more faithfully approximate the ground truth.
In this paper, we develop a novel weighted Laplacian method, which is partially inspired by the theory of graph Laplacian, to study recent popular graph problems, such as multilevel graph partitioning and balanced minimum cut problem, in a more convenient manner. Since the weighted Laplacian strategy inherits the virtu…
Extends graph theory to hypergraphs with manifold-valued nodes.
problem Representing complex N-ary relationships on manifolds.
method Defined function spaces and symmetric products for manifold-valued nodes and edges.
result Generalized hypergraph Laplacians to manifold-valued hypergraphs.
New outlier detection method using graph Laplacian spectrum boosts performance.
problem Detecting outliers in large datasets efficiently.
method Boosted outlier detection based on graph Laplacian spectrum.
result Outperforms existing methods on synthetic datasets.
In this paper, we study eigenvalues and eigenfunctions of -Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of -Laplacian, as we ident…
A new graph generator uses heat diffusion on graph Laplacians to create new graph structures.
problem Creating realistic and diverse graph structures for various applications.
method Adapting the Generator Matching paradigm to graph data, using graph Laplacian and heat kernel for diffusion.
result The method effectively generates graphs with structural properties of real and synthetic graphs.
Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.
problem Approximating eigenvalues of Laplace-Beltrami on manifolds with bounded Ricci curvature.
method Graph discretization of Riemannian manifolds with -approximation, proving eigenvalue convergence.
result Graph Laplacian eigenvalues converge uniformly to manifold Laplacian eigenvalues as parameters approach zero.
The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph -Laplacian. Unlike the …
Novel GNN for signed and directed networks using magnetic signed Laplacian.
problem Efficiently modeling signed and directed networks for tasks like clustering and link prediction.
method Introduced a magnetic signed Laplacian for directed signed graphs, used it to construct a spectral GNN.
result Demonstrated effective performance on tasks involving signed and directional information.
New invariant for special alternating links based on graph Laplacian.
problem Developing an invariant for special alternating links.
method Using the Laplacian matrix of the Tait graph, invariant is defined.
result A specific quadratic trace expression is invariant under flype moves.
Graph Laplacians and machine learning predict properties of finite graphs.
problem Understanding properties of finite graphs using spectral and topological methods.
method Combining graph Laplacians, spectral inequalities, machine learning, and topological data analysis.
result Neural networks can accurately predict graph properties like Ricci-flatness and spectral gaps.
We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci…
The paper extends Laplacian spectra approximations to vector bundles.
problem Approximating the spectrum of the connection Laplacian.
method Extending the graph connection Laplacian to vector bundles and proving spectrum approximation.
result The spectrum of the extended operator approximates the spectrum of the connection Laplacian.
Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.
problem Approximating eigenvalues of manifolds with cylindrical boundaries.
method Using truncated graph Laplacians constructed from -proximity graphs.
result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretatio…
In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality , which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, hea…
Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.
problem Understanding machine learning methods for data with varying intrinsic dimensions.
method Γ-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on intersecting manifolds of varying dimensions.
result Normalized Dirichlet energy converges to a tensorized Dirichlet energy that adapts to all dimensions simultaneously.
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…
The paper proves Lipschitz regularity of graph Laplacian eigenvectors on random data clouds.
problem Analyzing the regularity of solutions to graph Laplacian equations on random data points.
method Probabilistic coupling of random walks and interpolation method for point clouds to continuum.
result Graph Laplacian eigenvectors are essentially Lipschitz with constants depending on eigenvalues.
ELD compares graphs by their embedded Laplacian eigenvectors, resolving ambiguities.
problem Comparing graphs of different sizes and structures.
method ELD uses symmetrization and perturbation techniques to compare graph embeddings.
result ELD resolves ambiguities in graph comparisons, making it a natural pseudo-metric.
Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
This short note aims at (re)proving that the symmetrically normalized graph Laplacian $L=\Id - D^{-1/2}WD^{-1/2}$ (from a graph defined from a Gaussian weighting kernel on a sampled smooth manifold) converges towards the continuous Manifold Laplacian when the sampling become infinitely dense. The convergence rate with …
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.
problem Complexity of signed graphs and its relation to Alexander polynomials.
method Definition of graph complexity using Laplacian matrix and Mahler measure, linking to Alexander polynomials and Lehmer's question.
result Complexity growth of signed graphs is related to the growth rate of Alexander polynomials.
The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.
problem Analyzing spectral properties of graph Laplacians on manifolds with curvature constraints.
method Quantitative bounds on eigenvalues and eigenfunctions of graph Laplacians constructed from random variables on manifolds with uniform lower Ricci curvature bounds.
result Spectral convergence of graph Laplacians on manifolds with curvature bounds and in non-collapsed spaces.