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.
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.
Graphs approximate Laplacian spectra on manifolds.
problem Approximating Laplacian spectra on complex manifolds.
method Graph Laplacians on proximity graphs.
result Spectra of graph Laplacians approximate the Laplacian spectra of manifolds.
Optimal bounds for Laplacian eigenvalues on weighted graphs.
problem Finding lower bounds for Laplacian eigenvalues in weighted graphs.
method Formulating bounds in terms of graph geometry, specifically inradius of subsets.
result Optimal lower bounds for the first non-zero eigenvalue in finite volume and Dirichlet Laplacian on subsets with geometric conditions.
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.
Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.
problem Understanding ancient solutions on graphs with unbounded Laplacians.
method Generalizing Colding and Minicozzi's theorem and Hua's result to graphs with unbounded Laplacians.
result The dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth.
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.
Develops a new weighted Laplacian method for graph problems.
problem Graph partitioning and balanced minimum cut problems.
method Weighted Laplacian method based on graph theory and PDEs.
result Established equivalence relations among graph problems.
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…
The paper studies eigenvalues and Cheeger constants on symmetric graphs.
problem Characterizing eigenvalues and Cheeger constants on symmetric graphs.
method Characterization of the first eigenfunction via sign condition, and calculation of Cheeger constants using the limit of p-Laplacian eigenvalues. result Identifies Cheeger constants of symmetric graphs and their quotients.
Study uses graph Laplacians to analyze surface links.
problem Analyzing virtual genus of surface links.
method Laplacian matrices of weighted graphs in surfaces are used to define invariants.
result Obtained information about virtual genus.
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.
The paper improves spectral convergence rates for graph Laplacians.
problem Improving spectral convergence rates for graph Laplacians.
method Utilizing regularity of continuum eigenfunctions and strong pointwise consistency results.
result Eigenvalues and eigenvectors of graph Laplacian converge to continuum at rate O(n−1/(m+4)). 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.
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.
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.
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.
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.
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.
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 2-l…
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.
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.
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 (ε,ρ)-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 p-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.
Graph Laplacian spectrum serves as a robust feature representation.
problem Difficulties in analyzing and comparing graphs due to their structure.
method Proposes using the graph Laplacian spectrum (GLS) as a feature representation.
result Graph Laplacian spectrum (GLS) preserves structural information and is consistent under deformation and invariance under isomorphism.
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 (ε,ρ)-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 CDE′(n,K), 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.
The paper uses graph Laplacians and maximum principles to study learning problems on unknown manifolds.
problem Learning problems on unknown manifolds with noise.
method Maximum principle arguments and techniques from partial differential equations and the Calculus of variations.
result Asymptotic consistency guarantees for noise-corrupted, non-parametric regression.
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.