New MLG kernels account for multi-scale graph structures.
problem Existing graph kernels are either local or global, ignoring multi-scale structures.
method Builds a hierarchy of nested subgraphs and uses Feature Space Laplacian Graph kernels.
result MLG kernels can capture structure at multiple scales.
New kernel improves graph learning with fewer labeled data.
problem Limited kernels for node-level problems on graphs.
method Derived from a regularization framework, transductive kernel for graphs with node features.
result Improved learning on fewer training points and non-Euclidean data.
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.
New method circumvents curse of dimensionality in Laplacian estimation.
problem High-dimensional data challenges spectral clustering and diffusion maps.
method Kernelized Laplacian estimation via reproducing kernel Hilbert space.
result Non-asymptotic statistical rates show improved performance in high dimensions.
Proves error bounds for state representation in RL using graph spectral features.
problem Addressing the curse of dimensionality in RL with unknown transition graphs.
method Proves upper bounds on approximation error of linear value function approximation using learned spectral features of the state-graph.
result Error bounds scale with algebraic connectivity and eigenvector estimation error.
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.
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 proves convergence of graph Laplacian with kNN self-tuned kernels.
problem Theoretical and practical challenges in choosing kernel bandwidth for graph-based analysis.
method Develops and analyzes a new family of kNN self-tuned kernels for graph Laplacian convergence.
result Proves convergence of graph Laplacian to manifold Laplacian for new kNN self-tuned kernels.
New random feature maps for Laplacian and related kernels.
problem Challenges in approximating the Laplacian kernel and its generalizations.
method Developed random feature maps for Laplacian and related kernels, providing efficient sampling schemes.
result Demonstrated the efficacy of these random feature maps on real datasets.
Study on manifolds with kinks and Gaussian kernel behavior.
problem Understanding the asymptotic behavior of graph Laplacian on manifolds with singularities.
method Introduced manifolds with kinks, derived asymptotic behavior of Graph Laplacian with Gaussian kernel, and validated results numerically.
result Asymptotic behavior of the Graph Laplacian is determined by the inward sector of the tangent space.
A new method improves graph random features with quasi-Monte Carlo techniques.
problem Improving the accuracy of graph random features.
method Induces negative correlations in random walks using antithetic termination.
result Strong theoretical guarantees on lower-variance estimators of the Laplacian kernel.
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…
Unified framework for multi-user bandits using Laplacian kernels.
problem Multi-user contextual bandits with graph-related users and non-linear rewards.
method Joint penalty combining graph smoothness and individual roughness in a unified RKHS.
result Unified multi-user RKHS and effective dimension for regret bounds.
New GL-GP models learn covariance respecting domain geometry.
problem Suboptimal results from nonparametric regression on restricted domains.
method Graph Laplacian based Gaussian Processes (GL-GPs) with Nyström extension.
result Performance gains in various applications.
A novel graph spectral method for mixed categorical and numerical data.
problem Feature learning for mixed data types (numerical and categorical).
method Graph spectral decomposition of the graph Laplacian to model probabilistic dependence structure.
result Increased separability and clusterability of observations in the transformed feature space.
Paper develops a novel kernel-based method for MRI data recovery.
problem Reconstructing dynamic MRI data on manifolds.
method Kernel bi-linear modeling in reproducing kernel Hilbert spaces.
result Validated on synthetic dMRI data, the method outperforms state-of-the-art approaches.
Proposes a graph pooling method leveraging node proximity for hierarchical graph representation learning.
problem Efficiently exploiting the geometry of graph data for hierarchical representation learning.
method Combines node proximity with kernel representation of topology and node features for adaptive node signal similarities evaluation.
result Achieves state-of-the-art performance on graph classification benchmark datasets.
DHGAK aligns substructures for better graph kernel performance.
problem Limited performance of traditional graph kernels due to missing substructure similarities.
method Hierarchically aligns relational substructures in deep embedding space, assigning same feature maps in RKHS.
result DHGAK outperforms state-of-the-art graph kernels on various benchmarks.
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.
Paper develops a method to identify graphs and filters from filtered signals.
problem Learning graphs and filters from filtered signals.
method Developed an algorithm to jointly identify a graph and a graph-based filter (GBF) from multiple signal/data observations.
result The proposed algorithm outperforms current state-of-the-art methods.
Paper derives Li-Yau inequality for unbounded Laplacian on graphs.
problem Deriving Li-Yau inequality for unbounded Laplacian on graphs.
method Assumption of curvature-dimension inequality CDE′(n,K) and derivation of Li-Yau inequality. result First results on Li-Yau inequality for unbounded Laplacian on graphs.
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.
Enhances graph neural networks by considering feature similarities in node aggregation.
problem Ignoring node feature similarities in traditional graph aggregation schemes.
method Interprets node aggregation as kernel weighting, proposing a framework that considers feature similarities.
result Proposed framework outperforms traditional GCNs in real-world applications.
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.
End-to-end graph SVM with graph convolutions and RKHS.
problem Graph classification with complex feature spaces.
method End-to-end training of graph convolutions, kernel function, and SVM parameters.
result Outperforms existing deep learning models on graph classification tasks.
Laplacian matrix helps in reducing data dimensions and clustering.
problem Representing and clustering data using graphs and matrices.
method Using the Laplacian matrix to assign values to nodes based on their connectivity.
result The Laplacian matrix can be used to find a good embedding of data in a low-dimensional space and perform clustering.
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 …
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.
We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we …
We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set the bandwidth by optimizing the Laplacian's ability to preser…
Generative model controls heterophily in graph signals.
problem Controlling heterophily in graph signals for better model effectiveness.
method Combines graphon-based generator with spectral filtering of Gaussian node features.
result Establishes theoretical guarantees for heterophily control and convergence.
Inspired by a growing interest in analyzing network data, we study the problem of node classification on graphs, focusing on approaches based on kernel machines. Conventionally, kernel machines are linear classifiers in the implicit feature space. We argue that linear classification in the feature space of kernels comm…
A new graph kernel uses LCS and Wasserstein distance for better graph comparisons.
problem Graph learning methods can be limited by information from distant vertices and path length constraints.
method Proposes a Graph Kernel based on LCS similarity and Wasserstein distance in a novel metric space.
result The new kernel emphasizes comparisons between similar paths and reduces information loss.
Improved convergence rate for kNN graph Laplacians with adaptive bandwidth.
problem Enhancing the efficiency of graph-based data analysis methods.
method Introducing a new class of kNN graph with adaptive bandwidth and proving operator convergence rate.
result Operator convergence rate of O(N−2/(d+6)) for the kNN graph Laplacian, up to a log factor. STARK improves denoising of low-depth spatial transcriptomics images.
problem Denoising spatial transcriptomics images at ultra-low sequencing depths.
method Adaptive regularization with kernel ridge regression and graph Laplacian.
result STARK optimizes denoising performance over competing methods.
Study the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
problem Understanding the heat kernel on quaternionic anti-de Sitter spaces and related spaces.
method Detailed study of the geometry, derivation of the horizontal Laplacian and subelliptic heat kernel formulas, derivation of small time asymptotics.
result Explicit formulas for the horizontal Laplacian and subelliptic heat kernel of the quaternionic anti-de Sitter fibration.
The study examines the geometry of Lichnerowicz Laplacian's kernel on various spaces.
problem Understanding the kernel of the Lichnerowicz Laplacian on different types of spaces.
method Analytical method of Bochner to prove vanishing theorems for null space of Laplace operator.
result Applications to theories of infinitesimal Einstein deformations and stability of Einstein manifolds.
Paper develops a graph-based method for reconstructing spatio-temporal signals.
problem Reconstructing space-time varying signals on graphs given limited data.
method Multi-kernel Kriged Kalman Filter combining graph-aware kernels and online selection.
result Superior reconstruction performance compared to existing methods.
PCR-LE achieves optimal rates for nonparametric regression over Sobolev spaces.
problem Nonparametric regression over Sobolev spaces with random design.
method PCR-LE using Laplacian Eigenmaps on neighborhood graphs.
result PCR-LE achieves minimax rates of convergence for both estimation and goodness-of-fit testing.
Study subelliptic heat kernel on lifted sphere from octonionic projective space.
problem Analyzing sub-Laplacian on lifted sphere from octonionic projective space.
method Explicit formulas for heat kernel and Green function derived.
result Explicit formulas for heat kernel and Green function.
Unified feature maps for graph kernels improve efficiency without sacrificing accuracy.
problem Efficiently applying non-linear kernel methods to large-scale graph data.
method Constructing feature maps for graph kernels, analyzing feasibility, and proposing algorithms.
result Explicit feature maps can achieve similar accuracy to kernel trick methods but with significantly reduced computation time.
This paper reconstructs complex graph signals using kernel methods on manifolds.
problem Reconstructing complex graph signals from samples on graph vertices.
method Kernel methods on complex manifolds, embedding vertices into higher-dimensional spaces.
result Effective reconstruction of complex graph signals, outperforming conventional methods.
Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good under…
A faster graph kernel using optical random features.
problem High computation cost of graphlet kernel due to isomorphism test.
method Kernel random features, optical random features, mean kernel metric.
result The proposed method is orders of magnitude faster with similar or better accuracy.
Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
problem Uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
method Analysis of random walks on geometric and directed kNN graphs, using concentration tools and differential geometry.
result Uniform convergence of kNN Laplacians to diffusion Laplacian, without continuity of transition kernel. New graph kernels capture spatio-temporal interactions.
problem Lack of justified spatio-temporal graph kernels for graph problems.
method Derive graph kernels via SPDEs for spatio-temporal modelling.
result Non-separable spatio-temporal graph kernels outperform existing ones.
Study subelliptic heat kernel on octonionic anti-de Sitter space.
problem Heat kernel of octonionic anti-de Sitter space.
method Lift Laplacian of octonionic hyperbolic space and use sub-Laplacian.
result Two integral representations for subelliptic heat kernel.
Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive a…