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.
This paper approximates p-resistance for multi-class graph clustering.
problem Efficiently clustering graphs with p-Laplacian bias. method Developed an approximation to p-resistance for multi-class clustering. result Approximated p-resistance provides a computationally feasible alternative to p-Laplacian eigenvectors. Study approximates product of spheres using Laplacian eigenvalues.
problem Approximating product of spheres using Laplacian eigenvalues.
method Gromov-Hausdorff approximation with pinching condition on eigenvalues.
result Convergence to product of spheres achieved.
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.
Paper analyzes convergence of Laplacian eigenmaps on submanifolds with singularities.
problem Analyzing convergence of Laplacian eigenmaps on submanifolds with singularities.
method Using ε-neighborhood graphs constructed from random points on the submanifold, the paper provides a spectral approximation result for the Laplacian.
result The convergence rate for the eigenvalue of the Laplacian is \( O\left(\left(\log n/n
ight)^{1/(m+2)}
ight) \), where \( m \) and \( n \) are the dimension of the manifold and the sample size, respectively.
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 paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.
problem Developing a more accurate discrete Laplacian for 3D meshes.
method Proves the Euler-Lagrange equation for the Dirichlet energy using the associated discrete Laplacian of the dual construction.
result The associated discrete Laplacian is optimal in R3 compared to the primal construction. I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued…
The smallest eigenvectors of the graph Laplacian are well-known to provide a succinct representation of the geometry of a weighted graph. In reinforcement learning (RL), where the weighted graph may be interpreted as the state transition process induced by a behavior policy acting on the environment, approximating the …
For an embedded submanifold Σ⊂RN, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose…
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
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.
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.
Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…
Let (E,h) be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.
Paper proposes a method to reduce hallucinations in diffusion models using Laplacian score sharpening.
problem Hallucinations in diffusion models create incoherent or unrealistic samples.
method Post-hoc adjustment to the score function during inference using Laplacian approximation.
result Significantly reduces the rate of hallucinated samples across various data types.
Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistica…
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.
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.
This work generalizes a geometric Laplacian determinant description to higher dimensions.
problem Defining and understanding the Laplacian determinant in higher dimensions with non-Delaunay triangulations.
method Geometric description of the Laplacian determinant in higher dimensions, relating it to volume quantities derived from simplex geometry.
result Generalizes geometric Laplacian determinant description to higher dimensions, showing negative semidefiniteness and kernel of constants.
Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.
problem Proving an asymptotic expansion for spectral zeta functions on discrete tori.
method Inspired by Friedli and Karlsson's work, the authors derive an asymptotic expansion for the spectral zeta function on discrete tori.
result Similar asymptotic expansions hold for m=2 and higher dimensions, equivalent to the Epstein-Riemann conjecture.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.
problem Approximating the Willmore functional using nonlocal methods.
method Gamma-convergence and fractional Laplacian analysis in Fermi coordinates.
result Proves Γ-limsup estimate for the proposed nonlocal approximation. 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.
Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date, little work has been done to investigate the tradeoff between computational resourc…
Unified methods for fast column selection in various applications.
problem Efficiently selecting columns for low-rank approximations in data science and machine learning.
method Deterministic and randomized algorithms exploiting nuclear scores.
result Theoretical guarantees and performance bounds for column selection.
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.
The paper analyzes rates of approximation for eigenpairs of Laplace-Beltrami operators on manifolds.
problem Estimating eigenpairs of elliptic differential operators from manifold samples.
method Analyzes minimax rates for eigenvalue and eigenvector estimation using graph Laplacians.
result The minimax rate for H1(M)-sense approximation is n−2/(d+4). The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schroedinger operator on the limit…
DMPS uses diffusion maps and LAWGD for efficient generative modeling.
problem Efficiently modeling complex data distributions.
method Diffusion maps for manifold learning and LAWGD for sampling.
result DMPS outperforms other methods on moderate-dimensional data.
The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the…
We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.
The cost of computing the spectrum of Laplacian matrices hinders the application of spectral clustering to large data sets. While approximations recover computational tractability, they can potentially affect clustering performance. This paper proposes a practical approach to learn spectral clustering based on adaptive…
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the s…
Manifold regularization, such as laplacian regularized least squares (LapRLS) and laplacian support vector machine (LapSVM), has been widely used in semi-supervised learning, and its performance greatly depends on the choice of some hyper-parameters. Cross-validation (CV) is the most popular approach for selecting the …
We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances o…
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.
Extends manifold learning to non-Euclidean metrics.
problem Applying manifold learning to data in non-Euclidean spaces.
method Generalizes manifold learning to metric spaces and studies conditions for convergence.
result Conditions for the convergence of graph Laplacian in metric spaces.
Calculates Laplacian spectra on Calabi-Yau hypersurfaces.
problem Computing the spectrum of the Laplacian on complex manifolds.
method Numerical computation of eigenvalues and eigenmodes for line bundles.
result Agreement with exact results for P3 and a torus, first numerical results for Fermat quintic. This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold M. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators natur…
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. 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.
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.
A new method selects features for clustering without labels.
problem Identifying meaningful features in large datasets.
method Differentiable unsupervised feature selection using a gated Laplacian.
result The method improves clustering performance in noisy data.
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.
On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} Δω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of rando…