The paper studies Laplacians on smooth distributions and proves they are multipliers in C∗-algebras.
problem Understanding spectral properties of Laplacians on generalized smooth distributions.
method Survey of generalized smooth distributions, proof of Laplacian as a multiplier in foliation C∗-algebra. result Laplacians on smooth distributions define unbounded multipliers in foliation C∗-algebras. The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.
problem Studying spectral properties of the horizontal Laplacian
method Interpreting the horizontal Laplacian as a twisted Laplacian acting on a flat vector bundle
result The horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold
In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distributi…
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
Universal inequalities for Laplacian eigenvalues on convex domains.
problem Eigenvalue distribution of the Laplacian on convex domains.
method Established two universal inequalities.
result Two new inequalities for Laplacian eigenvalues.
Method estimates multiple related Gaussian distributions using Laplacian regularization.
problem Jointly estimate multiple related zero-mean Gaussian distributions.
method Laplacian regularized stratified model fitting with hyper-parameters to encourage covariance closeness.
result The method performs well, especially in low data regimes, as demonstrated in finance, radar, and weather.
The EM algorithm performs well for mixture models of Laplacian distributions.
problem Understanding the behavior of the EM algorithm for mixture models.
method Analysis of a simple mixture of Laplacian distributions and proof of convergence for the EM algorithm.
result The EM algorithm converges to the ground truth parameters almost surely with random initialization.
Study on Kohn Laplacian spectrum on sphere quotients.
problem Determining fundamental group from Kohn Laplacian spectrum.
method Weyl-type theorem, CR manifolds, Sobolev estimates.
result Fundamental group can be determined from Kohn Laplacian spectrum in 3D.
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…
We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Lapla…
The paper proves boundedness of envelopes in complex manifolds.
problem Regularity of envelopes in complex manifolds.
method Analyzes bounded functions and their envelopes in the context of cohomology classes and Laplacians.
result The α-psh envelope P(f) is locally bounded with locally bounded Laplacian on the ample locus of {α}. 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.
We study a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution. We prove that this Laplacian does not behave very well: it is not hypoelliptic in general and does not respect the bigrading on forms in a complex setting. We also discuss the consequences of these n…
Proposes a flexible deep learning model for complex distributions.
problem Complex shapes, strong skews, and multiple modes in output variable distributions.
method Uncountable Mixture of Asymmetric Laplacians (UMAL) deep learning framework.
result UMAL can estimate heterogeneous distributions without strong assumptions.
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.
The paper extends Bochner's technique to singular distributions on manifolds.
problem Analyzing the curvature and null space of Hodge Laplacian on singular distributions.
method Defining modified statistical connection, exterior derivative, and Weitzenbock type curvature operator.
result Derivation of Bochner-Weitzenbock type formula leading to vanishing theorems.
Paper interprets UMAP and t-SNE as probabilistic MAP inference.
problem Understanding and interpreting UMAP and t-SNE.
method Interprets UMAP and t-SNE as MAP inference methods corresponding to a probabilistic model of the graph Laplacian.
result Shows UMAP and t-SNE can be understood as probabilistic inference methods.
Efficiently fits stratified models with Laplacian regularization for categorical features.
problem Traditional stratified models fit separate models for each categorical value, which can be inefficient and lack flexibility.
method Proposes a distributed ADMM method for fitting stratified models with Laplacian regularization.
result Improves model performance and allows fitting for categorical values not in training data.
Proposes a tensor Laplacian-based method for better subspace clustering of non-uniformly distributed data.
problem LRR's inability to handle non-uniform data distribution and local information loss.
method Tensor Laplacian Regularized Low-Rank Representation (TLRR) using hypergraph model and tensor Laplacian algorithm.
result Higher accuracy and precision in subspace clustering compared to state-of-the-art methods.
Study contact manifold heat kernels under Riemannian metrics blow-up.
problem Analyze spectral invariants on contact manifolds.
method Examine Hodge Laplacian heat kernel behavior under Riemannian metrics.
result Contact versions of eta-invariant and analytic torsion are topological.
We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous t…
As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is th…
For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density fun…
Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.
problem Estimating regression functions on unknown smooth submanifolds.
method Random geometric graph structure, Bayesian priors based on random basis expansion in graph Laplacian eigenbasis.
result Posterior contraction rates are minimax optimal for any positive smoothness index.
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.
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). We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian.
Let M be a compact smooth manifold equipped with a positive smooth density μ and H be a smooth distribution endowed with a fiberwise inner product g. We define the Laplacian ΔH associated with (H,μ,g) and prove that it gives rise to an unbounded self-adjoint operator in L2(M,μ). Then, assuming that H …
We prove a central limit theorem for the components of the eigenvectors corresponding to the d largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized La…
The paper develops a method to sparsify magnetic Laplacians using multi-type spanning forests.
problem Sparsifying magnetic Laplacians for large and dense graphs.
method Sampling multi-type spanning forests using a determinantal point process.
result The method provides statistical guarantees for estimating the connection Laplacian.
GS-B3SE improves label shift estimation by smoothing priors on a graph.
problem Label shift adaptation when source and target distributions share conditional but not marginal probabilities.
method Graph-Smoothed Bayesian Black-Box Shift Estimator (GS-B3SE) places Laplacian-Gaussian priors on log-priors and confusion-matrix columns tied by a label-similarity graph. result GS-B3SE produces a tractable posterior with HMC or Newton-CG schemes, proving identifiability, contraction, and robustness. Proposes a novel approach for cluster-aware matching using Laplacian Optimal Transport.
problem Matching point clouds with intrinsic cluster structure requires robust region-to-region alignment over precise point-to-point correspondence.
method Laplacian Optimal Transport (LapOT) with regularization for cluster-aware matching and Refined Simultaneous Clustering (RSC) for consistent partitions.
result Laplacian Optimal Transport produces more consistent and meaningful alignments between point clouds.
In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is asymptotically uniformly distributed. We pay special attention to the case of locally …
We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimens…
A novel approach for augmenting histopathological images by blending Gaussian-Laplacian pyramids.
problem Data imbalance and inter-patient variability in histopathological images.
method Image blending using Gaussian-Laplacian pyramids to distribute inter-patient variability.
result Promising gains in performance compared to existing data augmentation techniques.
Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionali…
Study graph-based algorithms for multi-manifold clustering with sufficient conditions.
problem Clustering data from a union of manifolds with different dimensions and intersections.
method Investigate sufficient conditions for similarity graphs to capture geometric information.
result High probability error bounds for spectral approximation of tensorized Laplacian.
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.
Unified spectral clustering for sparse networks with heterogeneous degrees.
problem Efficiently detecting communities in sparse networks with varying degrees.
method Developed a parametrized regularized Laplacian matrix for spectral clustering.
result Improved parametrization accounts for network heterogeneity and community hardness.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.
problem Analyzing Riemannian submanifolds from point cloud data.
method Constructing deformed Hodge Laplacians and empirical operators from point clouds, proving convergence properties.
result Empirical spectral cluster contains the k-th Betti number and converges to harmonic k-forms. 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.
Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.
problem Analyzing sectoriality of Laplacian and Lichnerowicz Laplacian on asymptotically hyperbolic spaces.
method Proves sectoriality in weighted Hölder spaces using asymptotically hyperbolic metrics.
result Analytic semigroups apply, yielding well-posedness results for parabolic evolution equations.
New sampling method for heavy-tailed distributions using Langevin Algorithm.
problem Sampling from heavy-tailed distributions efficiently.
method Transformed Unadjusted Langevin Algorithm on specific transformations.
result Polynomial-order oracle complexities for certain heavy-tailed densities.
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…
Study local invariants and geometry of sub-Laplacian on H-type foliations.
problem Characterize the geometry and invariants of H-type foliations.
method Use Bott connection, scalar curvature, and new invariant to analyze sub-Riemannian geometry.
result Express second heat invariant as a linear combination of scalar curvature and new invariant.