Paper studies Laplace operator estimates in harmonic map heat flows.
problem Estimating Laplace operator in harmonic map heat flows outside singularities.
method Investigates estimates using spherical coordinates for T 2 \mathbb{T}^2 T 2 and T 3 \mathbb{T}^3 T 3 boundary conditions. result Provides higher-order estimates for the Ericksen--Leslie system.
Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.
problem Estimating eigenvalues of poly-Laplace operator on subgraphs of lattice graphs.
method Introduced discrete poly-Laplace operator, derived upper and lower bounds for eigenvalues.
result Poly-Laplace eigenvalues are at least squares of lower-order poly-Laplace eigenvalues.
QLA improves Bayesian uncertainty estimation for DNNs without increasing computational cost.
problem Overconfident out-of-distribution predictions from DNNs.
method Proposes Quadratic Laplace Approximation (QLA) to improve Bayesian uncertainty quantification.
result QLA yields modest yet consistent uncertainty estimation improvements over Linearized Laplace Approximation (LLA) on five regression datasets.
The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.
problem Analyzing heat-type equations on manifolds with specific boundary conditions.
method Proving Schauder estimates for the Laplace-Beltrami operator on manifolds with fibered boundaries and a Φ-metric.
result The proof of parabolic Schauder estimates for the Laplace-Beltrami operator.
Logistic Gaussian process (LGP) priors provide a flexible alternative for modelling unknown densities. The smoothness properties of the density estimates can be controlled through the prior covariance structure of the LGP, but the challenge is the analytically intractable inference. In this paper, we present approximat…
Improved bounds for estimating discrete distributions in KL divergence.
problem Estimating discrete distributions in KL divergence with accuracy.
method Used Laplace estimator and established concentration bounds.
result Deviation from mean scales as k / n \sqrt{k}/n k / n for n ≥ k n \ge k n ≥ k . A novel Laplace-approximated Bayesian Tensor Network Kernel Machine (LA-TNKM) provides principled uncertainty estimates.
problem How to provide principled uncertainty estimates for tensor network kernel machines.
method Employing a linearized Laplace approximation for Bayesian inference.
result Consistently matches or surpasses Gaussian Processes and BNNs across diverse UCI regression benchmarks.
Modeling stock returns and volatility using a bivariate gamma generalized Laplace law.
problem Analyzing stock returns and volatility using a new statistical model.
method Maximum likelihood estimation for a bivariate generalized Laplace distribution, simplifying to linear regression.
result Explicit estimators derived with nonstandard convergence rates for certain parameter configurations.
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on L 2 L^{2} L 2 --Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates. result Almost sharp local L p L^{p} L p --Bernstein inequalities for p ∈ [ 1 , ∞ ] p\in[1,\infty] p ∈ [ 1 , ∞ ] . Bayesian meta-reinforcement learning improves over point estimates with Laplace approximation.
problem Improving meta-reinforcement learning by providing full posterior distributions.
method Augmenting point estimates with Laplace approximation for full posterior distributions.
result Our method performs similarly to variational baselines with fewer parameters.
We propose a generalized double Pareto prior for Bayesian shrinkage estimation and inferences in linear models. The prior can be obtained via a scale mixture of Laplace or normal distributions, forming a bridge between the Laplace and Normal-Jeffreys' priors. While it has a spike at zero like the Laplace density, it al…
Bayesian deep learning method using subnetwork inference.
problem Improving deep neural networks' calibration and efficiency.
method Perform inference over a subset of model weights, keeping others as point estimates.
result Subnetwork inference enables accurate predictive posteriors without full network approximations.
It is shown that the estimates obtained by Manfredo P. do Carmo and Detang Zhou, in their paper "Eigenvalue estimate on complete noncompact Riemannian manifolds and applications", for the first eigenvalue of the Laplace-Beltrami operator on open manifolds, via an oscillation theorem, can be naturally extended for the s…
A new method for uncertainty estimation in neural networks using existing optimization steps.
problem Uncertainty quantification in deep neural networks.
method L2M: Practical posterior Laplace approximation with optimization-driven second moment estimation.
result L2M method yields reasonable results without requiring changes in models or extra computational steps.
Improved Kuznecov remainder estimates for generic metrics.
problem Estimating period integrals of Laplace eigenfunctions on manifolds.
method Two-term asymptotic expansion and elimination of oscillatory second term.
result Improved remainder estimates for Baire-generic metrics.
This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.
Estimates for harmonic forms on a 3-Torus, proving their existence.
problem Existence of nowhere vanishing harmonic 1-forms on a 3-Torus.
method Explicit computation of injectivity estimates using the Laplace operator on the 3-Torus and its perturbations.
result Existence of a nowhere vanishing harmonic 1-form on a perturbed metric on the 3-Torus.
Variational Laplace improves Bayesian neural networks performance.
problem Improving Bayesian neural networks performance.
method Develops variational Laplace for BNNs, exploiting curvature of likelihood.
result Variational Laplace outperforms other inference methods.
We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.
Study small perturbations on low energy Laplace eigenfunctions.
problem Understanding small changes in low energy Laplace eigenfunctions.
method Investigates nodal geometry and topology, focusing on low frequency regimes and small perturbations.
result Highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions.
We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities continue to hold for conformal metrics, and moreover, extend naturally to minimal su…
Paper improves estimates for discrete Laplace in hyperbolic geometry.
problem Establishing compactness for discrete Laplace in hyperbolic geometry.
method Explicit estimates for discrete Laplace based on Glickenstein-Thomas formulation.
result New proofs of long time existence for Calabi flows in hyperbolic geometry.
Variational Laplace improves Bayesian neural network performance without sampling.
problem Improving Bayesian neural network performance and calibration.
method Develops a new variational Laplace method for BNNs, exploiting curvature of likelihood.
result Variational Laplace outperforms standard VI methods in test performance and calibration.
Estimates Laplace eigenvalues and diameter for Lie group metrics.
problem Estimating Laplace eigenvalues and diameter for left-invariant metrics on compact Lie groups.
method Relates left-invariant metrics to positive definite matrices and uses eigenvalue properties.
result Partial answers to Eldredge's conjecture on Laplace eigenvalues and diameter.
Estimates the index of the Laplace operator on planar domains with Robin boundary condition.
problem Index estimates for planar domains with Robin boundary condition
method Combines conformal and spectral techniques with topology of the domain.
result Lower bounds for the index in terms of the number of boundary components.
New method optimizes hyperparameters in deep learning models efficiently.
problem Manual hyperparameter tuning in deep learning models is inefficient and requires expertise.
method Introduces lower bounds to the linearized Laplace approximation of the marginal likelihood using neural tangent kernels.
result Optimization of hyperparameters can be significantly accelerated using the method.
Paper introduces p-Laplace equations for curvature in conformal geometry.
problem Study of geometry and topology of manifolds.
method Introducing p-Laplace equations for intermediate Schouten curvature.
result Nonnegative intermediate Schouten curvature leads to estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups.
Optimizes survey design for private mean estimation with reduced variance.
problem Minimizing variance in private mean estimation with privacy constraints.
method Formulates optimal survey design as an optimization problem, determining optimal subsampling sizes to minimize variance.
result Identifies the first privacy-aware stratified sampling scheme that minimizes variance under different privacy mechanisms.
Meta-learning variance reduced via Laplace approximation for regression tasks.
problem High variance in meta-learning due to limited support data for each task.
method Laplace approximation to estimate posterior variance and reduce gradient estimate variance.
result Effective variance reduction in meta-learning, improving generalization performance.
A family of parsimonious shifted asymmetric Laplace mixture models is introduced. We extend the mixture of factor analyzers model to the shifted asymmetric Laplace distribution. Imposing constraints on the constitute parts of the resulting decomposed component scale matrices leads to a family of parsimonious models. An…
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…
LaLoRA prevents forgetting in LoRA fine-tuning.
problem Catastrophic forgetting in fine-tuned models.
method LaLoRA applies Laplace approximation to LoRA weights for regularization.
result Improved learning-forgetting trade-off with controllable regularization strength.
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p p p -Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p p p -Wasserstein distance between positive and negative parts of Laplace eigenfunctions. Study eigenvalues of a generalized p-Laplacian on forms.
problem Estimating the first nonzero eigenvalue of a weighted p-Laplacian.
method Introduced a weighted p-Laplace operator for differential forms and derived sharp lower bounds.
result Extended and improved eigenvalue estimates for the p-Laplacian.
Two private algorithms estimate Jaccard similarity efficiently.
problem Estimating Jaccard similarity while preserving user privacy.
method Extends MinHash with Generalized Randomized Response and Laplace Mechanism.
result Achieves privacy-utility trade-off with theoretical bounds and experiments.
In this paper, we would like to give an answer to \textbf{Problem 1} below issued firstly in [J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of th…
The study bounds Riesz transforms on manifolds with controlled curvature.
problem Bounding Riesz transforms on manifolds with controlled curvature.
method Established L p L^p L p -boundedness of local covariant Riesz transforms for differential forms. result Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
Let Δ M Δ_M Δ M be the Laplace operator on a compact n n n -dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u : Δ u + λ u = 0 u:Δu + λu =0 u : Δ u + λ u = 0 . In dimension n = 2 n=2 n = 2 we refine the Donnelly-Fefferman estimate by showing that H 1 ( { u = 0 } ) ≤ C λ 3 / 4 − β H^1(\{u=0 \})\le Cλ^{3/4-β} H 1 ({ u = 0 }) ≤ C λ 3/4 − β , β ∈ ( 0 , 1 / 4 ) β\in (0,1/4) β ∈ ( 0 , 1/4 ) . The proof employs the Donnelli-Fef…
A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the general inverse Gaussian distribution. This approach is mathematically elegant and relatively comput…
Let ( X , h ) (X,h) ( X , h ) be a compact and irreducible Hermitian complex space of complex dimension v > 1 v>1 v > 1 . In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…
Improved estimate for eigenvalues of minimal hypersurfaces in spheres.
problem Estimating the first non-zero eigenvalue of minimal hypersurfaces in spheres.
method Proved an improved lower bound for the first non-zero eigenvalue of the induced Laplace-Beltrami operator on minimal hypersurfaces in spheres.
result First explicitly computable improvement on the eigenvalue lower bound without additional assumptions.
The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.
problem Computing conformal mappings between Riemannian surfaces.
method Adapting the conjugate function method to Riemannian surfaces using h p hp h p -adaptive finite element methods. result Highly accurate numerical computations of conformal mappings on surfaces, including complex geometries.
New mechanism for pure differential privacy on functional summaries using Laplace-like process.
problem Challenges in achieving differential privacy for complex, structured functional summaries.
method Independent Component Laplace Process (ICLP) mechanism for infinite-dimensional Hilbert space.
result Effective enhancement of utility of private summaries through oversmoothing.
The study bounds heat kernel for manifolds with specific curvature conditions.
problem Estimating heat kernel for manifolds with Bakry-Émery Ricci curvature.
method Gaussian upper bound for heat kernel, proving L^1-Liouville property, deriving eigenvalue bounds.
result Established Gaussian upper bound for heat kernel, derived eigenvalue bounds.
New method estimates marginal likelihood for deep learning models using training data alone.
problem Estimation difficulties in marginal likelihood for model selection in deep learning.
method Scalable marginal likelihood estimation based on Laplace's method and Gauss-Newton approximations.
result Estimate outperforms cross-validation and manual tuning on various datasets.
We give various estimates of the first eigenvalue of the p p p -Laplace operator on closed Riemannian manifold with integral curvature conditions.
Bayesian tensor train kernel machine uses Laplace approximation for scalable GP regression.
problem Scalability limitations of Gaussian process regression.
method Bayesian tensor train kernel machine with Laplace approximation and variational inference.
result VI replaces cross-validation and offers up to 65x faster training.