A new method for density estimation using nearest neighbor Dirichlet mixtures.
problem Slow and unstable Bayesian density estimation methods.
method Nearest neighbor grouping, local Bayesian parametric models, Dirichlet prior, Monte Carlo sampling.
result Effective density estimation with improved computational efficiency.
We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the c…
Wave equation map reveals manifold's structure.
problem Reconstructing Lorentzian manifold from wave equation map.
method Analyzing Schwartz kernel and boundary light observation set.
result Full Lorentzian structure can be recovered under geometric assumptions.
For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.
problem Investigating intrinsic ultracontractivity for domains in negatively curved manifolds.
method Using volume doubling property, Poincaré inequality, and Li-Yau Gaussian estimate for the Dirichlet heat kernel.
result The reciprocal of the bottom of the spectrum and the supremum of the torsion function are comparable with the square of the capacitary width for small capacitary width.
The paper compares heat kernels on manifolds with Robin boundary conditions.
problem Comparing heat kernels on manifolds with different boundary conditions.
method Proving comparison theorems for heat kernels on geodesic balls and minimal submanifolds.
result Eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds.
Method learns Dirichlet-to-Neumann maps on graphs using Gaussian processes.
problem Coupling multiphysics simulations on graphs with conservation constraints.
method Gaussian processes combined with discrete exterior calculus and maximum likelihood estimation.
result Data-driven predictions with uncertainty quantification on entire graph.
Improved Gaussian process experts model for complex data.
problem Limitations of standard Gaussian processes: scalability and predictive performance.
method Proposes a new mixture model of Gaussian process experts based on kernel stick-breaking processes.
result Improved predictive performance compared to existing models.
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.
This paper proposes a Hilbert space embedding for Dirichlet Process mixture models via a stick-breaking construction of Sethuraman. Although Bayesian nonparametrics offers a powerful approach to construct a prior that avoids the need to specify the model size/complexity explicitly, an exact inference is often intractab…
Heat kernel estimates on manifolds with mixed boundary conditions.
problem Estimating heat kernels on manifolds with ends and mixed boundary conditions.
method Global harmonic function construction and h-transform technique. result Two-sided heat kernel estimates for Riemannian manifolds with mixed boundary conditions.
The paper develops heat kernel comparison theorems and applies them to spectral geometry.
problem Developing mathematical tools for spectral geometry.
method Established weighted heat kernel comparison theorems for manifolds with bounded radial curvatures.
result Two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian.
Study on Dirichlet process mixtures for clustering consistency.
problem Consistency of clustering with Dirichlet process mixtures.
method Analysis of posterior distribution as sample size increases, focusing on consistency for the number of clusters.
result Consistency for the number of clusters can be achieved with a properly adapted concentration parameter in a Bayesian setting.
Latent Dirichlet Allocation models discrete data as a mixture of discrete distributions, using Dirichlet beliefs over the mixture weights. We study a variation of this concept, in which the documents' mixture weight beliefs are replaced with squashed Gaussian distributions. This allows documents to be associated with e…
Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.
problem Understanding Brownian motions and heat kernel bounds on specific geometric manifolds.
method Sharp Laplacian comparison theorems and Cheeger-Yau type lower bounds for heat kernels.
result Sharp Cheeger-Yau type lower bounds for heat kernels and Dirichlet eigenvalues of metric balls.
We shed a new light on the L1-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the L1-Liouville property is strictly weaker t…
Study reveals how to determine area and curvature from fluid flow resonances.
problem Determining geometric properties from fluid flow data.
method Asymptotic expansion of heat kernel and Steklov spectral invariants.
result Area and total mean curvature can be inferred from Steklov eigenvalues.
Given a data set and a subset of labels the problem of semi-supervised learning on point clouds is to extend the labels to the entire data set. In this paper we extend the labels by minimising the constrained discrete p-Dirichlet energy. Under suitable conditions the discrete problem can be connected, in the large da…
Geometric theory connects machine learning classifiers to differential geometry.
problem Classifying data points in machine learning.
method Mapping binary classification to vector bundles and differential geometry.
result Harmonic interpolation solves RKHS interpolation problems.
New method reveals corners of drum shapes.
problem Determining the shape of drum corners from its sound.
method Locality principle and calculations of heat kernels.
result Corners are spectral invariants of the Laplacian.
Posterior regularization enhances Bayesian hierarchical mixture clustering by improving node separation.
problem High nodal variance in BHMC trees, leading to weak separation between nodes at higher levels.
method Employing Posterior Regularization to impose max-margin constraints on nodes at every level.
result Improves cluster separation in BHMC models, enhancing overall model performance.
In this paper we study convex stochastic search problems where a noisy objective function value is observed after a decision is made. There are many stochastic search problems whose behavior depends on an exogenous state variable which affects the shape of the objective function. Currently, there is no general purpose …
WBCP improves conformal prediction for distribution shifts using weighted Dirichlet posteriors.
problem Handling distribution shifts in conformal prediction.
method Generalizes Bayesian Quadrature Conformal Prediction (BQ-CP) to arbitrary importance-weighted settings.
result WBCP maintains coverage guarantees while providing richer uncertainty information.
Note establishes a local maximum principle for Ricci flow under curvature conditions.
problem Preserving nonnegativity of curvature along Ricci flow with unbounded curvature.
method Combining scaling invariant curvature condition with Dirichlet heat kernel estimates.
result Unified and more direct proof of localized maximum principle.
BKP R package models spatially varying binomial probabilities efficiently.
problem Modeling spatially varying binomial probabilities efficiently.
method Beta Kernel Process (BKP) combining localized kernel-weighted likelihoods with conjugate beta priors.
result Closed-form posterior inference without requiring latent variables or intensive MCMC sampling.
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…
In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p-Laplacian (1<p<∞) obtained by Matei [A.-M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the p-Laplacian …
Proposes a new calibration error estimator for deep neural networks.
problem Improves calibration of deep neural networks, especially for canonical calibration.
method Uses a Dirichlet kernel density estimate to create a low-bias, trainable calibration error estimator.
result Asymptotically converges to true Lp calibration error, enabling efficient estimation and mini-batch updates. New learning rates derived for Tikhonov-regularized problems without kernel assumptions.
problem Learning rates for Tikhonov-regularized learning problems.
method Minimax adaptive rates derived using Fourier isocapacitary condition and interpolation theory.
result Derivation of minimax adaptive rates without requiring kernel assumptions.
Unified analysis of kernel-based and locally adaptive bandit optimization methods.
problem Performance of bandit optimization algorithms in RKHS functions.
method Investigates the relationship between kernel regularity and algorithmic performance, characterizing spectral properties of various kernels.
result Unified framework for analyzing kernel-based and locally adaptive bandit algorithms, deriving explicit regret bounds.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic p≥2. In this cas…
Enhances Fourier estimator performance for asynchronous event-data.
problem Improving correlation and covariance estimation on event-data.
method Implement and test NUFFT methods with different averaging kernels.
result Demonstrates improved performance and relationship between averaging scales.
Existing multi-view learning methods based on kernel function either require the user to select and tune a single predefined kernel or have to compute and store many Gram matrices to perform multiple kernel learning. Apart from the huge consumption of manpower, computation and memory resources, most of these models see…
Analyzes conformal anomaly in five dimensions, identifying new boundary conformal invariants.
problem Analyzing the conformal anomaly in five dimensions.
method Detailed analysis of boundary conformal invariants, computation of heat kernel coefficients.
result Identification of a new conformal invariant involving extrinsic curvature.
Semi-supervised clustering is the task of clustering data points into clusters where only a fraction of the points are labelled. The true number of clusters in the data is often unknown and most models require this parameter as an input. Dirichlet process mixture models are appealing as they can infer the number of clu…
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.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
Solves a specific Dirichlet problem on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds.
method Solves the Dirichlet problem for Monge-Ampère type equations for (n−1)-plurisubharmonic functions on Hermitian manifolds. result Solves a specific Dirichlet problem on Hermitian manifolds.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
The paper improves interpolation in generative models by using specific base distributions.
problem Unexpected side effects in linear interpolations of normalizing flows.
method Enforces a specific manifold using Dirichlet and von Mises-Fisher base distributions.
result Superior performance in terms of bits per dimension, FID, and KID scores for interpolation.
LDTA expands LDA's topic modeling capacity with tree-structured priors.
problem Limited expressiveness of Dirichlet priors in LDA for complex topic relationships.
method Introduces Latent Dirichlet-Tree Allocation (LDTA) with Dirichlet-Tree (DT) priors, and develops universal mean-field variational inference and Expectation Propagation.
result LDTA enables expressive, tree-structured priors over topic proportions, expanding modeling capacity of LDA.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
Solves a specific Dirichlet problem on Riemannian manifolds.
problem Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds.
method Derives existence of C1,1-solutions under appropriate assumptions. result Existence of C1,1-solutions. Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
Enhances topic models to better handle polysemous words.
problem Lack of polysemy handling in Gaussian latent Dirichlet allocation.
method Introduces a hierarchical structure to capture polysemy in Gaussian latent Dirichlet allocation.
result Significantly improves polysemy detection and provides more parsimonious topic representations.
TPBS models improve robustness to overfitting with localized Dirichlet energy regularization.
problem Global Dirichlet energy-based regularization fails for TPBS models due to perfect interpolation.
method Propose local Dirichlet energy regularization and two inference estimators.
result TPBS models outperform neural networks in overfitting regimes and maintain competitive performance otherwise.