Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
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.
New portfolios outperform traditional methods by using factor weights.
problem Improving portfolio allocation in markets driven by factors.
method Factor-weighted Dirichlet portfolios outperform uniform Dirichlet portfolios.
result Factor-weighted portfolios outperform uniformly sampled portfolios in market returns.
Gradient estimate for harmonic functions with boundary condition proved.
problem Proving gradient estimates for harmonic functions with boundary conditions.
method Using weighted f-harmonic functions and infinite dimensional Bakry-Emery Ricci tensor. result Gradient estimates for positive f-harmonic functions with Dirichlet boundary condition. Solves constant mean curvature Dirichlet problem on catenoids with improved estimates.
problem Solving constant mean curvature Dirichlet problem on catenoidal necks.
method Found solutions in exponentially weighted Hölder spaces with non-integer weight.
result Improved estimate to γ=1 by comparing solutions with their limits on the disk.
Sharp bounds for Dirichlet sums lead to improved Bayesian algorithm analysis.
problem Improving Bayesian algorithm performance through precise deviation bounds.
method Novel integral representation of Dirichlet sum density, Gaussian approximation, complex analysis.
result Significantly sharpened regret bounds for Multinomial Thompson Sampling.
Dirichlet-Neumann duality for Riemannian submersions
problem Spectral geometry of Riemannian submersions
method Summation formula for reciprocal of basic Dirichlet eigenvalues
result Supersymmetric duality between basic Dirichlet and Neumann spectra
The paper explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted p-Laplacian first eigenvalues. Bayesian models that mix multiple Dirichlet prior parameters, called Multi-Dirichlet priors (MD) in this paper, are gaining popularity. Inferring mixing weights and parameters of mixed prior distributions seems tricky, as sums over Dirichlet parameters complicate the joint distribution of model parameters. This paper s…
Study on a weighted Suita conjecture for higher derivatives and their geometric properties.
problem Analyzing the Suita conjecture for higher derivatives with weights.
method Examining the set of points for equality in a weighted Suita conjecture and relating it to harmonic functions and Dirichlet problems.
result Relations between the set of points and integer-valued points of harmonic functions and Dirichlet problems for planar domains.
In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi metric, the K-energy, the degenerate complex Hessian equation. The second part is on…
Paper proves stability and Dirichlet problem for translating hypersurfaces.
problem Stability and Dirichlet problem for translating hypersurfaces.
method Analyzes translating solitons in en+k, proves stability conditions, and studies Dirichlet problem. result Proves the infimum of mean curvature is zero for translating solitons and conditions for stability.
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.
We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the p-Laplacian. In the case of the closed eigenvalue problem and the Neuma…
Estimates eigenvalues on weighted manifolds with curvature.
problem Estimating eigenvalues of Dirichlet and Neumann problems.
method Using Bakry-Émery Ricci curvature.
result Established a stability condition for h-minimal hypersurfaces.
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.
A new method uses a product of experts with Dirichlet variables to approximate complex distributions.
problem Approximating complex distributions with tractable models.
method A product of experts with auxiliary Dirichlet variables, using a Feynman identity to sample and optimize.
result The method efficiently approximates complex distributions using a product of experts and Dirichlet variables.
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 …
This paper proposes Dirichlet Variational Autoencoder (DirVAE) using a Dirichlet prior for a continuous latent variable that exhibits the characteristic of the categorical probabilities. To infer the parameters of DirVAE, we utilize the stochastic gradient method by approximating the Gamma distribution, which is a comp…
The article characterizes gradient ρ-Einstein solitons under specific conditions.
problem Characterizing gradient ρ-Einstein solitons with certain properties.
method Analyzing solitons with vector fields of bounded norm, finite weighted Dirichlet integral, and specific Ricci curvature restrictions.
result Non-trivial complete gradient ρ-Einstein solitons with finite weighted Dirichlet integral and certain Ricci curvature restrictions are of constant scalar curvature and steady.
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
Proposes a new model for clustering multiplex networks with compositional data.
problem Clustering multiplex networks with multiple types of relations and compositional data.
method Multiplex Dirichlet stochastic block model for compositional networks.
result Validated through simulation and applied to international export data.
Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.
problem Bounding asymptotics of a conformal invariant under degeneration of Riemann surfaces.
method Meyer-Vietoris formula, gluing, height function on moduli space, properness of height function, Steklov isospectral metrics, Laplacian with Dirichlet/Neumann boundary conditions.
result Properness of height function on moduli space of genus zero hyperbolic surfaces implies compactness theorem for Steklov isospectral metrics.
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
Efficiently approximates uncertainty in classification models using Dirichlet distributions.
problem Inefficient computation of uncertainty estimates in Bayesian deep learning.
method Revised Laplace Bridge method to construct a Dirichlet approximation of softmax output distributions.
result The Dirichlet approximation leads to more efficient computation and better uncertainty estimates.
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…
We study Riemannian manifolds with boundary under a lower N-weighted Ricci curvature bound for N at most 1, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …
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 on Bayesian transformers finds issues with weight-space inference and prior specification.
problem Challenges in obtaining meaningful uncertainty estimates for transformer models.
method Proposed a novel method based on implicit reparameterization of the Dirichlet distribution for variational inference on attention weights.
result Proposed method performs competitively with baselines in estimating predictive uncertainty.
Study connects curvature to graph theory and reveals differences.
problem Exploring differences between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
method Real (1,1)--forms and Weitzenböck curvature operator used to represent graph Dirichlet energy.
result Curvature differences illuminated between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
A new model clusters network nodes based on relative edge weights.
problem Clustering networks ignores node capacities, leading to biased results.
method Proposes a Dirichlet stochastic block model for composition-weighted networks.
result Validated on simulated and real-world networks, showing improved clustering accuracy.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…
DrNAS improves neural architecture search with Dirichlet distribution and progressive learning.
problem Efficiently search for neural architectures with improved generalization and exploration.
method Formulates architecture search as a distribution learning problem using Dirichlet distribution and gradient-based optimization. Introduces a progressive learning scheme to handle large-scale tasks.
result Achieves state-of-the-art results on CIFAR-10 and ImageNet, demonstrating improved generalization and exploration.
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.
Given a compact Riemannian manifold (M, g) and two positive functions ρ and σ, we are interested in the eigenvalues of the Dirichlet energy functional weighted by σ, with respect to the L 2 inner product weighted by ρ. Under some regularity conditions on ρ and σ, these eigenvalues are those of the operator …
The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.
problem Eigenvalue comparison theorems for Witten-Laplacian and weighted p-Laplacian on manifolds with modified Ricci curvature. method Established Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and weighted p-Laplacian on geodesic balls. result Successfully set up eigenvalue comparison theorems for the Witten-Laplacian and weighted p-Laplacian. RL learns to ignore factors in factor investing portfolios.
problem Combining factor investing and reinforcement learning for optimal portfolio allocation.
method RL agent learns through sequential allocations based on firms' characteristics using Dirichlet distributions.
result RL-based portfolios are very close to equally-weighted allocations, indicating agnostic factor learning.
The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.
problem Generalizing spin geometry to weighted manifolds and defining a new mass.
method Investigates spectral properties of the weighted Dirac operator and defines a new mass.
result Defines a new mass for weighted asymptotically Euclidean manifolds and shows its monotonicity under Ricci flow.
Precise estimation of uncertainty in predictions for AI systems is a critical factor in ensuring trust and safety. Deep neural networks trained with a conventional method are prone to over-confident predictions. In contrast to Bayesian neural networks that learn approximate distributions on weights to infer prediction …
We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the proces…
Bayesian neural networks improve with summary information and Dirichlet process.
problem Lack of prior knowledge in BNNs for complex architectures.
method Incorporates external summary information about predicted probabilities using a Dirichlet process.
result Improves model accuracy, uncertainty calibration, and robustness.
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower N-weighted Ricci curvature bounds with ε-range to study comparison geometry. result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
Capital distribution curve is defined as log-log plot of normalized stock capitalizations ranked in descending order. The curve displays remarkable stability over periods of time. Theory of exchangeable distributions on set partitions, developed for purposes of mathematical genetics and recently applied in non-parametr…
We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…
Graph neural networks over-smooth when layers increase, reducing discriminative power.
problem Over-smoothing in graph neural networks reduces model performance as the number of layers increases.
method Analyzed over-smoothing in general graph neural network architecture using Dirichlet energy.
result The Dirichlet energy of embeddings converges to zero, leading to loss of discriminative power.
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.