arXiv research

Sharp bounds for Dirichlet sums lead to improved Bayesian algorithm analysis.

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…

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

We describe a simple and efficient procedure for approximating the Lévy measure of a $\text{Gamma}(α,1)$ random variable. We use this approximation to derive a finite sum-representation that converges almost surely to Ferguson's representation of the Dirichlet process based on arrivals of a homogeneous Poisson process.…

In this paper, the elastic Dirichlet-to-Neumann map $Ξ_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(Ω,g)$ with smooth boundary $\partial Ω$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map $Ξ_g$. We …

Interior $C^{2}$ estimates for sum Hessian quotient equations on Riemannian manifolds

Presented a simple group presentation for degree four cactus group.

We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kaehler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivati…

The restricted class of quasicircles sometimes called the "Weil-Petersson-class" has been a subject of interest in the last decade. In this paper we establish a Sokhotski-Plemelj jump formula for WP-class quasicircles, for boundary data in a certain conformally invariant Besov space. We show that this Besov space is pr…

Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreas…

The paper introduces a new model to correct bias in treatment effect estimates due to sample selection.

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}$ It…

A Dirichlet $k$-partition of a domain $U \subseteq \mathbb{R}^d$ is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit va…

Bayes-UCBVI tackles reinforcement learning with a new upper confidence bound method.

New heat trace coefficients reveal curvature effects in polygonal domains.

In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of Differential Equations, 255 (2013), 220-233). In particular, if $l=2$ and $k$ is large enou…

The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb N,$ $k…

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} λ_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary…

The paper studies Gauss sums and their applications in algebra and topology.

The Special Lagrangian Potential Equation for a function $u$ on a domain $Ω\subset {\bf R}^n$ is given by ${\rm tr}\{\arctan(D^2 \,u) \} = θ$ for a contant $θ\in (-n {π\over 2}, n {π\over 2})$. For $C^2$ solutions the graph of $Du$ in $Ω\times {\bf R}^n$ is a special Lagrangian submanfold. Much has been understood abou…

New methods for predicting compositional data using conformal prediction.

In this paper (Part I) and its sequels (Part II and Part III), we analyze the structure of the space of solutions to the epsilon-Dirichlet problem for the Yang-Mills equations on the 4-dimensional disk, for small values of the coupling constant epsilon. These are in one-to-one correspondence with solutions to the Diric…

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper m…

Derive Dirichlet scalar curvature energy functional variation formula

Adaptive Bayesian model for covariate-dependent power spectra analysis.

Solves a specific Dirichlet problem on Hermitian manifolds.

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

Study classifies ruled surfaces critical to Dirichlet energy.

Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.

LDTA expands LDA's topic modeling capacity with tree-structured priors.

Solves a specific Dirichlet problem for Lagrangian mean curvature equations.

Solves a specific Dirichlet problem on Riemannian manifolds.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

Enhances topic models to better handle polysemous words.

TPBS models improve robustness to overfitting with localized Dirichlet energy regularization.

Study on geometry of Dirichlet distributions using Fisher-Rao metric.

Graph Laplacians adapt to different manifold dimensions, while Dirichlet energies converge to a tensorized Dirichlet energy.

Discriminative classifier for compositional data using hierarchical mixture of Generalized Dirichlet models.

The paper finds universal inequalities for eigenvalues on hyperbolic spaces.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

Paper solves Dirichlet problem for $p$-convex hypersurfaces with curvature constraints.

Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.

New portfolios outperform traditional methods by using factor weights.

A new method detects outliers using ensembles of Dirichlet process mixtures.

Dirichlet pruning compresses neural networks by removing unimportant units.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

We derive and approximate the conjugate prior of Dirichlet and beta distributions.

Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space