The paper shows how discrete Dirichlet partitions converge to continuum partitions.
problem Consistency of Dirichlet partitions on graphs and domains.
method Variational formulation and Γ-convergence of discrete to continuum Dirichlet energies.
result Discrete Dirichlet energies Γ-converge to continuum Dirichlet energies.
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…
Proves an Euler-type formula for Möbius strip partitions.
problem No specific problem stated; focuses on a mathematical formula.
method Analyzes partitions of the Möbius strip.
result Proves an Euler-type formula for Möbius strip partitions.
Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.
problem Determining Courant-sharp eigenvalues on a Möbius strip.
method Analyzing the eigenvalues and nodal patterns of the Möbius strip.
result Only the first and second eigenvalues are Courant-sharp on the Möbius strip.
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…
The study shows that random surfaces built from polygons converge to a Poisson-Dirichlet partition.
problem Understanding geometric properties of random surfaces constructed from polygons.
method Uniformly pairing polygon sides to form surfaces, analyzing degree sequences and geometric properties using probabilistic techniques.
result Several geometric properties of the graph are universal, converging to a Poisson-Dirichlet partition as no∞. Study on length distribution of random multicurves on large genus surfaces converging to Poisson-Dirichlet distribution.
problem Length statistics of random multicurves on large genus hyperbolic surfaces.
method Analytical proof of convergence to Poisson-Dirichlet distribution as genus tends to infinity.
result Mean lengths of the three longest components converge to specific percentages of total length as genus increases.
DPMM-CFL clusters clients for federated learning without fixed K, improving performance.
problem Improving federated learning performance under non-IID client heterogeneity.
method DPMM-CFL uses a Dirichlet Process Mixture Model to infer both cluster number and client assignments.
result DPMM-CFL optimizes per-cluster federated objectives and jointly infers cluster number and assignments.
In this note we provide detailed derivations of two versions of small-variance asymptotics for hierarchical Dirichlet process (HDP) mixture models and the HDP hidden Markov model (HDP-HMM, a.k.a. the infinite HMM). We include derivations for the probabilities of certain CRP and CRF partitions, which are of more general…
We present a Bayesian nonparametric framework for multilevel clustering which utilizes group-level context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-m…
CLDA improves topic modeling for large, dynamic datasets.
problem Dynamic topic modeling for large, diverse text streams.
method Data decomposition followed by topic modeling on segments, then clustering.
result Very fast runtime and insight into topic composition over time.
Novel threefold partitioning of L2-spinor space on cone links.
problem Understanding equivariant Riemann-Roch defects in complex spaces with conic singularities.
method Investigates supertraces over local cohomology groups and spectral asymmetry.
result Novel complex equivariant ξT and ηT invariants defined. HIRM models noisy, sparse, heterogeneous relational data using hierarchical clustering and Dirichlet processes.
problem Modeling noisy, sparse, and heterogeneous relational data.
method Hierarchical Chinese restaurant process and Dirichlet process mixture for clustering and modeling relation values.
result HIRM generalizes standard models and discovers relational structure in real-world datasets.
The Gibbs-type Indian buffet process extends feature allocation models with power-law behaviors.
problem Feature allocation models with power-law behaviors.
method Gibbs-type random measures parameterization.
result Properties of Gibbs-type partitions allow for black-box posterior inference.
Consensus Monte Carlo clusters big data with shared anchors.
problem Clustering and feature allocation in large datasets.
method Bayesian nonparametric models, Dirichlet process, Indian buffet process, consensus Monte Carlo.
result Valid for various sampling models and priors.
The paper proves a new semiclassical limit result for abstract Schrödinger operators.
problem Abstract Schrödinger operators on locally compact spaces.
method Probabilistic proof using the principle of not feeling the boundary.
result A new semiclassical limit result for partition functions.
Bayesian nonparametric method for hierarchical clustering.
problem Hierarchical non-overlapping clustering of a dataset.
method Combining nCRP and HDP for complex latent mixture features.
result Solid empirical results compared to existing algorithms.
A new method for multiclass calibration using vector quantization.
problem Challenges in multiclass calibration, especially in high-stakes settings.
method Compositional approach via Vector Quantization (VQ) to learn region-specific calibration maps.
result Significant improvements in local calibration with competitive global calibration and predictive performance.
New EPM models improve model shrinkage in edge partition models.
problem Overfitting and inappropriate model shrinkage in EPMs.
method Proposed two novel EPM models: CEPM and DEPM, incorporating constrained and Dirichlet priors respectively.
result IDEPM model shows state-of-the-art performance in generalization and prediction.
Exemplar-based clustering methods have been shown to produce state-of-the-art results on a number of synthetic and real-world clustering problems. They are appealing because they offer computational benefits over latent-mean models and can handle arbitrary pairwise similarity measures between data points. However, when…
New families of non-tiling domains satisfy Pólya's conjecture.
problem Finding non-tiling domains that satisfy Pólya's conjecture.
method Analyzing partitioning and eigenvalue orders of domains.
result Existence of families of non-tiling domains satisfying Pólya's conjecture.
We define the beta diffusion tree, a random tree structure with a set of leaves that defines a collection of overlapping subsets of objects, known as a feature allocation. A generative process for the tree structure is defined in terms of particles (representing the objects) diffusing in some continuous space, analogou…
Formulas for heat equation solutions on manifolds with boundary.
problem Solving heat equations on manifolds with boundaries.
method Path integral formulas for approximating solutions.
result Approximations of solutions by integrals over path spaces.
New model clusters set-valued data without specifying cluster number.
problem Clustering set-valued data and unknown number of clusters.
method Dirichlet Process mixture of Poisson random finite sets, MCMC inference.
result Model discovers extremely unbalanced clusters.
We developed a new quantum annealing (QA) algorithm for Dirichlet process mixture (DPM) models based on the Chinese restaurant process (CRP). QA is a parallelized extension of simulated annealing (SA), i.e., it is a parallel stochastic optimization technique. Existing approaches [Kurihara et al. UAI2009, Sato et al. UA…
Bayesian nonparametric model for image segmentation using a generalized Swendsen-Wang algorithm.
problem Unsupervised image segmentation of pixels into homogeneous regions.
method Combination of Potts-like spatial smoothness and prior on partitions controlled by a Bayesian nonparametric model.
result Competitive performance in terms of RAND index compared to popular methods.
Proposes a nonparametric tensor factorization for sparse data.
problem Handling sparse tensor data with structural and interpretability benefits.
method Hierarchical Gamma processes and Poisson random measures for tensor-valued process, Dirichlet processes for sampling entry indices, Gaussian processes for values.
result Demonstrates superior performance on benchmark datasets.
Study finds only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
problem Determining Courant-sharp eigenvalues for compact flat surfaces.
method Analyzing flat Klein bottle and cylinders, proving only first and second eigenvalues are Courant-sharp.
result Only first and second eigenvalues are Courant-sharp for flat Klein bottle and cylinders.
We propose a restricted collapsed draw (RCD) sampler, a general Markov chain Monte Carlo sampler of simultaneous draws from a hierarchical Chinese restaurant process (HCRP) with restriction. Models that require simultaneous draws from a hierarchical Dirichlet process with restriction, such as infinite Hidden markov mod…
We describe the combinatorial stochastic process underlying a sequence of conditionally independent Bernoulli processes with a shared beta process hazard measure. As shown by Thibaux and Jordan [TJ07], in the special case when the underlying beta process has a constant concentration function and a finite and nonatomic …
The paper examines the stability of Killing cylinders in hyperbolic space.
problem Stability of Killing cylinders in hyperbolic space.
method Explicit computation of Morse index for Jacobi operator on various support surfaces.
result Delaunay surfaces can be bifurcated from Killing cylinders supported on geodesic planes.
Bayesian mixture models are widely applied for unsupervised learning and exploratory data analysis. Markov chain Monte Carlo based on Gibbs sampling and split-merge moves are widely used for inference in these models. However, both methods are restricted to limited types of transitions and suffer from torpid mixing and…
Investigates a conjugate prior for Dirichlet distribution.
problem No specific problem stated; focuses on mathematical investigation.
method Investigates a conjugate class for the Dirichlet distribution within the exponential family.
result Identifies a conjugate prior for the Dirichlet distribution.
New algorithm speeds Bayesian nonparametric model inference.
problem Slow inference in Bayesian nonparametric models.
method Decompose random measures into finite and infinite sub-measures; use different algorithms for each.
result Hybrid algorithm improves scalability and mixing.
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 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.
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.
New method simplifies Bayesian inference for multi-Dirichlet priors.
problem Inference for models with hierarchical Multi-Dirichlet priors is tricky.
method Auxiliary variable scheme simplifies joint distribution of model parameters.
result Efficient inference schemes derived using the auxiliary variable scheme.
Study solves Dirichlet problem for higher-dimensional submanifolds.
problem Existence and uniqueness of graphical maximal submanifolds.
method General existence and uniqueness results for any codimension.
result General existence and uniqueness results for graphical maximal submanifolds of higher codimension.
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.
The paper studies higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues.
problem Analyzing eigenvalues of higher order Dirichlet-to-Neumann maps on graphs.
method Introducing and studying higher order Dirichlet-to-Neumann maps on graphs, deriving estimates on eigenvalues.
result Raulot-Savo-type estimates on the eigenvalues of the DtN maps.
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. Paper shows approximate Dirichlet domain works as well as exact one for tiling hyperbolic balls.
problem Empirical success of SnapPea's length spectrum algorithm despite using approximate data.
method Showed under certain conditions, approximate Dirichlet domain can perform equivalently to exact one.
result Empirical success of SnapPea's length spectrum algorithm explained.
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.
Many solutions found for a ball boundary problem.
problem Finding solutions for a Dirichlet problem on balls.
method Infinitely many solutions provided.
result Many solutions found for a Dirichlet problem on balls.