Paper explores Polya's characterization of positive-definite kernels and random feature maps.
problem Characterizing positive-definite kernels and their random feature maps.
method Study Polya's criterion and derive novel kernels; compare random Fourier and binning feature maps.
result Random binning feature map yields a closer Euclidean inner product to the kernel.
Optimal bounds on rational points on algebraic curves established.
problem Bounding the number of rational points on algebraic curves of degree d. method Combination of smooth parametrizations and Pólya's criterion.
result Optimal upper bound Cd2H2/d(logH)κ with constants C and κ. 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.
Falsehood of Pólya's conjecture for spheres shown.
problem Disproving Pólya's eigenvalue conjecture for spheres.
method Comparison of Laplace spectrum and Weyl function of spheres.
result No analogue of Pólya's conjecture holds for spheres.
Proves Pólya's conjecture for thin products and Riemannian manifolds.
problem Proving Pólya's conjecture for specific geometric domains.
method Analyzes thin products and Riemannian manifolds, proving inequalities for eigenvalues.
result Proves Pólya's conjecture for thin products and related Riemannian manifolds.
The paper characterizes Pólya's conjecture for spheres and hemispheres, deriving inequalities and bounds.
problem Characterizing Pólya's conjecture for eigenvalues on spheres and hemispheres.
method Analyzing eigenvalues of the Laplace-Beltrami operator on spheres and hemispheres, deriving inequalities and bounds.
result Pólya's conjecture holds for hemispheres in the Neumann case but not in the Dirichlet case when n>2. Efficient GP classification using Polya-Gamma data augmentation.
problem Scalable Gaussian Process Classification.
method Stochastic variational approach with closed-form updates.
result Up to two orders of magnitude faster than state-of-the-art.
Study fine Pólya-Szegő inequalities in metric spaces with applications.
problem Fine Pólya-Szegő rearrangement inequalities in metric spaces.
method Theory of Sobolev and BV functions, synthetic Ricci bounds, isoperimetric inequality.
result New geometric and functional inequalities under Ricci lower bounds.
The paper proves inequalities for lattice eigenvalues, answering a question.
problem Finding bounds for Dirichlet Laplace eigenvalues on integer lattices.
method Proving analogues of existing inequalities for eigenvalues.
result Answers a question posed by Chung and Oden about eigenvalues on integer lattices.
The paper extends inequalities to closed Riemannian manifolds.
problem Extending inequalities from Euclidean domains to Riemannian manifolds.
method Proving explicit bounds using geometric quantities.
result Explicit bounds on Riemannian manifolds are derived.
Pólya's theorem extended to meromorphic functions on Riemann surfaces.
problem Distribution of zeros of iterated derivatives of meromorphic functions.
method Recasting local arguments into translation surfaces and using flat metrics.
result Asymptotic distribution of zeros on compact Riemann surfaces.
New BAM model connects tensor factorization and topic models using Polya Urns.
problem Efficiently modeling and analyzing nonnegative tensors and topic distributions.
method Dynamic generative model BAM based on Poisson process and Polya-Bayes process.
result Developed efficient simulation algorithms for NTF and topic models.
An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya--Szegö symmetrization principle for functions on Rn. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which i…
Efficient inference for nonparametric Hawkes processes using Pólya-Gamma augmentation.
problem Efficient inference for nonparametric Hawkes processes.
method Pólya-Gamma augmentation, EM algorithm, mean-field variational inference.
result The proposed algorithms can recover well the underlying prompting characteristics efficiently.
Improved Thompson Sampling for logistic contextual bandits reduces regret.
problem Minimizing regret in logistic contextual bandits with sequential actions and binary rewards.
method Polya-Gamma augmented Thompson Sampling (PG-TS) with efficient inference and Gibbs sampler.
result Achieves state-of-the-art performance on simulated and real data, reducing regret.
Study establishes Pólya-Szegő inequalities on submanifolds with small total mean curvature.
problem Analyzing Sobolev functions on submanifolds with curvature constraints.
method Developed Pólya-Szegő-type inequalities and derived corollaries.
result Proved sharp p-Log-Sobolev inequality for minimal submanifolds. We propose a new data-augmentation strategy for fully Bayesian inference in models with binomial likelihoods. The approach appeals to a new class of Polya-Gamma distributions, which are constructed in detail. A variety of examples are presented to show the versatility of the method, including logistic regression, negat…
In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain Ω in Rn. It is well known that the k-th Dirichlet eigenvalue λk obeys the Weyl asymptotic formula, that is, \[ λ_k\sim\frac{4π^2}{(ω_n\mathrm{vol}Ω)^\frac{2}{n}}k^\frac…
For a given bounded domain Ω⊂Rn with C1-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} wher…
Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.
problem Lower bounds for higher eigenvalues of the poly-Laplacian operator.
method Sharp inequalities and eigenvalue bounds in low and arbitrary dimensions.
result Improved lower bounds for eigenvalues of the poly-Laplacian in arbitrary dimensions.
Privacy-preserving Bayesian inference framework for sensitive data.
problem Protecting sensitive information in Bayesian data analysis.
method Differential privacy framework for Variational Bayes, tailored to CE and non-CE models.
result Effective privatization of VB for CE models and improved privacy for non-CE models.
Develops a data augmentation method for models with gamma functions.
problem Models with gamma functions lack natural conjugate priors, complicating inference and prediction.
method Derives Pólya Inverse Gamma distributions and applies them to scalable EM and MCMC algorithms.
result Provides scalable algorithms for inference and prediction in models with gamma functions.
Upper bound found for Steklov eigenvalues counting function.
problem Counting Steklov eigenvalues on compact manifolds with boundary.
method Used Weyl's law and Pólya's Conjecture in the Steklov case.
result Obtained an upper bound for the counting function.
Proposes a new model for better speech segmentation.
problem Improving speech segmentation accuracy.
method Integrates recurrent explicit duration variables into rSLDS and uses Pólya-gamma augmentation for inference.
result Demonstrates improved segmentation on various datasets.
The study proves Lieb-Thirring inequalities on hyperbolic manifolds.
problem Proving Lieb-Thirring inequalities on manifolds with negative constant curvature.
method Analytical proof of inequalities on hyperbolic manifolds.
result Discrete spectrum below the continuous spectrum (d−1)2/4,∞). Bayesian method improves few-shot classification accuracy.
problem Few-shot classification with small labeled datasets.
method Gaussian process classifier with Pólya-Gamma augmentation and one-vs-each softmax.
result Improved accuracy and uncertainty quantification.
Bayesian test assesses conditional independence between variables.
problem Quantifying dependence or independence between variables given a third.
method Uses Polya tree priors on conditional probability densities.
result Provides a Bayesian measure of conditional dependence or independence.
Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
In dynamic topic modeling, the proportional contribution of a topic to a document depends on the temporal dynamics of that topic's overall prevalence in the corpus. We extend the Dynamic Topic Model of Blei and Lafferty (2006) by explicitly modeling document level topic proportions with covariates and dynamic structure…
A new sampler speeds up LDA topic modeling for big data.
problem Training LDA on large corpora is slow and requires dense memory storage.
method Uses a Pólya-urn-based approximation in a sparse partially collapsed sampler.
result The new sampler is faster and asymptotically exact.
Efficiently infers Gaussian process density models with Gibbs sampling and variational methods.
problem Density estimation for complex, nonparametric models.
method Augmented likelihood with latent variables, Gibbs sampling, and variational mean field approximations.
result Efficient inference for Gaussian process density models with up to thousands of data points.
Study of spectral gaps in non-smooth spaces with bounded Ricci curvature.
problem Analyzing spectral gaps in non-smooth metric measure spaces.
method Establishing a Polya-Szego type inequality and applying it to show spectral gaps for the p-Laplace operator.
result Sharp spectral gap results for the p-Laplace operator on various non-smooth spaces.
Paper tackles anomaly detection in e-commerce using Bayesian semi-supervised tensor decomposition.
problem Detecting anomalies in seller-reviewer data in e-commerce.
method Bayesian semi-supervised tensor decomposition with Polya-Gamma data augmentation and partial natural gradient learning.
result Semi-supervised approach outperforms state-of-the-art unsupervised baselines.
Many practical modeling problems involve discrete data that are best represented as draws from multinomial or categorical distributions. For example, nucleotides in a DNA sequence, children's names in a given state and year, and text documents are all commonly modeled with multinomial distributions. In all of these cas…
In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, Pólya-Szegö and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is char…
The paper proves gradient and comparison inequalities for RCD spaces.
problem Gradient and comparison inequalities for RCD spaces.
method Elliptic Dirichlet problems and Talenti-type comparison.
result Sharp, rigid, and stable Talenti-type comparison results.
In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in Euclidean space and on the sphere.
Hyper-parameters play a major role in the learning and inference process of latent Dirichlet allocation (LDA). In order to begin the LDA latent variables learning process, these hyper-parameters values need to be pre-determined. We propose an extension for LDA that we call 'Latent Dirichlet allocation Gibbs Newton' (LD…
This note proves a Gaussian version of a Pólya-Szegö conjecture using rearrangement techniques.
problem Finding the domain with the minimum Gaussian principal frequency when the Gaussian torsional rigidity is fixed.
method Adapted Kohler-Jobin rearrangement technique to the Gauss space, considering a modified torsional rigidity and rearranging layers to half-spaces.
result The Gaussian principal frequency is minimized for the half-space when the Gaussian torsional rigidity is fixed.
Bayesian framework captures correlations in discrete environments for better decision-making.
problem Capturing correlations in discrete state-action domains for better decision-making.
method Bayesian learning framework based on Pólya-Gamma augmentation.
result Superior predictive performance compared to correlation-agnostic models.
We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg …
New method infers network couplings from spin trajectories in continuous time.
problem Inferring network couplings from observed spin trajectories in continuous time.
method Introducing latent variables to linearize and make likelihood quadratic, deriving EM and variational algorithms.
result Demonstrated performance on simulated data and biologically plausible network.
This paper is devoted to exploring the relationship between the [1,n)∋p-capacity and the surface-area in Rn≥2 which especially shows: if Ω⊂Rn is a convex, compact, smooth set with its interior Ω∘=∅ and the mean curvature H(∂Ω,⋅)>0 of its boundary $\p…
Bayesian NHMMs with non-homogeneous emissions for rainfall modeling.
problem Modeling temporal non-homogeneity in discrete-time hidden Markov models.
method Polya-Gamma data augmentation for Bayesian inference.
result Efficient MCMC sampling for complex NHMMs.
Bayesian Tensor Ring factorization improved for scalability and handling of discrete data.
problem Scalability issues and handling of discrete data in Bayesian Tensor Ring factorization.
method Proposes a novel Bayesian Tensor Ring model with a nonparametric Multiplicative Gamma Process prior and Pólya-Gamma augmentation for discrete data. Developed efficient Gibbs sampler and online EM algorithm for scalability.
result Significantly improved scalability and handling of discrete data compared to previous methods.
New MAB model incentivizes user arm-pulling with self-reinforcing preferences.
problem Balancing exploration and exploitation in recommender systems with incentivized user preferences.
method Proposes a new MAB model with random arm selection and two policies: At-Least-n Explore-Then-Commit and UCB-List. result Achieves O(logT) expected regret and O(logT) expected payment over a time horizon T. Universal inequalities for Laplacian eigenvalues on discrete groups.
problem Proving inequalities for Laplacian eigenvalues on discrete groups.
method Analyzing Laplacian eigenvalues with Dirichlet boundary conditions on subsets of discrete groups.
result Yang-type universal inequalities for Cayley graphs of amenable groups and the d-regular tree.
Self-poisoning in adaptive OOD detectors is explained with a sharp threshold theory and certified calibration.
problem Self-poisoning in adaptive OOD detectors.
method Modeling bank impurity as a generalized Pólya urn, proving almost-sure convergence to a mean-field equilibrium.
result A certified admission gate removes the transition at every contamination rate, controlling false positives label-free.