A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussi…
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48 results for “Lebesgue quadrature”
An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, and , two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for and ) from two eigenproblems, the projections of - and…
A novel approach to interpolation, classification, and clustering using Radon-Nikodym derivatives.
problem Interpolation, classification, and clustering problems in data analysis.
method Radon-Nikodym approach with Lebesgue quadrature for optimal clustering.
result The approach changes both probabilities and the probability space with new observations.
New method certifies neural network function space norms from point evaluations.
problem Certifying neural network function space norms from point evaluations alone.
method Combining interval arithmetic enclosures, adaptive marking/refinement, and quadrature-based aggregation.
result Certified computation of , , and norms.
Kernel quadrature with DPPsstat.ML
Kernel quadrature uses DPPs for sampling with tight error bounds.
problem Efficiently sampling nodes for quadrature rules in RKHS.
method Nodes sampled from a truncated and saturated DPP kernel.
result Tighter quadrature error bounds using DPPs.
New quadrature method using randomly pivoted Cholesky outperforms existing techniques.
problem Efficiently approximating integrals of functions in reproducing kernel Hilbert spaces.
method Nodes drawn by randomly pivoted Cholesky algorithm.
result Randomly pivoted Cholesky quadrature is fast and achieves comparable accuracy to more computationally intensive methods.
Bayesian quadrature uses probabilistic models for estimating intractable integrals.
problem Estimating intractable integrals in complex models.
method Probabilistic, model-based approach using Gaussian processes.
result Comprehensive review and systematic taxonomy of Bayesian quadrature methods.
Paper studies geometric properties of nonlinear Lebesgue spaces.
problem Geometric and analytic properties of nonlinear Lebesgue spaces.
method Formalizes pointwise description of geometric properties using a nonlinear Fubini-Lebesgue theorem.
result Definition of length structure, Alexandrov curvature bounds, and speed for absolutely continuous curves in nonlinear Lebesgue spaces.
Characterizes Lebesgue points using nearest neighbor methods.
problem Consistency of classification algorithms based on nearest neighbors.
method Characterization of Lebesgue points via 1-Nearest Neighbor regression.
result Proves convergence of 1-Nearest Neighbor classification algorithms in metric spaces.
SOBER: Highly Parallel Bayesian Optimization and Bayesian Quadrature over Discrete and Mixed Spacescs.LG
SOBER optimizes and quadrates efficiently in parallel for diverse tasks.
problem Scalability of batch Bayesian optimization and quadrature for expensive functions.
method Reformulates batch selection as a quadrature problem, balancing exploitation and exploration.
result SOBER outperforms 11 baselines on 12 tasks.
New analysis proves consistency for adaptive Bayesian quadrature methods.
problem No theoretical guarantees for adaptive Bayesian quadrature methods.
method Introduces weak adaptivity and proves consistency for a broad class of adaptive Bayesian quadrature rules.
result Proves consistency and derives non-tight but informative convergence rates for adaptive Bayesian quadrature methods.
This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We pr…
Improved Gaussian Process regression using TQFF over RFF and Gaussian QFF.
problem Limited performance of Quadrature Fourier Features (QFF) in approximating highly oscillatory functions.
method Developed Trigonometric Quadrature Fourier Features (TQFF) using a novel non-Gaussian quadrature rule.
result TQFF provides better approximation accuracy and fewer features compared to RFF and Gaussian QFF.
Study generalizes Lebesgue curves to new space-filling and fractal sets.
problem Generating space-filling curves from planar substitutions.
method Generalized Lebesgue's construction to new curves and fractal sets.
result Some substitutions create relatively dense fractal-like sets.
Bayesian quadrature improves integration efficiency with invariant priors.
problem Efficient numerical integration with known structure.
method Invariance priors for bijective transformations in input domain.
result Superior performance in synthetic and real-world applications.
Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior varianc…
Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior varianc…
RLF uses Riemann-Lebesgue cutting for better regression.
problem Improving regression accuracy through novel tree splitting.
method Develops Riemann-Lebesgue Tree (RLT) for partitioning response intervals.
result RLF achieves larger variance reduction compared to CART.
Improved kernel quadrature with convex weights using subsampling.
problem Constructing quadrature rules with small worst-case error.
method Combining spectral properties of the kernel with recombination results.
result Effective algorithms for constructing convex quadrature rules with i.i.d. samples.
The sigma-point filters, such as the UKF, which exploit numerical quadrature to obtain an additional order of accuracy in the moment transformation step, are popular alternatives to the ubiquitous EKF. The classical quadrature rules used in the sigma-point filters are motivated via polynomial approximation of the integ…
In this study, a numerical quadrature for the generalized inverse Gaussian distribution is derived from the Gauss-Hermite quadrature by exploiting its relationship with the normal distribution. The proposed quadrature is not Gaussian, but it exactly integrates the polynomials of both positive and negative orders. Using…
The aim of this article is to design a moment transformation for Student- t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable…
Unified quadrature framework for large-scale kernel machines.
problem Efficiently approximating kernel functions for large-scale machine learning.
method Deterministic and randomized interpolatory rules for numerical integration of kernel functions.
result The proposed method reduces the number of nodes needed for accurate kernel approximation.
This paper introduces repulsive Monte Carlo methods for computing the sliced Wasserstein distance.
problem Computing the integral of a function on the unit sphere using Monte Carlo methods.
method The approach involves using determinantal point processes and repelled point processes to create quadratures for the sliced Wasserstein distance.
result The UnifOrtho estimator is recommended for the computation of the sliced Wasserstein distance in large dimensions.
In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes (). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study th…
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
Novel approach for estimating conditional expectations using Bayesian quadrature.
problem Estimating conditional expectations with costly evaluations.
method Probabilistic numerical methods incorporating prior smoothness knowledge.
result Fast convergence rate and uncertainty quantification.
New Fourier features improve high-precision approximation in large-scale problems.
problem Designing scalable, high-precision Fourier features for large-scale kernel methods.
method Introducing a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions.
result Improved approximation bounds with new Fourier features.
Improved Nyström approximation for kernel quadrature with theoretical guarantees.
problem Efficiently approximating positive definite kernels for large datasets.
method Refined sampling and subspace selection in Nyström approximation.
result Novel theoretical guarantees for non-i.i.d. landmark points in kernel quadrature.
Computation of moments of transformed random variables is a problem appearing in many engineering applications. The current methods for moment transformation are mostly based on the classical quadrature rules which cannot account for the approximation errors. Our aim is to design a method for moment transformation for …
Parallelized Bayesian quadrature improves sample efficiency and inference.
problem Efficient Bayesian inference and model evidence calculation.
method Batch Bayesian quadrature with kernel recombination for parallel sampling.
result Empirically, outperforms state-of-the-art methods in various datasets.
Real intersection theory (I)math.AG
We introduce the notion of Lebesgue currents. They are a special type of currents involving Lebesgue measure. We apply it to define the intersection of singular cycles, which provides the foundation to the real intersection theory.
Adaptive quadrature improves Bayesian inference through active learning.
problem Efficiently estimating posterior densities in Bayesian inference.
method Sequential node selection using acquisition functions, combining interpolative surrogate models and quadrature rules.
result Positive estimation of marginal likelihood with improved accuracy.
Study shows twist tori equidistribute in moduli space, with other families having singular distributions.
problem Statistical behavior of twist tori in moduli space of hyperbolic surfaces.
method Analyzing expanding families of twist tori and their limiting distributions.
result Equidistribution of twist tori to a Lebesgue measure, with other families having singular distributions.
A new method calculates accurate SABR model option prices and deltas.
problem Inaccurate and arbitrageable SABR model option prices and deltas.
method Gaussian quadrature integration scheme for the normal SABR model.
result Accurate and arbitrage-free SABR model option prices and deltas calculated with 49 points.
DBQPG improves policy gradient estimation with fewer samples.
problem Accurate policy gradient estimation with limited samples.
method Deep Bayesian Quadrature Policy Gradient (DBQPG).
result DBQPG provides more accurate and less variable gradient estimates.
Uniqueness found for elliptic equations with drift on manifolds.
problem Finding unique solutions to elliptic equations with drift on manifolds.
method Investigation in weighted Lebesgue spaces, focusing on conditions for uniqueness.
result Sharp conditions on drift term for uniqueness in polynomial volume growth manifolds.
Improved kernel herding algorithm for faster quadrature rule convergence.
problem Slow convergence speed of standard kernel herding algorithm.
method Improved gradient approximation to obtain sparser solutions.
result The cosine of the angle between negative gradient and approximate gradient determines convergence speed.
Bayesian quadrature improves conformal prediction for better risk assessment.
problem Improving risk assessment for machine learning models.
method Revisiting conformal prediction from a Bayesian perspective and proposing Bayesian quadrature.
result Provides interpretable guarantees and a richer representation of likely losses.
Automated model selection using Bayesian quadrature improves efficiency.
problem Slow convergence and unreliability of Monte Carlo methods for model comparison.
method Automated algorithm maximizing mutual information between posterior probability and model likelihoods.
result More accurate model posterior estimates with fewer likelihood evaluations.
Efficiently marginalizes over Gaussian Process kernels for better model flexibility and uncertainty.
problem Inefficient marginalization over Gaussian Process kernels for large datasets.
method Bayesian Quadrature scheme with maximum mean discrepancies and invariances between Spectral Mixture kernels.
result Achieves more accurate predictions and better calibrated uncertainty than state-of-the-art baselines.
The paper improves error bounds for Bayesian quadrature in noisy settings.
problem Improving error bounds for Bayesian quadrature in noisy settings.
method Develops a two-step meta-algorithm to relate average-case quadrature error to -function approximation error.
result Provides new average-case results for various kernels and noise settings.
Bayesian Quadrature improves ensembling for neural networks with dispersed likelihood peaks.
problem Ensembling neural networks struggles with dispersed, narrow peaks in likelihood surfaces.
method Uses Bayesian Quadrature to construct weighted ensembles of architectures.
result Empirically outperforms state-of-the-art baselines in test likelihood, accuracy, and expected calibration error.
We present a framework for accelerating a spectrum of machine learning algorithms that require computation of bilinear inverse forms , where is a positive definite matrix and a given vector. Our framework is built on Gauss-type quadrature and easily scales to large, sparse matrices. Further, it …
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that th…
Several numerical approximation strategies for the expectation-propagation algorithm are studied in the context of large-scale learning: the Laplace method, a faster variant of it, Gaussian quadrature, and a deterministic version of variational sampling (i.e., combining quadrature with variational approximation). Exper…
BayesSum improves Bayesian quadrature for discrete domains, requiring fewer samples.
problem Estimating intractable expectations over discrete domains.
method BayesSum is a Bayesian quadrature extension for discrete domains, leveraging prior information through Gaussian processes.
result BayesSum requires fewer samples than Monte Carlo, achieving faster convergence rates.