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 …
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48 results for “matrix quadrature”
Proposes a new regularizer for semi-supervised learning on multilayer graphs.
problem Semi-supervised learning on multilayer graphs with labeled and unlabeled data.
method Generalized matrix mean regularizer and matrix-free numerical scheme.
result The regularizer outperforms state-of-the-art methods numerically.
Develops numerical method for joint probability estimation from random processes.
problem Estimating joint probability distribution from random processes.
method Formulates and solves generalized eigenvalue problems for two random processes, then uses projections of eigenvectors to build a joint distribution estimator.
result Develops a new type of probability correlation, , for random processes.
Let be a Lie algebra valued differential -form on a manifold satisfying the structure equations where is solvable. We show that the problem of finding a smooth map , where is an -dimensional so…
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.
The paper analyzes kernel-based quadrature in misspecified settings, providing convergence rates and robustness conditions.
problem Analyzing kernel-based quadrature in settings where the test integrand is less smooth than the RKHS.
method Convergence analysis based on two assumptions: constant weights or minimum distance between design points.
result Derives convergence rates and conditions for robustness in Bayesian quadrature under misspecification.
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.
The study develops a quadrature method for the generalized hyperbolic distribution using finite normal-mixture approximation.
problem Efficiently approximating and computing expectations under the generalized hyperbolic distribution.
method Derived a numerical quadrature from Gauss-Hermite quadrature, approximated the distribution as a finite normal variance-mean mixture.
result Accurately computed expectations and sampled generalized hyperbolic random variates using the proposed method.
Improved sigma-point filters reduce quadrature error bias.
problem Quadrature error in sigma-point filters leads to poorly calibrated estimates.
method Bayes-Sard quadrature method for sigma-point filters.
result Better-calibrated state estimates with improved RMSE.
Paper addresses stability in multi-asset American option pricing.
problem Stability in multi-asset American option pricing problems.
method Semi-discretization approach followed by full discretization.
result Stability conditions found for numerical solution.
Develops a new quadrature method for Lebesgue integrals.
problem Finding optimal values and weights for non-Gaussian processes.
method Solves a generalized eigenvalue problem to find value-nodes and weights.
result Advantages in analyzing irregular and stochastic processes.
Study of Hamilton-Jacobi Theory with symmetries and integrability by quadratures.
problem Hamilton-Jacobi equation in systems with symmetries.
method Constructing complete solutions and solving reconstruction equations.
result Explicit expressions for exponential curves in Lie groups, valid for all elements in the Lie algebra.
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.
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.
This paper improves filtering of non-linear systems with heavy-tailed noise.
problem Improving filtering accuracy for non-linear systems with heavy-tailed noise.
method Developed a moment transformation for Student-t distributed random variables using Student-t process quadrature.
result The method outperforms state-of-the-art moment transforms in numerical examples.
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.
Gaussian process quadrature improves moment transformation accuracy.
problem Computing moments of transformed Gaussian variables with error accounting.
method Bayesian quadrature (Gaussian process quadrature) for numerically estimating integrals.
result Proposed method outperforms classical quadrature methods in accuracy.
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…
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.
Bayesian Quadrature speeds up integration by selecting batches of points instead of single points.
problem Efficiently parallelizing Bayesian Quadrature for integration over non-negative integrands.
method Developed methods to select batches of points at each step, based on recent batch Bayesian Optimization.
result Significantly reduces computation time, especially for expensive integrands.
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.
Improved Bayesian quadrature for constrained functions.
problem Performing inference of constrained functions in Bayesian inference.
method Bayesian framework with explicit approximation schemes for constraints, log transformation for high dynamic range, and optimization of hyperparameters in original space.
result Model achieves superior estimates using less time than existing procedures.
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.
Kernel quadratures can be consistent even when the integrand is less smooth than assumed.
problem Kernel quadratures assume smoothness of integrands, but this assumption is often violated in practice.
method Derives convergence rates for kernel quadratures in misspecified settings, relating them to the lesser smoothness of the integrand.
result Kernel quadratures can be consistent even when the integrand is less smooth than assumed, providing alternatives to Monte Carlo integration.
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.
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.
Kernel Quadrature improves numerical integration with adaptive tempering.
problem Optimizing sampling distribution for Kernel Quadrature to reduce integration error.
method Adaptive tempering and sequential Monte Carlo approach to find optimal sampling distribution.
result Significant reduction in integration error (up to 4 orders of magnitude) achieved with the proposed method.
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.
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.
Study improves kernel quadrature for infinitely wide models with faster approximation and estimation rates.
problem Efficiently approximating and estimating expectations in infinitely wide models.
method Developed general kernel quadrature (GKQ) for parameter distributions, achieving faster rates.
result Achieved a fast approximation rate of and a fast estimation rate of .
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.
Kernel quadrature improves CRPS estimation for probabilistic time-series forecasting.
problem Intractable integrations in CRPS evaluation metrics lead to improper rankings of forecasting models.
method Introduced kernel quadrature approach for unbiased CRPS estimation and scalable computation.
result Our approach consistently outperforms existing CRPS estimators.
The paper studies symmetries and conservation laws of non-diagonalisable hydrodynamic systems.
problem Integrating non-diagonalisable hydrodynamic systems of partial differential equations.
method Analysis of gl-regular Nijenhuis operators, splitting Theorem for symmetries and conservation laws, relationship between symmetries and conservation laws.
result The system of partial differential equations is integrable in quadratures.
A new method optimizes Fourier pricing for multi-asset options using adaptive quadrature.
problem Efficiently pricing multi-asset options in Lévy models.
method Optimized damping parameters and hierarchical adaptive quadrature.
result Significant speed-up in computational time for up to six dimensions.
Combines control variates and adaptive importance sampling for Monte Carlo integration.
problem Improving Monte Carlo integration accuracy with control variates and adaptive sampling.
method A quadrature rule combining control variates and adaptive importance sampling.
result Non-asymptotic bound on the probabilistic error of the procedure.
ALOQ optimizes policies in simulators by adjusting unobservable environment variables.
problem Learning optimal policies in simulators without considering unobservable environment variables.
method ALOQ uses Bayesian optimisation and Bayesian quadrature to address this problem.
result ALOQ learns more efficiently and robustly than existing methods.