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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
Bayesian quadrature improves integration on Riemannian manifolds.
problem Efficiently computing integrals on nonlinear geometric data.
method Probabilistic numerical methods, specifically Bayesian quadrature, on Riemannian manifolds.
result Bayesian quadrature reduces the number of function evaluations compared to Monte Carlo 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.
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…
There is renewed interest in formulating integration as an inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. A…
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.
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.
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.
In this paper we develope, in a geometric framework, a Hamilton-Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some appr…
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.
This article is concerned with Gaussian process quadratures, which are numerical integration methods based on Gaussian process regression methods, and sigma-point methods, which are used in advanced non-linear Kalman filtering and smoothing algorithms. We show that many sigma-point methods can be interpreted as Gaussia…
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.
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 O ( e − p ) O(e^{-p}) O ( e − p ) and a fast estimation rate of O ~ ( 1 / n ) \widetilde{O}(1/n) O ( 1/ n ) . Improved option pricing for SABR model using Gauss-Hermite quadrature.
problem Improving accuracy of option pricing in the SABR model.
method Using Gauss-Hermite quadrature for numerical integration of the integrated variance.
result New method provides accurate option prices across all strike prices.
TQ separates sampling and integration for high-dimensional integrals.
problem High-dimensional integration challenges in science.
method Tree Quadrature (TQ) constructs a surrogate model using regression trees.
result TQ outperforms existing methods in up to 15 dimensions.
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.
Unified method for efficient pricing of multivariate options.
problem Efficient pricing of complex financial options under multivariate models.
method Unified method using quadrature integration of multi-asset BSM prices, state space rotation.
result Unified method provides accurate and efficient pricing for basket, spread, and Asian options.
Bayesian quadrature improves numerical efficiency and uncertainty representation.
problem Computing integrals of multiple related functions efficiently and accurately.
method Extending Bayesian quadrature to handle multiple related functions, proving convergence rates.
result The method provides increased numerical efficiency and more faithful uncertainty representation.
Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super- n \sqrt{n} n convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging settings where integrands are expensive to evaluate or where integrands are high d…
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.
This work introduces a fixed-point optimization for variational inference.
problem Improving quantified uncertainty in predictions by optimizing a simplified distribution over parameters.
method Projective integral updates for high-dimensional variational inference.
result Efficient quasirandom quadrature sequence for mean-field distributions, leading to quasi-Newton variational Bayes (QNVB).
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, P f [ i ] ; g [ j ] P_{f^{[i]};g^{[j]}} P f [ i ] ; g [ j ] , for random processes. Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.
problem Classifying geodesic curves on the Heisenberg group.
method Completely integrable Hamiltonian system, classification of geodesic curves.
result Complete classification of geodesic curves on the Heisenberg group.
The paper improves probabilistic herding methods using Gibbs distributions.
problem Improving integration accuracy over Monte Carlo quadrature in infinite-dimensional RKHS.
method Developed a Gibbs distribution over quadrature nodes to minimize MMD.
result The Gibbs distribution outperforms i.i.d. Monte Carlo in integration accuracy.
Active multi-source Bayesian quadrature improves efficiency in expensive function evaluations.
problem Efficiently solving integrals of expensive-to-evaluate functions using multiple related sources of information.
method Constructing cost-sensitive multi-source acquisition rates as an extension to vanilla Bayesian quadrature.
result Active multi-source Bayesian quadrature allocates budget more efficiently than vanilla Bayesian quadrature.
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.
The paper introduces new methods for Asian option pricing using Laguerre quadrature.
problem Developing accurate pricing models for Asian options.
method Utilizes Laguerre quadrature and diffusion kernel approach.
result Demonstrates new techniques to solve complex Asian option pricing equations.
New theory extends Hamilton-Jacobi for contact systems, ensuring integrability.
problem Integrability of contact Hamiltonian systems.
method Developed a Hamilton-Jacobi theory for fibered phase spaces, applied to contact systems, studied HJE solutions.
result Complete pseudo-isotropic solutions ensure integrability by quadratures for contact systems.
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.
Kernel-based algorithms improve integral estimation with near-geometric speed.
problem Estimating integrals with target measures that are nearly atomic.
method Weighted kernel herding and sequential Bayesian quadrature.
result Near-geometric rate of convergence for nearly atomic target measures.
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 L 2 L^2 L 2 -function approximation error. result Provides new average-case results for various kernels and noise settings.
Bayesian neural networks speed up numerical integration.
problem Scalability of Bayesian quadrature methods.
method Bayesian Stein networks using neural networks and Laplace approximation.
result Orders of magnitude speed-up on benchmark functions and real-world problems.
We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation e…
Faster training of neural ODEs using Gauß-Legendre quadrature.
problem Training neural ODEs is slow due to solving ODEs numerically.
method Use Gauß-Legendre quadrature to solve integrals faster than ODE-based methods.
result Faster training of neural ODEs, especially for large models.
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.
This paper provides a dictionary of closed-form kernel mean embeddings.
problem Challenges in deriving closed-form kernel mean embeddings.
method Comprehensive dictionary and practical tools for deriving new embeddings.
result Provides a Python library with minimal implementations of embeddings.
We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic dif…