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.
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.
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.
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.
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.
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.
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…
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.
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.
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.
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.
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 ) . 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.
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.
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 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…
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.
Kernel interpolation improved with continuous volume sampling.
problem Approximating functions from RKHS using weighted sums of kernel translates.
method Continuous volume sampling for choosing node locations.
result Proved almost optimal bounds for interpolation and quadrature under VS.
Paper proposes a new estimator for nested expectations with faster convergence.
problem Estimating nested expectations is computationally challenging.
method Nested kernel quadrature estimators with proof of faster convergence rate.
result The proposed method requires fewer samples for accurate estimation.
Proposes an efficient method for GPLVM with arbitrary kernels.
problem GPLVM's limitation with standard kernel functions and computational bottlenecks.
method Uses the unscented transformation to handle arbitrary kernels efficiently.
result Comparable or better performance with linear computational complexity.
Positive weights improve kernel quadrature's accuracy.
problem Improving kernel quadrature weights to be positive and stable.
method Using convex geometry to approximate the kernel mean embedding with positive weights.
result Positive weights lead to improved kernel quadrature bounds with Monte-Carlo-beating rates.
SOBER framework optimizes Bayesian optimization tasks efficiently.
problem Challenges in parallel Bayesian optimization.
method Probabilistic Lifting with Kernel Quadrature.
result Versatile and flexible batch Bayesian optimization.
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.
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.
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.
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 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…
Efficiently approximates kernel mean embeddings using Nyström method.
problem Computational cost of kernel mean embeddings in large-scale settings.
method Nyström method for approximating a small random subset of the dataset.
result Upper bound on approximation error with sufficient subsample size conditions.
The paper studies estimating the normalizing constant using queries to a black-box function in RKHS.
problem Estimating the normalizing constant of a function in a reproducing kernel Hilbert space.
method Combines Bayesian quadrature and Bayesian optimization approaches, considering different levels of difficulty based on the parameter λ.
result The difficulty of estimating the normalizing constant varies between Bayesian quadrature and Bayesian optimization, even with noisy function evaluations.
New method improves kernel approximation for larger datasets.
problem Efficiently approximate kernel functions for large datasets.
method Monte Carlo integration for numerical approximation of kernel functions.
result Improved convergence behavior and empirical support for better kernel estimates.
Novel method interprets black box predictions using Fisher kernels and SBQ.
problem Interpreting black box models' predictions.
method Fisher kernels as feature embeddings, Sequential Bayesian Quadrature for selection.
result Method efficiently handles any subset of test predictions.
No-trick kernel adaptive filtering uses deterministic features for scalability and robustness.
problem Scalability issues in kernel methods for large datasets.
method Deterministic feature-map construction using polynomial-exact solutions.
result Deterministic features outperform random Fourier features in performance and scalability.
New filters improve radar target inference in complex scenarios.
problem Improving radar target inference in highly non-linear system models.
method Developed inverse cubature Kalman filter (I-CKF), inverse quadrature Kalman filter (I-QKF), and inverse cubature-quadrature Kalman filter (I-CQKF) for non-linear systems.
result Numerical experiments show improved estimation accuracy compared to existing methods.
A new method slices and sums radial kernels faster.
problem Fast computation of large kernel sums in kernel methods.
method Random projections to 1D subspaces and QMC for selecting projections.
result QMC-slicing outperforms existing methods on test datasets.
Paper proposes no-regret algorithms for private GP bandit optimization.
problem Private Gaussian process bandit optimization.
method Combines uniform kernel approximator with random perturbations for differentially private GP bandit algorithms.
result Provable no-regret algorithms for stationary kernel functions in two DP settings.
New method connects leverage scores and kernel density, revealing a decreasing relationship.
problem Understanding the relationship between leverage scores and kernel density.
method Introducing regularized Christoffel functions to study leverage scores for kernel methods.
result Quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels.
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.
The paper analyzes greedy algorithms for MMD minimization, showing their efficiency and approximation error.
problem Minimizing Maximum Mean Discrepancy (MMD) for probability measure quantization.
method Iterative algorithms including kernel herding, greedy MMD minimization, and Sequential Bayesian Quadrature (SBQ).
result The greedy algorithms have a lower approximation error than SBQ, but are significantly faster.
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.
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.
A conservative drifting method improves generative modeling by using KDE gradients, proving convergence rates.
problem Improving generative modeling by addressing non-conservatism issues.
method Proposes a conservative drifting method using kernel density estimator gradients to address non-conservatism.
result Proves finite-particle convergence rates for the conservative method, providing explicit quadrature constants.
SLEIPNIR improves Gaussian process regression with derivatives, scaling up efficiently and accurately.
problem Scaling Gaussian process regression with derivatives for large datasets.
method Quadrature Fourier features for feature expansion, proving error bounds.
result Deterministic, non-asymptotic, exponentially fast decaying error bounds for approximated kernel and posterior.
Efficiently approximates integrals using a subset of samples from a target distribution in RKHS.
problem Approximating integrals with a target distribution using limited pointwise evaluations.
method Proposes a procedure using a small random subset of samples from the target distribution, either uniformly or using approximate leverage scores.
result Upper bound on approximation error for both sampling strategies, achieving optimal rate with reduced evaluations.
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 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.