arXiv research

New quadrature method using randomly pivoted Cholesky outperforms existing techniques.

We study quadrature rules for functions from an RKHS, using nodes sampled from a determinantal point process (DPP). DPPs are parametrized by a kernel, and we use a truncated and saturated version of the RKHS kernel. This link between the two kernels, along with DPP machinery, leads to relatively tight bounds on the qua…

Improved kernel quadrature with convex weights using subsampling.

Unified quadrature framework for large-scale kernel machines.

Efficiently marginalizes over Gaussian Process kernels for better model flexibility and uncertainty.

Improved Nyström approximation for kernel quadrature with theoretical guarantees.

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.

Improved kernel herding algorithm for faster quadrature rule convergence.

Parallelized Bayesian quadrature improves sample efficiency and inference.

New Fourier features improve high-precision approximation in large-scale problems.

SOBER optimizes and quadrates efficiently in parallel for diverse tasks.

Kernel quadrature improves CRPS estimation for probabilistic time-series forecasting.

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…

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super-$\sqrt{n}$ 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.

Kernel interpolation improved with continuous volume sampling.

Paper proposes a new estimator for nested expectations with faster convergence.

Positive weights improve kernel quadrature's accuracy.

SOBER framework optimizes Bayesian optimization tasks efficiently.

The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio $s/d$, where $s$ and $d$ encode the smoothness and dimension of the integrand. However, an empirical investigation re…

An infinitely wide model is a weighted integration $\int \varphi(x,v) d μ(v)$ of feature maps. This model excels at handling an infinite number of features, and thus it has been adopted to the theoretical study of deep learning. Kernel quadrature is a kernel-based numerical integration scheme developed for fast approxi…

Improved Gaussian Process regression using TQFF over RFF and Gaussian QFF.

The paper improves probabilistic herding methods using Gibbs distributions.

This paper provides a dictionary of closed-form kernel mean 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.

The paper studies estimating the normalizing constant using queries to a black-box function in RKHS.

We consider the problem of improving kernel approximation via randomized feature maps. These maps arise as Monte Carlo approximation to integral representations of kernel functions and scale up kernel methods for larger datasets. Based on an efficient numerical integration technique, we propose a unifying approach that…

New filters improve radar target inference in complex scenarios.

A new method slices and sums radial kernels faster.

Paper proposes no-regret algorithms for private GP bandit optimization.

The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geom…

Gaussian Process Latent Variable Model (GPLVM) is a flexible framework to handle uncertain inputs in Gaussian Processes (GPs) and incorporate GPs as components of larger graphical models. Nonetheless, the standard GPLVM variational inference approach is tractable only for a narrow family of kernel functions. The most p…

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we in…

Bayesian quadrature uses probabilistic models for estimating intractable integrals.

The paper analyzes greedy algorithms for MMD minimization, showing their efficiency and approximation error.

A conservative drifting method improves generative modeling by using KDE gradients, proving convergence rates.

SLEIPNIR improves Gaussian process regression with derivatives, scaling up efficiently and accurately.

Efficiently approximates integrals using a subset of samples from a target distribution in RKHS.

Bayesian quadrature improves integration efficiency with invariant priors.

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…

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…

Adaptive Bayesian quadrature (ABQ) is a powerful approach to numerical integration that empirically compares favorably with Monte Carlo integration on problems of medium dimensionality (where non-adaptive quadrature is not competitive). Its key ingredient is an acquisition function that changes as a function of previou…

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…

New sampling methods improve Shapley value estimation for machine learning models.