11 citations · 40 across the 18 of their papers we have counts for
27 papers
Wasserstein bounds for non-linear Gaussian filters
Toni Karvonen, Simo Särkkä
Most Kalman filters for non-linear systems, such as the unscented Kalman filter, are based on Gaussian approximations. We use Poincaré inequalities to bound the Wasserstein distanc…
Multilevel Bayesian Quadrature
Kaiyu Li, Daniel Giles, Toni Karvonen +2
Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator…
Orthonormal Expansions for Translation-Invariant Kernels
Filip Tronarp, Toni Karvonen
We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of …
Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posed
Toni Karvonen, Chris J. Oates
Gaussian process regression underpins countless academic and industrial applications of machine learning and statistics, with maximum likelihood estimation routinely used to select…
Asymptotic Bounds for Smoothness Parameter Estimates in Gaussian Process Interpolation
Toni Karvonen
It is common to model a deterministic response function, such as the output of a computer experiment, as a Gaussian process with a Matérn covariance kernel. The smoothness paramete…
Error analysis for a statistical finite element method
Toni Karvonen, Fehmi Cirak, Mark Girolami
The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the unknown true…
ProbNum: Probabilistic Numerics in Python
Jonathan Wenger, Nicholas Krämer, Marvin Pförtner +9
Probabilistic numerical methods (PNMs) solve numerical problems via probabilistic inference. They have been developed for linear algebra, optimization, integration and differential…
Scale estimation and rate-unbiasedness for Gaussian processes under smoothness misspecification
Toni Karvonen, François Bachoc
Gaussian process regression is used throughout statistics and machine learning for prediction and uncertainty quantification. A Gaussian process is specified by its mean and covari…
Black Box Probabilistic Numerics
Onur Teymur, Christopher N. Foley, Philip G. Breen +2
Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quanti…
Small Sample Spaces for Gaussian Processes
Toni Karvonen
It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process is controlled by a certain nuclear dominance condition.…
Bumps and polynomials in RKHSs of translation-invariant kernels
Toni Karvonen
We use the uncertainty principle of harmonic analysis and the Beurling-Malliavin theorem to describe when a reproducing kernel Hilbert space of a translation-invariant positive-def…
A probabilistic Taylor expansion with Gaussian processes
Toni Karvonen, Jon Cockayne, Filip Tronarp +1
We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of deri…
Integration in reproducing kernel Hilbert spaces of Gaussian kernels
Toni Karvonen, Chris J. Oates, Mark Girolami
The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numer…
Sampling based approximation of linear functionals in Reproducing Kernel Hilbert Spaces
Gabriele Santin, Toni Karvonen, Bernard Haasdonk
In this paper we analyze a greedy procedure to approximate a linear functional defined in a Reproducing Kernel Hilbert Space by nodal values. This procedure computes a quadrature r…
Semi-Exact Control Functionals From Sard's Method
Leah F. South, Toni Karvonen, Chris Nemeth +2
The numerical approximation of posterior expected quantities of interest is considered. A novel control variate technique is proposed for post-processing of Markov chain Monte Carl…
Taylor Moment Expansion for Continuous-Discrete Gaussian Filtering and Smoothing
Zheng Zhao, Toni Karvonen, Roland Hostettler +1
The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an Itô stochastic differ…
Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions
Toni Karvonen, George Wynne, Filip Tronarp +2
Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically ne…
Kernel-based interpolation at approximate Fekete points
Toni Karvonen, Simo Särkkä, Ken'ichiro Tanaka
We construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion…
Worst-case optimal approximation with increasingly flat Gaussian kernels
Toni Karvonen, Simo Särkkä
We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new per…
On the positivity and magnitudes of Bayesian quadrature weights
Toni Karvonen, Motonobu Kanagawa, Simo Särkkä
This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate…
Improved Calibration of Numerical Integration Error in Sigma-Point Filters
Jakub Prüher, Toni Karvonen, Chris J. Oates +2
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 t…
Symmetry Exploits for Bayesian Cubature Methods
Toni Karvonen, Simo Särkkä, Chris. J. Oates
Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional flexibility, c…
On stability of a class of filters for non-linear stochastic systems
Toni Karvonen, Silvère Bonnabel, Eric Moulines +1
This article develops a comprehensive framework for stability analysis of a broad class of commonly used continuous and discrete time-filters for stochastic dynamic systems with no…
A Bayes-Sard Cubature Method
Toni Karvonen, Chris J. Oates, Simo Särkkä
This paper focusses on the formulation of numerical integration as an inferential task. To date, research effort has largely focussed on the development of Bayesian cubature, whose…