Related papers: Optimal randomized multilevel algorithms for infin…
We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the…
We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…
We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…
We consider $L^2$-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear…
We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration…
We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration…
The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $n^{-1/2}$. However, re-weighting of random points can sometimes be used to improve the convergence order.…
Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem.…
We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend…
In the present paper we study quasi-Monte Carlo rules for approximating integrals over the $d$-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for…
We study integration and $L^2$-approximation in the worst-case setting for deterministic linear algorithms based on function evaluations. The underlying function space is a reproducing kernel Hilbert space with a Gaussian kernel of tensor…
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required…
We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level algorithms, based on the conjugate gradient method and the…
We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical…
Learning mappings between infinite-dimensional function spaces has achieved empirical success in many disciplines of machine learning, including generative modeling, functional data analysis, causal inference, and multi-agent reinforcement…
We present an algorithm for multivariate integration over cubes that is unbiased and has optimal order of convergence (in the randomized sense as well as in the worst case setting) for all Sobolev spaces $H^{r, mix}([0,1]^d)$ and…
We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where…
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…
We present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings, The construction has been used in…
In the multidimensional setting, we consider the errors-in-variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimator based on projection…