Related papers: Minimal Subsampled Rank-1 Lattices for Multivariat…
We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-$s$ integration lattice. The main result is the fact that any (non-)linear reconstruction algorithm taking…
This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by K\"{a}mmerer and Volkmer (2019) that algorithms based on multiple rank-1 lattices achieve…
In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize so-called reconstructing multiple rank-1 lattices, which combine several preferable…
A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant,…
In this paper we present the first known deterministic algorithm for the construction of multiple rank-1 lattices for the approximation of periodic functions of many variables. The algorithm works by converting a potentially large…
We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study…
We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded…
The mean squared error and regularized versions of it are standard loss functions in supervised machine learning. However, calculating these losses for large data sets can be computationally demanding. Modifying an approach of J. Dick and…
We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randomly…
In this paper, we propose a novel algorithm for function approximation in a weighted Korobov space based on shifted rank-1 lattice rules. To mitigate aliasing errors inherent in lattice-based Fourier coefficient estimation, we employ…
We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights…
The problem of low rank approximation is ubiquitous in science. Traditionally this problem is solved in unitary invariant norms such as Frobenius or spectral norm due to existence of efficient methods for building approximations. However,…
We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for…
We develop a general framework for estimating the $L_\infty(\mathbb{T}^d)$ error for the approximation of multivariate periodic functions belonging to specific reproducing kernel Hilbert spaces (RHKS) using approximants that are…
We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need…
In this paper, we investigate the sample complexity of recovering tensors with low symmetric rank from symmetric rank-one measurements. This setting is particularly motivated by the study of higher-order interactions and the analysis of…
Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate…
We consider in this paper the problem of noisy 1-bit matrix completion under a general non-uniform sampling distribution using the max-norm as a convex relaxation for the rank. A max-norm constrained maximum likelihood estimate is…
The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications. In order to deal with high-dimensional problems, efficient sparse fast Fourier transform algorithms are essential tools. The second…
Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying…