Related papers: How sharp are error bounds? --lower bounds on quad…
This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and…
We consider differentially private approximate singular vector computation. Known worst-case lower bounds show that the error of any differentially private algorithm must scale polynomially with the dimension of the singular vector. We are…
Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the…
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s --…
In this paper we analyze the approximation of multivariate integrals over the Euclidean plane for functions which are analytic. We show explicit upper bounds which attain the exponential rate of convergence. We use an infinite grid with…
Randomized quadratures for integrating functions in Sobolev spaces of order $\alpha \ge 1$, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the…
This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
In practical conjugate gradient (CG) computations it is important to monitor the quality of the approximate solution to $Ax=b$ so that the CG algorithm can be stopped when the required accuracy is reached. The relevant convergence…
The regret bound of an optimization algorithms is one of the basic criteria for evaluating the performance of the given algorithm. By inspecting the differences between the regret bounds of traditional algorithms and adaptive one, we…
We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the…
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…
The detection of insufficiently resolved or ill-conditioned integrand structures is critical for the reliability assessment of the quadrature rule outputs. We discuss a method of analysis of the profile of the integrand at the quadrature…
We study in this paper the function approximation error of multivariate linear extrapolation. The sharp error bound of linear interpolation already exists in the literature. However, linear extrapolation is used far more often in…
Some extremalities for quadrature operators are proved for convex functions of higher order. Such results are known in the numerical analysis, however they are often proved under suitable differentiability assumptions. In our considerations…
In this paper, we consider the Gauss quadrature formulae corresponding to some modifications of anyone of the four Chebyshev weights, considered by Gautschi and Li in \cite{gauli}. As it is well known, in the case of analytic integrands,…
Efficient global optimization is a widely used method for optimizing expensive black-box functions such as tuning hyperparameter, and designing new material, etc. Despite its popularity, less attention has been paid to analyzing the…
This paper studies generalization error bounds for Transformer models. Based on the offset Rademacher complexity, we derive sharper generalization bounds for different Transformer architectures, including single-layer single-head,…
We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw-Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n^{-s-1}) for some s > 0,…
Corrected trapezoidal rules are proved for $\int_a^b f(x)\,dx$ under the assumption that $f"\in L^p([a,b])$ for some $1\leq p\leq\infty$. Such quadrature rules involve the trapezoidal rule modified by the addition of a term…