Related papers: An adaptive partition of unity method for multivar…
Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by…
Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization…
In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the…
In this paper, we propose a proximal gradient method and an accelerated proximal gradient method for solving composite optimization problems, where the objective function is the sum of a smooth and a convex, possibly nonsmooth, function. We…
This paper considers optimization of smooth nonconvex functionals in smooth infinite dimensional spaces. A H\"older gradient descent algorithm is first proposed for finding approximate first-order points of regularized polynomial…
Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation…
We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for…
We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions…
Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered datasets in d-dimensional space. It is non-separable approximation, as it is…
The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end a variant of Chernoff's product formula is…
This paper presents a novel method for generating a single polynomial approximation that produces correctly rounded results for all inputs of an elementary function for multiple representations. The generated polynomial approximation has…
In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to…
We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral…
In this paper a local approximation method on the sphere is presented. As interpolation scheme we consider a partition of unity method, such as the modified spherical Shepard's method, which uses zonal basis functions (ZBFs) plus spherical…
In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…
Given cell-average data values of a piecewise smooth bivariate function $f$ within a domain $\Omega$, we look for a piecewise adaptive approximation to $f$. We are interested in an explicit and global (smooth) approach. Bivariate…
In many areas of machine learning, it becomes necessary to find the eigenvector decompositions of large matrices. We discuss two methods for reducing the computational burden of spectral decompositions: the more venerable Nystom extension…
In this paper we present a variant of the proximal forward-backward splitting iteration for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The…
The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge…
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result…