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Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…
Nonlinear expressions are often approximated by piecewise affine (PWA) functions to simplify analysis or reduce computational costs. To reduce computational complexity, multivariate functions can be represented as compositions of functions…
An overview is given of the methods for treating complicated problems without small parameters, when the standard perturbation theory based on the existence of small parameters becomes useless. Such complicated problems are typical of…
Derivative-matching approximations are constructed as power series built from functions. The method assumes the knowledge of special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented…
A general framework with a series of different methods is proposed to improve the estimate of convex function (or functional) values when only noisy observations of the true input are available. Technically, our methods catch the bias…
The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet…
Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach…
We introduce the notion of the generalized-analytical function of the poly-number variable, which is a non-trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in…
An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive…
We propose a general method for constructing system-dependent basis functions for correlated quantum chemical calculations. Our construction combines features from several traditional approaches: plane waves, localized basis functions, and…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
In this note, we will present global equisingular approximations of quasi-plurisubharmonic functions with stable analytic pluripolar sets on compact complex manifolds.
Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems.…
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply…
Motivated by applications for simulating quantum many body functions, we propose a new universal ansatz for approximating anti-symmetric functions. The main advantage of this ansatz over previous alternatives is that it is bi-Lipschitz with…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen \& Kuijlaars states that fast and stable approximation from equispaced samples is…
We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such…
This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. These approximations are obtained using the techniques called \emph{generalized simplex Hessian} and…