Tractability Of Approximation Research Articles

Tractability of approximation by general shallow networks

In this paper, we present a sharper version of the results in the paper [H. N. Mhaskar, Dimension independent bounds for general shallow networks, Neural Netw. 123 (2020) 142–152]. Let [Formula: see text] and [Formula: see text] be compact metric spaces. We consider approximation of functions of the form [Formula: see text], [Formula: see text], by [Formula: see text]-networks of the form [Formula: see text], [Formula: see text], [Formula: see text]. Defining the dimensions of [Formula: see text] and [Formula: see text] in terms of covering numbers, we obtain dimension independent bounds on the degree of approximation in terms of [Formula: see text], where also the constants involved are all dependent at most polynomially on the dimensions. Applications include approximation by power rectified linear unit networks, zonal function networks, certain radial basis function networks as well as the important problem of function extension to higher-dimensional spaces.

EC-(t1,t2)-tractability of approximation in weighted Korobov spaces in the worst case setting

We investigate multivariate approximation problems in the weighted Korobov spaces in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We discuss the exponential convergence-(t1,t2)-weak tractability (EC-(t1,t2)-WT) under the absolute or normalized error criterion. Especially, we give the sufficient and necessary conditions for EC-WT and EC-(t1,1)-WT with t1<1.

Tractability of approximation in the weighted Korobov space in the worst-case setting — a complete picture

In this paper, we study tractability of L2-approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class Λall consisting of all continuous linear functionals and from the class Λstd, which only consists of function evaluations. We provide necessary and sufficient conditions on the weights of the function space for quasi-polynomial tractability, uniform weak tractability, weak tractability and (σ,τ)-weak tractability. Together with the already known results for strong polynomial and polynomial tractability, our findings provide a complete picture of the weight conditions for all current standard notions of tractability.

On tractability of approximation in special function spaces

We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show that, depending on the norm for measuring the error, the problems are strongly polynomially or quasi-polynomially tractable even in the model of computation where functional evaluations have the cost exponential in the number of active variables.

Dependence of Computational Models on Input Dimension: Tractability of Approximation and Optimization Tasks

The role of input dimension <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex> </formula> is studied in approximating, in various norms, target sets of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex></formula> -variable functions using linear combinations of adjustable computational units. Results from the literature, which emphasize the number <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n$</tex></formula> of terms in the linear combination, are reformulated, and in some cases improved, with particular attention to dependence on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex> </formula> . For worst-case error, upper bounds are given in the factorized form <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\xi(d)\kappa(n)$</tex> </formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\kappa$</tex> </formula> is nonincreasing (typically <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\kappa(n) \sim n^{-1/2}$</tex></formula> ). Target sets of functions are described for which the function <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\xi$</tex></formula> is a polynomial. Some important cases are highlighted where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\xi$</tex> </formula> decreases to zero as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d \to \infty$</tex></formula> . For target functions, extent (e.g., the size of domains in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBR}^d$</tex> </formula> where they are defined), scale (e.g., maximum norms of target functions), and smoothness (e.g., the order of square-integrable partial derivatives) may depend on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex> </formula> , and the influence of such dimension-dependent parameters on model complexity is considered. Results are applied to approximation and solution of optimization problems by neural networks with perceptron and Gaussian radial computational units.

Complexity of Gaussian-radial-basis networks approximating smooth functions

Complexity of Gaussian-radial-basis-function networks, with varying widths, is investigated. Upper bounds on rates of decrease of approximation errors with increasing number of hidden units are derived. Bounds are in terms of norms measuring smoothness (Bessel and Sobolev norms) multiplied by explicitly given functions a ( r , d ) of the number of variables d and degree of smoothness r . Estimates are proven using suitable integral representations in the form of networks with continua of hidden units computing scaled Gaussians and translated Bessel potentials. Consequences on tractability of approximation by Gaussian-radial-basis function networks are discussed.