scholarly journals Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions

2018 ◽  
Vol 26 (1) ◽  
pp. 37 ◽  
Author(s):  
O.V. Kozynenko
Keyword(s):  
Approximation Order ◽  
Order Of Approximation ◽  
Piecewise Constant ◽  
Differentiable Functions ◽  
Convex Partitions ◽  
Saturation Order ◽  
Piecewise Constant Approximation

We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.

ReLU Networks Are Universal Approximators via Piecewise Linear or Constant Functions

2020 ◽  
Vol 32 (11) ◽  
pp. 2249-2278
Author(s):  
Changcun Huang
Keyword(s):  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Piecewise Linear Approximation ◽  
Piecewise Constant ◽  
Deep Layers ◽  
Univariate Function ◽  
Arbitrary Precision ◽  
Piecewise Constant Approximation ◽  
Neural Unit ◽  
Universal Approximators

This letter proves that a ReLU network can approximate any continuous function with arbitrary precision by means of piecewise linear or constant approximations. For univariate function [Formula: see text], we use the composite of ReLUs to produce a line segment; all of the subnetworks of line segments comprise a ReLU network, which is a piecewise linear approximation to [Formula: see text]. For multivariate function [Formula: see text], ReLU networks are constructed to approximate a piecewise linear function derived from triangulation methods approximating [Formula: see text]. A neural unit called TRLU is designed by a ReLU network; the piecewise constant approximation, such as Haar wavelets, is implemented by rectifying the linear output of a ReLU network via TRLUs. New interpretations of deep layers, as well as some other results, are also presented.

scholarly journals Approximation and shape preserving properties of the nonlinear Favardsza-Szász-Mirakjan operator of max-product kind

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 55-72 ◽  
Author(s):  
Barnabás Bede ◽  
Lucian Coroianu ◽  
Sorin Gal
Keyword(s):  
Nonlinear Operator ◽  
Approximation Error ◽  
Pointwise Estimate ◽  
Approximation Order ◽  
Order Of Approximation ◽  
Bernstein Type ◽  
Shape Preserving ◽  
Explicit Constant ◽  
Open Question ◽  
Shape Preserving Properties

Starting from the study of the Shepard nonlinear operator of max-prod type in [6], [7], in the book [8], Open Problem 5.5.4, pp. 324-326, the Favard-Sz?sz-Mirakjan max-prod type operator is introduced and the question of the approximation order by this operator is raised. In the recent paper [1], by using a pretty complicated method to this open question an answer is given by obtaining an upper pointwise estimate of the approximation error of the form C?1(f;?x/?n) (with an unexplicit absolute constant C>0) and the question of improving the order of approximation ?1(f;?x/?n) is raised. The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages : it produces an explicit constant in front of ?1(f;?x/?n) and it can easily be extended to other max-prod operators of Bernstein type. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ?1(f;?) cannot be improved. However, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order ?1 (f;1/n) is obtained. Finally, some shape preserving properties are obtained.

Adaptive display images

2019 ◽  
Vol 18 (01) ◽  
pp. 1-23
Author(s):  
Meipeng Zhi ◽  
Yuesheng Xu
Keyword(s):  
Computational Complexity ◽  
Error Estimate ◽  
Visual Quality ◽  
Natural Image ◽  
Approximation Accuracy ◽  
Piecewise Constant ◽  
Computer Screen ◽  
Real Scene ◽  
Piecewise Constant Approximation ◽  
Optimal Formula

We develop a numerical method for construction of an adaptive display image from a given display image which is an artificial scene displayed in a computer screen. The adaptive display image is encoded on an adaptive pixel mesh obtained by a merging scheme from the original pixel mesh. The cardinality of the adaptive pixel mesh is significantly less than that of the original pixel mesh. The resulting adaptive display image is the best [Formula: see text] piecewise constant approximation of the original display image. Under the assumption that a natural image, the real scene that we see, belongs to a Besov space, we provide the optimal [Formula: see text] error estimate between the adaptive display image and its original natural image. Experimental results are presented to demonstrate the visual quality, the approximation accuracy and the computational complexity of the adaptive display image.

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