Improved Piecewise Constant Approximation Method for Compressing Data Streams

2019 ◽  
Author(s):  
Atik Mahbub ◽  
Farhana Haque ◽  
Habibul Bashar ◽  
Mohammad Rezwanul Huq
Keyword(s):  
Approximation Method ◽  
Data Streams ◽  
Piecewise Constant ◽  
Piecewise Constant Approximation

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.

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.

Approximation of solutions to second order nonlinear Picard problems with Carathéodory right-hand side

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Jacek Gulgowski
Keyword(s):  
Boundary Value Problems ◽  
Approximation Method ◽  
Measurable Function ◽  
Approximation Scheme ◽  
Second Order ◽  
Dimensional Problem ◽  
Piecewise Constant ◽  
Finite Dimensional ◽  
Right Hand ◽  
The Right

AbstractWe present an approximation method for Picard second order boundary value problems with Carathéodory righthand side. The method is based on the idea of replacing a measurable function in the right-hand side of the problem with its Kantorovich polynomial. We will show that this approximation scheme recovers essential solutions to the original BVP. We also consider the corresponding finite dimensional problem. We suggest a suitable mapping of solutions to finite dimensional problems to piecewise constant functions so that the later approximate a solution to the original BVP. That is why the presented idea may be used in numerical computations.

scholarly journals Fast piecewise-constant approximation of images

1991 ◽  
Author(s):  
Hayder Radha ◽  
Martin Vetterli ◽  
Riccardo Leonardi
Keyword(s):  
Piecewise Constant ◽  
Piecewise Constant Approximation

NON-PARABOLIC MARCHING ALGORITHM FOR SOUND FIELD CALCULATION IN THE OCEAN WAVEGUIDE

1996 ◽  
Vol 04 (04) ◽  
pp. 399-423 ◽  
Author(s):  
A. VORONOVICH
Keyword(s):  
Sound Propagation ◽  
A Priori ◽  
Numerical Algorithms ◽  
Sound Field ◽  
Computer Code ◽  
Couple Mode ◽  
Field Calculation ◽  
Piecewise Constant ◽  
S Matrix ◽  
Piecewise Constant Approximation

An algorithm is presented for calculating sound field in the inhomogeneous ocean waveguide. It does not involve parabolic approximation and can be considered as principally exact (at least for 2D inhomogeneities of the sound speed field). On the other hand, it is “marching” and can be easily implemented as a computer code (note, that marching in this case proceeds in “backward” direction, i.e. towards the source). Those features of the code are similar to couple mode algorithm (COUPLE) developed originally by R. Evans. The principal difference is that suggested code does not assume piecewise constant approximation of the waveguide properties with respect to horizontal coordinates. As a result, the horizontal steps of marching can be increased significantly. The estimate of the efficiency of the approach as compared to stepwise couple modes method is given. The results of the code testing with the help of benchmark problem as well as calculation of sound propagation through a strong inhomogeneity formed by the sub-arctic front are presented. The present version of the code can be used to calculate entries of scattering matrix (S-matrix) for the ocean waveguide as well as travel times of different modes (derivatives of phases of corresponding entries with respect to frequency). A priori restrictions on S-matrix (reciprocity and energy conservation) are also given, and some objective quantitative criterion of the accuracy of the numerical algorithms formulated in terms of S-matrix is suggested.

Sign in / Sign up

Export Citation Format

Share Document