Recognition of handwritten digits using a neural network based on the piecewise linear function

1994 ◽  
Author(s):  
Hongyu Liu ◽  
Yialei Wang ◽  
Peimao Sun ◽  
Tianyun Zhang ◽  
Rong Jiang
Keyword(s):  
Neural Network ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function

A GENERALIZED CHUA CELL FOR REALIZING ANY CONTINUOUS n-SEGMENT PIECEWISE-LINEAR FUNCTION

2001 ◽  
Vol 11 (09) ◽  
pp. 2517-2527 ◽  
Author(s):  
M. BUCOLO ◽  
L. FORTUNA ◽  
M. FRASCA ◽  
M. G. XIBILIA
Keyword(s):  
Neural Network ◽  
Nonlinear Dynamics ◽  
Linear Function ◽  
Wide Class ◽  
Piecewise Linear ◽  
Cellular Neural Network ◽  
Piecewise Linear Function ◽  
Fundamental Unit

In this letter it is proven that any piecewise-linear (PWL) function can be realized as the sum of a definite number of basic PWL functions, like that implemented in the classical Cellular Neural Network (CNN) structure by L. Chua. This allows us to consider the basic Chua's PWL function as a fundamental unit to build up a generic PWL function and, thus, to extend the class of nonlinear dynamics that can be realized by CNNs to a more wide class including PWL approximation of any nonlinearity. Moreover, an example of the cell realization of such a CNN is reported.

scholarly journals MINLP formulations for continuous piecewise linear function fitting

2021 ◽  
Author(s):  
Noam Goldberg ◽  
Steffen Rebennack ◽  
Youngdae Kim ◽  
Vitaliy Krasko ◽  
Sven Leyffer
Keyword(s):  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Function Fitting ◽  
Computational Performance ◽  
Integer Nonlinear Programming ◽  
Minlp Model ◽  
Necessary Constraints ◽  
Continuous Piecewise Linear Function

AbstractWe consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

scholarly journals DESIGN OF TIME DELAYED CHAOTIC CIRCUIT WITH THRESHOLD CONTROLLER

2011 ◽  
Vol 21 (03) ◽  
pp. 725-735 ◽  
Author(s):  
K. SRINIVASAN ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
M. LAKSHMANAN ◽  
SUDESHNA SINHA
Keyword(s):  
Numerical Simulations ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Operational Amplifiers ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Oscillator ◽  
Threshold Values ◽  
Chaotic Circuit

A novel time delayed chaotic oscillator exhibiting mono- and double scroll complex chaotic attractors is designed. This circuit consists of only a few operational amplifiers and diodes and employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of piecewise linear function facilitates controlling the shape of the attractors. This is demonstrated by constructing the phase portraits of the attractors through numerical simulations and hardware experiments. Based on these studies, we find that this circuit can produce multi-scroll chaotic attractors by just introducing more number of threshold values.

MAXIMUM DYNAMIC RANGE OF BIFURCATION OF CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 471-481 ◽  
Author(s):  
A. A. A. NASSER ◽  
E. E. HOSNY ◽  
M. I. SOBHY
Keyword(s):  
Linear Function ◽  
Dynamic Range ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Bifurcation Parameter ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Simulation Results

This paper includes a method for detecting the maximum possible range of bifurcations based upon the multilevel oscillation technique. An application of the method to Chua's circuit, and new simulation results using the slope of the piecewise-linear function as a bifurcation parameter are presented.

scholarly journals Use of the Shor’s r-Algorithm in Linear Robust Optimization Problems

2021 ◽  
pp. 29-42
Author(s):  
P. Stetsyuk ◽  
М. Stetsyuk ◽  
D. Bragin ◽  
N. Мolodyk
Keyword(s):  
Linear Programming ◽  
Robust Optimization ◽  
Convex Function ◽  
Linear Function ◽  
Penalty Function ◽  
Piecewise Linear ◽  
Auxiliary Problem ◽  
Unconstrained Minimization ◽  
Piecewise Linear Function ◽  
Nonsmooth Penalty

The paper is devoted to the description of a new approach to the construction of algorithms for solving linear programming problems (LP-problems), in which the number of constraints is much greater than the number of variables. It is based on the use of a modification of the r-algorithm to solve the problem of minimizing a nonsmooth function, which is equivalent to LP problem. The advantages of the approach are demonstrated on the linear robust optimization problem and the robust parameters estimation problem using the least moduli method. The developed octave programs are designed to solve LP problems with a very large number of constraints, for which the use of standard software from linear programming is either impossible or impractical, because it requires significant computing resources. The material of the paper is presented in three sections. In the first section for the problem of minimizing a convex function we describe a modification of the r-algorithm with a constant coefficient of space dilation in the direction of the difference of two successive subgradients and an adaptive method for step size adjustment in the direction of the antisubgradient in the transformed space of variables. The software implementation of this modification is presented in the form of Octave function ralgb5a, which allows to find or approximation of the minimum point of a convex function, or approximation of the maximum point of the concave function. The code of the ralgb5a function is given with a brief description of its input and output parameters. In the second section, a method for solving the LP problem is presented using a nonsmooth penalty function in the form of maximum function and the construction of an auxiliary problem of unconstrained minimization of a convex piecewise linear function. The choice of the finite penalty coefficient ensures equivalence between the LP-problem and the auxiliary problem, and the latter is solved using the ralgb5a program. The results of computational experiments in GNU Octave for solving test LP-problems with the number of constraints from two hundred thousand to fifty million and the number of variables from ten to fifty are presented. The third section presents least moduli method that is robust to abnormal observations or "outliers". The method uses the problem of unconstrained minimization of a convex piecewise linear function, and is solved using the ralgb5a program. The results of computational experiments in GNU Octave for solving test problems with a large number of observations (from two hundred thousand to five million) and a small number of unknown parameters (from ten to one hundred) are presented. They demonstrate the superiority of the developed programs over well-known linear programming software such as the GLPK package. Keywords: robust optimization, linear programming problem, nonsmooth penalty function, r-algorithm, least modulus method, GNU Octave.

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