LEARNING BY CRITERION OPTIMIZATION ON A UNITARY UNIMODULAR MATRIX GROUP

2008 ◽  
Vol 18 (02) ◽  
pp. 87-103 ◽  
Author(s):  
SIMONE FIORI
Keyword(s):  
Optimization Problems ◽  
Learning Theories ◽  
Neural Systems ◽  
Matrix Group ◽  
Unimodular Group ◽  
Unimodular Matrix ◽  
Low Dimension ◽  
Low Dimensional ◽  
Complex Valued ◽  
Special Case

The present manuscript aims at illustrating fundamental challenges and solutions arising in the design of learning theories by optimization on manifolds in the context of complex-valued neural systems. The special case of a unitary unimodular group of matrices is dealt with. The unitary unimodular group under analysis is a low dimensional and easy-to-handle matrix group. Notwithstanding, it exhibits a rich geometrical structure and gives rise to interesting speculations about methods to solve optimization problems on manifolds. Also, its low dimension allows us to treat most of the quantities involved in computation in closed form as well as to render them in graphical format. Some numerical experiments are presented and discussed within the paper, which deal with complex-valued independent component analysis.

Fast Unimodular Counting

2000 ◽  
Vol 9 (3) ◽  
pp. 277-285 ◽  
Author(s):  
JOHN MOUNT
Keyword(s):  
High Dimension ◽  
Nonnegative Integer ◽  
Unit Cost ◽  
Fixed Number ◽  
Integral Vector ◽  
Fixed Dimension ◽  
Unimodular Matrix ◽  
Low Dimension ◽  
Integer Solutions ◽  
Special Case

This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of ‘contingency tables’ the run-time is 2O(√dlogd) (where d is the dimension of the polytope). The method is complementary to Barvinok's in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints, whereas Barvinok's algorithms are effective on problems of low dimension and an arbitrary number of constraints.

ONE IMPROVED AGENT GENETIC ALGORITHM — RING-LIKE AGENT GENETIC ALGORITHM FOR GLOBAL NUMERICAL OPTIMIZATION

2009 ◽  
Vol 26 (04) ◽  
pp. 479-502 ◽  
Author(s):  
BIN LIU ◽  
TEQI DUAN ◽  
YONGMING LI
Keyword(s):  
Genetic Algorithm ◽  
Numerical Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Genetic Operators ◽  
Global Numerical Optimization ◽  
Candidate Solution ◽  
Low Dimension ◽  
Multi Agent ◽  
Agent Structure

In this paper, a novel genetic algorithm — dynamic ring-like agent genetic algorithm (RAGA) is proposed for solving global numerical optimization problem. The RAGA combines the ring-like agent structure and dynamic neighboring genetic operators together to get better optimization capability. An agent in ring-like agent structure represents a candidate solution to the optimization problem. Any agent interacts with neighboring agents to evolve. With dynamic neighboring genetic operators, they compete and cooperate with their neighbors, and they can also use knowledge to increase energies. Global numerical optimization problems are the most important ones to verify the performance of evolutionary algorithm, especially of genetic algorithm and are mostly of interest to the corresponding researchers. In the corresponding experiments, several complex benchmark functions were used for optimization, several popular GAs were used for comparison. In order to better compare two agents GAs (MAGA: multi-agent genetic algorithm and RAGA), the several dimensional experiments (from low dimension to high dimension) were done. These experimental results show that RAGA not only is suitable for optimization problems, but also has more precise and more stable optimization results.

Parameter Sensitivity Analysis of a Large-Scale System With Several Components Interacting in Series

1989 ◽  
Author(s):  
H. Torab
Keyword(s):  
Sensitivity Analysis ◽  
Large Scale ◽  
Optimization Problems ◽  
Parameter Sensitivity ◽  
Engineering Optimization ◽  
Parameter Sensitivity Analysis ◽  
Large Scale Systems ◽  
In Series ◽  
Special Case ◽  
Engineering Optimization Problems

Abstract Parameter sensitivity for large-scale systems that include several components which interface in series is presented. Large-scale systems can be divided into components or sub-systems to avoid excessive calculations in determining their optimum design. Model Coordination Method of Decomposition (MCMD) is one of the most commonly used methods to solve large-scale engineering optimization problems. In the Model Coordination Method of Decomposition, the vector of coordinating variables can be partitioned into two sub-vectors for systems with several components interacting in series. The first sub-vector consists of those variables that are common among all or most of the elements. The other sub-vector consists of those variables that are common between only two components that are in series. This study focuses on a parameter sensitivity analysis for this special case using MCMD.

Asymptotic Behavior of Periodic Cohen-Grossberg Neural Networks with Delays

2009 ◽  
Vol 21 (12) ◽  
pp. 3444-3459 ◽  
Author(s):  
Wei Lin
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Lyapunov Method ◽  
Autonomous Systems ◽  
Neural Systems ◽  
Stability Criteria ◽  
Volterra System ◽  
Numerical Examples ◽  
The Stability ◽  
Special Case

Without assuming the positivity of the amplification functions, we prove some M-matrix criteria for the [Formula: see text]-global asymptotic stability of periodic Cohen-Grossberg neural networks with delays. By an extension of the Lyapunov method, we are able to include neural systems with multiple nonnegative periodic solutions and nonexponential convergence rate in our model and also include the Lotka-Volterra system, an important prototype of competitive neural networks, as a special case. The stability criteria for autonomous systems then follow as a corollary. Two numerical examples are provided to show that the limiting equilibrium or periodic solution need not be positive.

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

Low-dimensional versus high-dimensional chaos in brain function – is it an and/or issue?

2001 ◽  
Vol 24 (5) ◽  
pp. 823-824 ◽  
Author(s):  
Márk Molnár
Keyword(s):  
Brain Function ◽  
Sensory Information ◽  
Event Related Potentials ◽  
Cognitive Effort ◽  
Neural Systems ◽  
Sensory Information Processing ◽  
Related Potentials ◽  
Dimensional Complexity ◽  
Low Dimensional ◽  
The Brain

We discuss whether low-dimensional chaos and even nonlinear processes can be traced in the electrical activity of the brain. Experimental data show that the dimensional complexity of the EEG decreases during event-related potentials associated with cognitive effort. This probably represents increased nonlinear cooperation between different neural systems during sensory information processing.

scholarly journals An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Wensheng Jia ◽  
Xiaoling Qiu ◽  
Dingtao Peng
Keyword(s):  
Exact Solution ◽  
Bounded Rationality ◽  
Equilibrium Problems ◽  
Approximation Theorem ◽  
Optimization Problems ◽  
Vector Equilibrium Problems ◽  
Special Cases ◽  
Full Rationality ◽  
Vector Equilibrium ◽  
Special Case

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

Groove Bottom Contours of Minimum Stress Concentration for Antiplane Shear Deformation

1985 ◽  
Vol 52 (2) ◽  
pp. 379-384 ◽  
Author(s):  
B. H. Eldiwany ◽  
L. T. Wheeler
Keyword(s):  
Stress Concentration ◽  
Numerical Methods ◽  
Shear Deformation ◽  
Half Space ◽  
Optimization Problems ◽  
Integral Form ◽  
Antiplane Shear ◽  
Space Boundary ◽  
Minimum Stress ◽  
Special Case

Results from free streamline hydrodynamics are exploited in order to solve optimization problems for antiplane shear deformation, in which the stress concentration is to be minimized. These problems pertain to the optimum shapes for grooves cut into a half-space. We obtain results, which from the standpoint of the hydrodynamics problem, complement those presently in the literature. The solution is given in an integral form which in general must be evaluated by numerical methods, but that reduces to elliptic integrals for the special case of a notch whose faces meet the half-space boundary at right angles.

A note on the existence of nontrivial homomorphism En(R) → En−1(R)

2017 ◽  
Vol 16 (01) ◽  
pp. 1750010
Author(s):  
Hong You
Keyword(s):  
Group Actions ◽  
Negative Answer ◽  
Group Homomorphism ◽  
Matrix Group ◽  
Local Rings ◽  
Division Rings ◽  
Nontrivial Group ◽  
Low Dimensional

We show that there is no nontrivial group homomorphism [Formula: see text] over commutative local rings and division rings for [Formula: see text], respectively. It gives a negative answer to Ye’s problem (see [S. K. Ye, Low-dimensional representations of matrix group actions on CAT(0) spaces and manifolds, J. Algebra 409 (2014) 219–243]) for the above rings.

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