A Study Of Neural Network Models For The Least Absolute Deviation Problem

Least absolute deviation problems include least absolute value problem with linear constraint,least absolute deviation problem subject to linear equalities constraint and general least absolute deviation problem.Their solutions are spare and insensitive to the outliers,and they are good at solving the signal with non-Gaussian noise.Then,they are widely used in science and engineer field,such as regression analysis.function approximation,parameter estimation,filter design,signal processing,image restoration,robot control,speech enhancement,etc.Therefore,it is very necessary and worthy for theory and application to research the efficient methods for these problems.As we all know,least absolute deviation problems are non-smooth.Furthermore,practical application problems are always large-scale,complex and need to be solved in real-time.However,the traditional numerical algorithms cannot satisfy the require-ments in engineering technology since their computing time required for a solution is greatly dependent on the structure and the dimension of the problem,and the com-plexity of the algorithm used.On the contrary,neural networks where the optimization procedures are truly distributed and in parallel can solve the large-scale problems in real-time.Though there had existed some neural network models for least absolute de-viation problems,they have complex structure,many neurons and high computational complexity.In order to overcome these disadvantages,this paper researches further the neural network models for least absolute deviation problems by optimization and Lyapunov theory.Its main work and innovation are as follows:1.For least absolute value problem with linear constraint.,its optimal conditions are transformed into the double projection equations by introducing new variables and using projection theory.And an alternative continuous neural network is proposed,which provides the new method and approach for the problem.Meanwhile,the discrete edition of the model is obtained from Euler approach.It is proved that the proposed models are stable in the sense of Lyapunov and their output.variables converge to the exact optimal solution of original problem for any starting point.Compared with the seven existing neural networks for the optimization problem,the proposed neural networks require the fewer neurons,have a lower complexity and are suitable to be implemented in parallel.2.For least absolute deviation problem with linear equality constraint,the equiv-alent conditions of its optimality are given using saddle point theorem,and a one-layer neural network is constructed.Using Lyapunov method,the proposed neural network is proved to be stable in the sense of Lyapunov and globally convergent.Compared with the six existing neural networks for least absolute deviation with linear equality constraint,the presented model has fewer neurons and a lower complexity.3.Introducing new variables and using variable substitutions,the equivalent con-ditions of optimality of generalized least absolute deviation are given,and a simplified neural network is established.Compared with the four existing neural networks for generalized least absolute deviation,the proposed model has the least neurons and lower complexity.Then,the relationship between the equilibrium point of the model and the exact optimal solution of the original problem is showed,and its stability and convergency are proved.4.For generalized least absolute deviation with linear equality constraint,the equivalent conditions of its optimality are given by utilizing projection theory and variable substitutions.Then,the neural network is designed for it,which is stable in the sense of Lyapunov and globally convergent.Compared with the five existing neural networks for the optimization problem,the presented model requires the fewest neurons and has the least computational complexity.Numerical simulations demonstrate the advantages and effectiveness of the pro-posed network models.And they are applied to practical application problems such as image processing,good results are obtained.