An Optimization Network for Matrix Inversion

Inverse matrix calculation can be considered as an optimization. We have demonstrated that this problem can be rapidly solved by highly interconnected simple neuron-like analog processors. A network for matrix inversion based on the concept of Hopfield's neural network was designed, and implemented with electronic hardware. With slight modifications, the network is readily applicable to solving a linear simultaneous equation efficiently. Notable features of this circuit are potential speed due to parallel processing, and robustness against variations of device parameters. INTRODUCTION Highly interconnected simple analog processors which mmnc a biological neural network are known to excel at certain collective computational tasks. For example, Hopfield and Tank designed a network to solve the traveling salesman problem which is of the np -complete class,l and also designed an AID converter of novel architecture2 based on the Hopfield's neural network model?' 4 The network could provide good or optimum solutions during an elapsed time of only a few characteristic time constants of the circuit. The essence of collective computation is the dissipative dynamics in which initial voltage configurations of neuron-like analog processors evolve simultaneously and rapidly to steady states that may be interpreted as optimal solutions. Hopfield has constructed the computational energy E (Liapunov function), and has shown that the energy function E of his network decreases in time when coupling coefficients are symmetric. At the steady state E becomes one of local minima. In this paper we consider the matrix inversion as an optimization problem, and apply the concept of the Hopfield neural network model to this problem. CONSTRUCTION OF THE ENERGY FUNCTIONS Consider a matrix equation AV=I, where A is an input n Xn matrix, V is the unknown inverse matrix, and I is the identity matrix. Following Hopfield we define n energy functions E Ie' k = 1, 2, ... , n, n n n E 1 = (1I2)[(~ A 1j Vj1 -1)2 + (~A2) Vj1 )2 + ... + (~Anj Vj1)2] )-1 j-1 n n E2 = (1/2)[(~A1)V)2l + (~A2)V)2-1)2 + )=1 )=1 © American Institute of Physics 1988 )-1 n + (~An)V}2)2] }-1 397