A condition for a unique equilibrium point in a recurrent neural network
F. Hunt, D. Pearson · 2002
This paper gives a sufficient condition on the weight matrix W of a recurrent neural network of the form x/spl dot/=-x+/spl sigma/(Wx)+I to have a unique equilibrium point for all values of I. It uses the following argument:-if the net has multiple equilibria for some value of I then F:x/spl rarr/-x+/spl sigma/(Wx) is non-injective. This in turn means that F is singular for some value of the vector x. F singular can be written as the vanishing of a particular determinant involving the elements of the vector x, which implies the vanishing of a particular function of x on a unit box. Since this function achieves its extremal values on the vertices of this box, if the values at the vertices all have the same sign, then the function does not vanish on this box, hence the network has a unique equilibrium point. The condition is the best possible, in that arbitrarily close to a matrix violating the condition is a weight matrix for which the system has multiple equilibria for some I.