Global exponential convergence and stability of gradient-based neural network for online matrix inversion (Q734897)

Starting point of online matrix inversions is the equation \(AX-I=0\) and the recurrent gradient based neural network \(\dot{X}(t) = -\gamma (AX(t)-I)\) for a design parameter \(\gamma\). To prove convergence and stability of this neural network, the authors use the Lyapunov function \(E(t) = \) tr\((\tilde{X}^TA^TA\tilde{X})/2\) for \(\tilde{X}(t) = X(t) - X^*\) and the exact inverse \(X^*\). For nonsingular \(A\) this neural network is shown to be exponentially convergent for any start \(X(0)\), while for singular \(A\), the method is still globally stable. In experiments this convergence behavior is verified and in case of singular \(A\), the computed solution \(X\) is found to equal the Moore-Penrose pseudoinverse of \(A\).