Recurrent Neural Networks for Computing Pseudoinverses of Rank-Deficient Matrices

Abstract

Three recurrent neural networks are presented for computing the pseudoinverses of rank-deficient matrices. The first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of Moore--Penrose inversion under the condition of zero initial states. The second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self-connections and constant connections in each row or column. The third recurrent neural network consists of two layers of neuron arrays corresponding, respectively, to a pseudoinverse matrix and a Lagrangian matrix with constant connections. All three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse. The proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank-deficient matrices.