Recurrent Neural Networks for Computing Pseudoinverses of Rank-Deficient Matrices

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.