Principal component analysis using neural network

Abstract

The authors present their analysis of the differential equation dX(t/dt = AX(t) - XT(t) BX(t)X(t), where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X ∈ Rn; showing that the equation characterizes a class of continuous type full-feedback artificial neural network; We give the analytic expression of the solution; discuss its asymptotic behavior; and finally present the result showing that, in almost all cases, one and only one of following cases is true. 1. For any initial value X0∈Rn, the solution approximates asymptotically to zero vector. In this case, the real part of each eigenvalue of A is non-positive. 2. For any initial value X0 outside a proper subspace of Rn, the solution approximates asymptotically to a nontrivial constant vector Ỹ(X0). In this case, the eigenvalue of A with maximal real part is the positive number λ = ‖ Ỹ(X0) ‖ B2 and B is the corresponding eigenvector. 3. For any initial value X0 outside a proper subspace of Rn, the solution approximates asymptotically to a non-constant periodic function Ỹ(X0, t). Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.