Approximation of functions with spatial inhomogeneity based on “true” ortho-ridgelet neural network

Abstract

To approximate the multivariate functions with spatial inhomogeneity, in this paper we proposed an ortho-ridgelet neural network (ORNN) model. By taking orthonormal ridgelet, which is a “true” ridgelet function different with the “classic” ridgelet, as the activation function of the hidden neurons, the network is characterized of more efficient representation of a set of functions with linear and curvilinear singularities. Although the ortho-ridgelet is initially more complicated to be understood than ridgelet, it can be easily applied to function approximation theory in a more effective manner. Moreover, we extended the ORNN model to the time-varied situation, and the time-varied ortho-ridgelet neural network (TV-ORRN) is established for time-varied function approximation, with an incremental learning of the ridgelets output layer. The theoretical analysis is presented and some experiments are taken, and the experimental results prove its efficiency in approximation of multivariate functions with linear and curvilinear singularities, in both the stationary and time-varied environment.