Toroidal Approximate Identity Neural Networks Are Universal Approximators

Abstract

The idea of approximation functions on the rotation group has important applications in many fields of science and engineering. This study is devoted to explore the universal approximation capability of a class of three layer feedforward artificial neural networks on the special orthogonal rotation group SO(2). To do this end, we propose the concept of SO(2) approximate identity. Moreover, we prove a theorem that provides a connection between SO(2) approximate identity and uniform convergence in the space of continuous functions on the rotation group SO(2). Furthermore, we apply this theorem to set a main theorem. The main theorem shows that three layer feedforward SO(2) approximate identity neural networks are universal approximators in the space of continuous functions on the rotation group SO(2). The construction of the proof of the main theorem utilizes a method based on the notion of epsilon-net.