Approximation capabilities of measure-preserving neural networks

Abstract

Measure-preserving neural networks are well-developed invertible models, however, their approximation capabilities remain unexplored. This paper rigorously analyzes the approximation capabilities of existing measure-preserving neural networks including NICE and RevNets. It is shown that for compact U⊂R^D with D≥2, the measure-preserving neural networks are able to approximate arbitrary measure-preserving map ψ:U→R^D which is bounded and injective in the L^p-norm. In particular, any continuously differentiable injective map with ±1 determinant of Jacobian is measure-preserving, thus can be approximated.

Keywords: Approximation theory; Dynamical systems; Measure-preserving; Neural networks.