Approximation capabilities of measure-preserving neural networks

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