Approximation capabilities of measure-preserving neural networks

TL;DR

This paper investigates the approximation capabilities of measure-preserving neural networks like NICE and RevNets, demonstrating they can approximate a broad class of measure-preserving maps under certain conditions.

Contribution

It provides a rigorous analysis showing measure-preserving neural networks can approximate any bounded, injective, measure-preserving map in $L^p$-norm, expanding understanding of their theoretical power.

Findings

Neural networks can approximate measure-preserving maps on compact sets.

Any continuously differentiable injective map with Jacobian determinant ±1 can be approximated.

The analysis applies to models like NICE and RevNets.

Abstract

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.