Approximation capabilities of measure-preserving neural networks
This work addresses a theoretical gap for researchers in machine learning and dynamical systems by rigorously analyzing the approximation properties of invertible models, though it is incremental as it builds on existing networks without introducing new methods.
The paper tackles the problem of understanding the approximation capabilities of measure-preserving neural networks, such as NICE and RevNets, and shows that these networks can approximate arbitrary bounded, injective measure-preserving maps in the L^p-norm for compact subsets in dimensions D≥2, including continuously differentiable injective maps with ±1 Jacobian determinant.
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 $ψ: U\to \R^D$ 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.