Approximation capabilities of measure-preserving neural networks

Area of Science:

• Machine Learning
• Deep Learning
• Geometric Deep Learning

Background:

• Measure-preserving neural networks are known for their invertibility.
• Their capacity to approximate complex functions has not been thoroughly investigated.
• Existing models like NICE and RevNets are examples of such networks.

Purpose of the Study:

• To rigorously analyze the approximation capabilities of measure-preserving neural networks.
• To determine if these networks can approximate arbitrary measure-preserving maps.
• To extend the understanding of invertible neural networks in function approximation.

Main Methods:

• Theoretical analysis of measure-preserving neural networks.
• Examination of NICE and RevNets architectures.
• Mathematical proofs concerning approximation bounds in L^p-norm for compact sets in R^D.

Main Results:

• Demonstrated that measure-preserving neural networks can approximate arbitrary bounded and injective measure-preserving maps in L^p-norm over compact sets U in R^D (D>=2).
• Established that continuously differentiable injective maps with a Jacobian determinant of +/-1 are measure-preserving and thus approximable.
• Provided theoretical guarantees for the approximation capabilities of these networks.

Conclusions:

• Measure-preserving neural networks possess significant approximation power for a class of measure-preserving maps.
• The findings expand the theoretical understanding of invertible neural networks and their potential applications.
• This research opens avenues for designing more powerful generative models and understanding complex dynamical systems.