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Integral representations of shallow neural network with rectified power unit activation function.

Ahmed Abdeljawad¹, Philipp Grohs²

¹Johann Radon Institute, Altenberger Straße 69, A-4040 Linz, Austria.

Neural Networks : the Official Journal of the International Neural Network Society

|September 27, 2022

Summary

Infinite neural networks with RePU activation functions can represent functions that are κ-order Lipschitz and have a finite R-norm. This research extends prior work on ReLU networks and explores sparse representations.

Keywords:

Integral representation Radon transform Rectified power unit Shallow neural network

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Area of Science:

Machine Learning
Neural Networks
Functional Analysis

Background:

Infinite width neural networks are powerful function approximators.
RePU activation functions offer greater smoothness than ReLU, improving applicability.
Previous studies focused on ReLU and L2 norm regularization.

Purpose of the Study:

Characterize functions representable by infinite neural networks with RePU activation.
Investigate the impact of L(2/p) norm regularization on network coefficients.
Extend existing function representation theory for neural networks.

Main Methods:

Analysis of function spaces representable by neural networks.
Utilizing L(2/p) (quasi)norm regularization for network coefficients.
Characterization based on function properties like Lipschitz continuity and R-norm.

Main Results:

Established necessary and sufficient conditions for function representation using RePU networks.
Demonstrated that functions must be κ-order Lipschitz with finite R-norm.
Extended representation capabilities beyond ReLU networks and L2 regularization.

Conclusions:

RePU activation functions expand the class of representable functions in infinite networks.
L(2/p) regularization is key to achieving these representations.
Findings provide insights into creating sparse neural network representations.