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Deep Neural Networks for Image-Based Dietary Assessment

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

Uniform approximation with quadratic neural networks.

Ahmed Abdeljawad¹

¹Johann Radon Institute of Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria.

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

|July 23, 2025

Summary

Deep neural networks with Rectified Quadratic Unit (ReQU) activation can effectively approximate Hölder-regular functions. The number of neurons needed depends on function smoothness and desired accuracy, showing ReQU

Keywords:

Function approximation Hölder spaces Neural network Quadratic activation function

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Area of Science:

Artificial Intelligence
Numerical Analysis
Functional Analysis

Background:

Deep neural networks (DNNs) are powerful function approximators.
Hölder-regular functions are a key class in analysis, characterized by a degree of smoothness.
The choice of activation function significantly impacts DNN approximation capabilities.

Purpose of the Study:

To investigate the approximation power of DNNs with the Rectified Quadratic Unit (ReQU) activation function.
To determine the network complexity (number of neurons and layers) required for approximating Hölder-regular functions.
To analyze the influence of function smoothness and activation function properties on approximation accuracy.

Main Methods:

Constructive proof methodology.
Approximation of local Taylor expansions using deep ReQU networks.
Analysis of approximation error with respect to the uniform norm.

Main Results:

DNNs with ReQU activation can approximate any function in the R-ball of r-Hölder-regular functions (H^r,R) up to any accuracy epsilon.
The number of neurons required is bounded by O(epsilon^{-d/2r}) for a fixed number of layers.
Approximation effectiveness is linked to function smoothness and ReQU characteristics.

Conclusions:

Deep neural networks with ReQU activation offer efficient approximation for Hölder-regular functions.
The ReQU activation's properties are well-suited for capturing the behavior of these functions.
Findings generalize to Rectified Power Unit (RePU) activations (p>=2), suggesting broad applicability.