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Deep ReLU neural networks in high-dimensional approximation.

Dinh Dũng¹, Van Kien Nguyen²

¹Information Technology Institute, Vietnam National University, Hanoi 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam.

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

|August 15, 2021

Summary

We analyzed deep Rectified Linear Unit (ReLU) neural networks for function approximation in high dimensions. We established tight bounds on their computational complexity, considering network size and depth for accurate function approximation.

Keywords:

Computation complexity Continuous piece-wise linear functions Deep ReLU neural network High-dimensional approximation Hölder–Zygmund space of mixed smoothness Sparse-grid sampling

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Area of Science:

Computational Mathematics
Machine Learning Theory
Numerical Analysis

Background:

Deep ReLU neural networks are powerful function approximators.
Understanding their computational complexity is crucial, especially in high dimensions.
Approximating functions in Hölder-Zygmund spaces presents unique challenges.

Purpose of the Study:

To investigate the computational complexity of deep ReLU neural networks for function approximation.
To analyze approximation errors in isotropic Sobolev space norms.
To derive dimension-dependent bounds for network size and depth.

Main Methods:

Explicit construction of deep ReLU neural networks for function approximation.
Utilizing sparse-grid sampling recovery based on Faber series.
Establishing tight upper and lower bounds on computational complexity.

Main Results:

Derived explicit constructions for deep ReLU networks approximating Hölder-Zygmund functions.
Proved tight, dimension-dependent upper and lower bounds for approximation complexity.
Characterized complexity in terms of network size, depth, dimension (d), and accuracy (ɛ).

Conclusions:

Deep ReLU networks offer efficient function approximation for high-dimensional Hölder-Zygmund spaces.
The study provides precise insights into the scaling of computational complexity with dimension and accuracy.
Sparse-grid methods and Faber series are key to achieving these approximation results.