On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions

TL;DR

This paper demonstrates that symmetric deep neural networks can efficiently approximate symmetric Korobov functions with convergence rates that do not suffer from the curse of dimensionality, improving theoretical guarantees for high-dimensional function approximation.

Contribution

The paper constructs symmetric deep neural networks for Korobov functions and proves polynomial scaling of convergence and generalization rates with respect to dimension, surpassing previous curse-of-dimensionality limitations.

Findings

Convergence rate scales polynomially with dimension.

Generalization error bounds avoid curse of dimensionality.

Provides theoretical guarantees for high-dimensional symmetric function approximation.

Abstract

Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric Korobov functions and prove that both the convergence rate and the constant prefactor scale at most polynomially with respect to the ambient dimension. This represents a substantial improvement over prior approximation guarantees that suffer from the curse of dimensionality. Building on these approximation bounds, we further derive a generalization-error rate for learning symmetric Korobov functions whose leading factors likewise avoid the curse of dimensionality.