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

2025/11/16 by Yulong Lu, Tong Mao, Lu, Yulong +5
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.2511.12398

openalex publication_date 2025/11/16 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

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.

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Related

• Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms
• The Representation Power of Neural Networks: Breaking the Curse of Dimensionality
• A Functional Perspective on Learning Symmetric Functions with Neural Networks
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• Deep Neural Network Approximation Theory