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

2025-11-15

AI Summary

This work addresses the deep neural network (DNN) approximation of high-dimensional symmetric Korobov functions, aiming to mitigate the curse of dimensionality. We propose an explicitly constructed class of permutation-invariant deep neural networks whose architecture intrinsically respects the spatial symmetry of Korobov functions. Under the Korobov space assumption, we establish rigorous upper bounds on both approximation and generalization errors—both scaling polynomially, rather than exponentially, in the input dimension. To our knowledge, this is the first result achieving a polynomial dependence of the leading constant in the convergence rate on dimension, thereby yielding dimension-independent approximation guarantees. The key innovation lies in embedding functional symmetry priors directly into the network architecture and leveraging this inductive bias to derive tight, dimension-explicit error bounds. Our framework provides the first theoretically sound and constructively implementable DNN approach for learning high-dimensional symmetric functions.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Approximating symmetric Korobov functions using deep neural networks
• Overcoming curse of dimensionality in convergence rate and prefactor
• Deriving generalization-error bounds for learning symmetric Korobov functions

Innovation

• Symmetric deep neural networks approximate symmetric Korobov functions
• Convergence rate scales polynomially with ambient dimension
• Generalization error avoids curse of dimensionality