Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms
This provides foundational improvements in function approximation theory for machine learning, with broad implications for high-dimensional problems.
The paper tackles the approximation of Korobov functions using deep neural networks, achieving a nearly optimal 'super-convergence' rate that overcomes the curse of dimensionality and outperforms traditional methods, with non-asymptotic error bounds in L_p and H^1 norms.
This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions, effectively overcoming the curse of dimensionality. The approximation results presented in this paper are measured with respect to $L_p$ norms and $H^1$ norms. Our achieved approximation rate demonstrates a remarkable "super-convergence" rate, outperforming traditional methods and any continuous function approximator. These results are non-asymptotic, providing error bounds that consider both the width and depth of the networks simultaneously.