Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov space

TL;DR

This paper analyzes how deep learning adapts to the intrinsic anisotropic smoothness of functions in Besov spaces, showing it can avoid the curse of dimensionality and outperform linear methods.

Contribution

It provides a theoretical analysis of deep learning's approximation and estimation abilities in anisotropic Besov spaces, highlighting its adaptivity and optimality without requiring low-dimensional input structures.

Findings

Deep learning's errors depend on average smoothness parameters.

It can avoid the curse of dimensionality with highly anisotropic functions.

Deep learning outperforms kernel methods for anisotropic smoothness.

Abstract

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse of dimensionality can be avoided if the smoothness of the target function is highly anisotropic. Unlike existing studies, our analysis does not require a low-dimensional structure of the input data. We also investigate the minimax optimality of deep learning and compare its performance with that of the kernel method (more generally, linear estimators). The results show that deep learning has better dependence on the input dimensionality if the target function possesses anisotropic smoothness, and it achieves an adaptive rate for functions with spatially inhomogeneous smoothness.