Dimension-independent learning rates for high-dimensional classification problems

In this paper, we study the problem of approximating and estimating classification functions that have their decision boundary in the [Formula: see text] space. Functions of [Formula: see text] type arise naturally as solutions of regularized neural network learning problems and neural networks can approximate these functions without the curse of dimensionality. We modify existing results to show that every [Formula: see text] function can be approximated by a neural network with bounded weights. Thereafter, we prove the existence of a neural network with bounded weights approximating a classification function. We leverage these bounds to quantify the estimation rates. Finally, we present a numerical study that analyzes the effect of different regularity conditions on the decision boundaries.