Regression estimation often suffers from the curse of dimensionality. In the present paper we circumvent this problem by introducing a class of models called hierarchical interaction models where the values of a function m ∶ ℝ d  → ℝ are computed in a feed-forward manner in several layers, where in each layer a function of at most d* inputs produced by the previous layer is computed. We introduce regression estimates based on neural networks with two hidden layers and apply them to estimation of regression functions from a class of hierarchical interaction models. Under smoothness condition imposed on all functions occurring in the model we show that the rate of convergence of these estimates depends on d*, which is typically much smaller than d.
['Michael Kohler', 'Adam Krzyzak']
The rates of convergence of neural network estimates of hierarchical interaction regression models