For regression functions that are locally low-dimensional, sparse neural network estimates achieve the rate n^{-2p/(2p+d*)}, with the exponent depending only on the local dimension d* and not the input dimension d.
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Estimation of a function of low local dimensionality by deep neural networks
For regression functions that are locally low-dimensional, sparse neural network estimates achieve the rate n^{-2p/(2p+d*)}, with the exponent depending only on the local dimension d* and not the input dimension d.