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.
Then log ( N (ǫ, F (sparse) M ∗,L,r,α , ‖ · ‖∞, [−a,a ]d) ) ≤ c33 · log(n) · M ∗ holds for any n > 1 and a constant c33 > 0 independent of n and M ∗
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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.