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- W4286891736 abstract "We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in $L^p$) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers. We prove embedding theorems between these spaces for different values of $p$. Furthermore, we derive sharp embeddings of these approximation spaces into Holder spaces. We find that, analogous to the case of classical function spaces (such as Sobolev spaces, or Besov spaces) it is possible to trade smoothness (i.e., approximation rate) for increased integrability. Combined with our earlier results in [arXiv:2104.02746], our embedding theorems imply a somewhat surprising fact related to learning functions from a given neural network space based on point samples: if accuracy is measured with respect to the uniform norm, then an optimal learning algorithm for reconstructing functions that are well approximable by ReLU neural networks is simply given by piecewise constant interpolation on a tensor product grid." @default.
- W4286891736 created "2022-07-25" @default.
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- W4286891736 date "2021-10-28" @default.
- W4286891736 modified "2023-09-27" @default.
- W4286891736 title "Sobolev-type embeddings for neural network approximation spaces" @default.
- W4286891736 doi "https://doi.org/10.48550/arxiv.2110.15304" @default.
- W4286891736 hasPublicationYear "2021" @default.
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