  Radial basis function neural networks (\emph{RBFNN}) are {well-known} for
their capability to approximate any continuous function on a closed bounded set
with arbitrary precision given enough hidden neurons. In this paper, we
introduce the first algorithm to construct coresets for \emph{RBFNNs}, i.e.,
small weighted subsets that approximate the loss of the input data on any
radial basis function network and thus approximate any function defined by an
\emph{RBFNN} on the larger input data. In particular, we construct coresets for
radial basis and Laplacian loss functions. We then use our coresets to obtain a
provable data subset selection algorithm for training deep neural networks.
Since our coresets approximate every function, they also approximate the
gradient of each weight in a neural network, which is a particular function on
the input. We then perform empirical evaluations on function approximation and
dataset subset selection on popular network architectures and data sets,
demonstrating the efficacy and accuracy of our coreset construction.
