Neural networks with values in a Banach space
I submitted a preprint on two-layer neural networks that take values in a Banach space.
It extends the results of
Bach (2017),
E, Ma and Wu (2019),
E and Wojtowytsch (2020)
and others on approximation rates of infinitely wide scalar-valued two-layer neural networks and establishes
Monte-Carlo rates in Bochner spaces. The most unexpected result (for me) is that in the vector-valued case
continuity of such neural networks can only be established with respect to the weak$^*$ topology in the
target space. This turns out to be a significant restriction for networks with the ReLU activation function.
The preprint can be found here:
arxiv:2105.02095