  Sparse neural networks have received increasing interest due to their small
size compared to dense networks. Nevertheless, most existing works on neural
network theory have focused on dense neural networks, and the understanding of
sparse networks is very limited. In this paper, we study the loss landscape of
one-hidden-layer sparse networks. First, we consider sparse networks with a
dense final layer. We show that linear networks can have no spurious valleys
under special sparse structures, and non-linear networks could also admit no
spurious valleys under a wide final layer. Second, we discover that spurious
valleys and spurious minima can exist for wide sparse networks with a sparse
final layer. This is different from wide dense networks which do not have
spurious valleys under mild assumptions.
