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.

与密集网络相比，稀疏神经网络由于体积小而受到越来越多的关注。然而，现有的关于神经网络理论的大部分工作都集中在密集神经网络上，对稀疏网络的理解非常有限。在本文中，我们研究了单层稀疏网络的损失情况。首先，我们考虑具有密集最后一层的稀疏网络。我们证明了线性网络在特殊的稀疏结构下不能有虚假谷，非线性网络也可以在宽的最后一层下不允许虚假谷。其次，我们发现对于具有稀疏最终层的宽稀疏网络，可能存在虚假山谷和虚假最小值。这与在温和假设下没有虚假谷的宽密集网络不同。