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

On the Landscape of One-hidden-layer Sparse Networks and Beyond