Neural Networks with Complex-Valued Weights Have No Spurious Local Minima

We study the benefits of complex-valued weights for neural networks. We prove that shallow complex neural networks with quadratic activations have no spurious local minima. In contrast, shallow real neural networks with quadratic activations have infinitely many spurious local minima under the same conditions. In addition, we provide specific examples to demonstrate that complex-valued weights turn poor local minima into saddle points. The activation function CReLU is also discussed to illustrate the superiority of analytic activations in complex-valued neural networks.