  Due to the success of deep learning to solving a variety of challenging
machine learning tasks, there is a rising interest in understanding loss
functions for training neural networks from a theoretical aspect. Particularly,
the properties of critical points and the landscape around them are of
importance to determine the convergence performance of optimization algorithms.
In this paper, we provide full (necessary and sufficient) characterization of
the analytical forms for the critical points (as well as global minimizers) of
the square loss functions for various neural networks. We show that the
analytical forms of the critical points characterize the values of the
corresponding loss functions as well as the necessary and sufficient conditions
to achieve global minimum. Furthermore, we exploit the analytical forms of the
critical points to characterize the landscape properties for the loss functions
of these neural networks. One particular conclusion is that: The loss function
of linear networks has no spurious local minimum, while the loss function of
one-hidden-layer nonlinear networks with ReLU activation function does have
local minimum that is not global minimum.
