  We show that for any convex differentiable loss, a deep linear network has no
spurious local minima as long as it is true for the two layer case. This
reduction greatly simplifies the study on the existence of spurious local
minima in deep linear networks. When applied to the quadratic loss, our result
immediately implies the powerful result in [Kawaguchi 2016]. Further, with the
work in [Zhou and Liang 2018], we can remove all the assumptions in [Kawaguchi
2016]. This property holds for more general "multi-tower" linear networks too.
Our proof builds on [Laurent and von Brecht 2018] and develops a new
perturbation argument to show that any spurious local minimum must have full
rank, a structural property which can be useful more generally.
