Depth creates no more spurious local minima in linear networks

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 by Kawaguchi (2016). Further, with the recent work by Zhou& 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 the work in (Laurent & 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