  We give a formal and complete characterization of the explicit regularizer
induced by dropout in deep linear networks with squared loss. We show that (a)
the explicit regularizer is composed of an $\ell_2$-path regularizer and other
terms that are also re-scaling invariant, (b) the convex envelope of the
induced regularizer is the squared nuclear norm of the network map, and (c) for
a sufficiently large dropout rate, we characterize the global optima of the
dropout objective. We validate our theoretical findings with empirical results.
