  We analyze speed of convergence to global optimum for gradient descent
training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1}
\cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence
at a linear rate is guaranteed when the following hold: (i) dimensions of
hidden layers are at least the minimum of the input and output dimensions; (ii)
weight matrices at initialization are approximately balanced; and (iii) the
initial loss is smaller than the loss of any rank-deficient solution. The
assumptions on initialization (conditions (ii) and (iii)) are necessary, in the
sense that violating any one of them may lead to convergence failure. Moreover,
in the important case of output dimension 1, i.e. scalar regression, they are
met, and thus convergence to global optimum holds, with constant probability
under a random initialization scheme. Our results significantly extend previous
analyses, e.g., of deep linear residual networks (Bartlett et al., 2018).
