The training dynamics of linear networks are well studied in two distinct setups: the lazy regime and balanced/active regime, depending on the initialization and width of the network. We provide a surprisingly simple unifying formula for the evolution of the learned matrix that contains as special cases both lazy and balanced regimes but also a mixed regime in between the two. In the mixed regime, a part of the network is lazy while the other is balanced. More precisely the network is lazy along singular values that are below a certain threshold and balanced along those that are above the same threshold. At initialization, all singular values are lazy, allowing for the network to align itself with the task, so that later in time, when some of the singular value cross the threshold and become active they will converge rapidly (convergence in the balanced regime is notoriously difficult in the absence of alignment). The mixed regime is the `best of both worlds': it converges from any random initialization (in contrast to balanced dynamics which require special initialization), and has a low rank bias (absent in the lazy dynamics). This allows us to prove an almost complete phase diagram of training behavior as a function of the variance at initialization and the width, for a MSE training task.

Any generic planar closed curve with n crossings can be turned into a simple closed curve by applying O(n3/2) homotopy moves without ever increasing the number of self-crossings; this improves over the O(n2) upper bound from Steinitz [Ency. Math. Wiss. III 1916], and matches the best lower bound. We prove the existence of a positive move that decreases the depth-sum potential at every step. Using similar techniques, we show that any 2-terminal plane graph with n vertices can be reduced to a single edge between the terminals using O(n3/2) electrical transformations, consisting of degree-1 reductions, series-parallel reductions, and ΔY-transformations; this proves a conjecture of Feo and Provan that was open for more than 30 years.