that the components of the phase retrieval inspired approach to neural network training have been introduced, we can highlight some of the features that make it attractive. These derive from the greater power that projections potentially have to offer over the gradient moves that underlie nearly all current algorithms. Consider the activation function constraint . Not only is $f$ not required to be a piecewise regular function, it need not even be a function! A good example is the modification of the step “function" shown in Figure 2. Interpreted as a locus of pre/post-activation pairs $(z=y-b,x)$, a projection $(z,x)\to (z_A,x_A)$ is easily computed even if there is a gap in the domain of the function definition as shown. Such a gap is analogous to the margin parameter normally reserved for the output layer in a classifier. By introducing a margin in the activation, individual neurons will be forced to make unambiguous choices during training, a feature that may improve generalization. After training, if a pre-activation $z$ falls within the gap, $f$ would be interpreted as the standard step function with discontinuity at $z=0$.

Another advantage of projections is that they provide a direct mechanism for imposing structural properties that gradient 