A max-affine spline approximation of neural networks using the Legendre
  transform of a convex-concave representation
Adam Perrett Danny Wood Gavin Brown
This work presents a novel algorithm for transforming a neural network into a
spline representation. Unlike previous work that required convex and
piecewise-affine network operators to create a max-affine spline alternate
form, this work relaxes this constraint. The only constraint is that the
function be bounded and possess a well-define second derivative, although this
was shown experimentally to not be strictly necessary. It can also be performed
over the whole network rather than on each layer independently. As in previous
work, this bridges the gap between neural networks and approximation theory but
also enables the visualisation of network feature maps. Mathematical proof and
experimental investigation of the technique is performed with approximation
error and feature maps being extracted from a range of architectures, including
convolutional neural networks.