  This paper presents a concise mathematical framework for investigating both
feed-forward and backward process, during the training to learn model weights,
of an artificial neural network (ANN). Inspired from the idea of the two-step
rule for backpropagation, we define a notion of F-adjoint which is aimed at a
better description of the backpropagation algorithm. In particular, by
introducing the notions of F-propagation and F-adjoint through a deep neural
network architecture, the backpropagation associated to a cost/loss function is
proven to be completely characterized by the F-adjoint of the corresponding
F-propagation relatively to the partial derivative, with respect to the inputs,
of the cost function.
