Backpropagation and F-adjoint

This paper presents a concise mathematical framework for investigating both feed-forward and backward propagation processes, during the training to learn model weights, of a deep neural network (DNN). Inspired from the idea of the two-step rule for back-propagation, presented by the author in \cite{bougham2023}, we define a notion of F-adjoint which is aimed at a better description of the backprpagation process. In particular, by introducing the notions of F-propagation and F-adjoint with respect to any deep neural network architecture, the backpropagation associated to any cost/loss function is proven to be completely characterized by the F-adjoint of the corresponding F-parpagation relatively to the partial derivative, with respect to the inputs, of the cost function.