Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Backpropagation and F-adjoint

Version 1 : Received: 26 April 2023 / Approved: 27 April 2023 / Online: 27 April 2023 (05:01:36 CEST)

How to cite: Boughammoura, A. Backpropagation and F-adjoint. Preprints 2023, 2023041046. Boughammoura, A. Backpropagation and F-adjoint. Preprints 2023, 2023041046.


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.


Artificial neural networks;Backpropagation;Two-step rule;F-propagation;F-adjoint


Computer Science and Mathematics, Applied Mathematics

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