# Backpropagation and F-adjoint

arXiv: https://arxiv.org/abs/2304.13820 · HTML: https://mlsanity.com/paper/2304.13820/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2023.

## Built on by (most cited first, 2 of 2)

- [Learning by the F-adjoint](https://mlsanity.com/paper/2407.11049/) (2024)
- [Backpropagation as a Nilpotent Linear System](https://mlsanity.com/paper/2607.11289/) (2026)

## Abstract (arXiv)

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.