# Learning by the F-adjoint

arXiv: https://arxiv.org/abs/2407.11049 · HTML: https://mlsanity.com/paper/2407.11049/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 8 Jul 2024.

## Builds on (most cited first, 7 of 7)

- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Direct Feedback Alignment Provides Learning in Deep Neural Networks](https://mlsanity.com/paper/1609.01596/) (2016)
- [Equivalence of Equilibrium Propagation and Recurrent Backpropagation](https://mlsanity.com/paper/1711.08416/) (2017)
- [Activation Relaxation: A Local Dynamical Approximation to Backpropagation in the Brain](https://mlsanity.com/paper/2009.05359/) (2020)
- [A Two-Step Rule for Backpropagation](https://mlsanity.com/paper/2304.13537/) (2023)
- [Backpropagation and F-adjoint](https://mlsanity.com/paper/2304.13820/) (2023)

## Built on by (most cited first, 1 of 1)

- [Backpropagation as a Nilpotent Linear System](https://mlsanity.com/paper/2607.11289/) (2026)

## Abstract (arXiv)

A recent paper by Boughammoura (2023) describes the back-propagation algorithm in terms of an alternative formulation called the F-adjoint method. In particular, by the F-adjoint algorithm the computation of the loss gradient, with respect to each weight within the network, is straightforward and can simply be done. In this work, we develop and investigate this theoretical framework to improve some supervised learning algorithm for feed-forward neural network. Our main result is that by introducing some neural dynamical model combined by the gradient descent algorithm, we derived an equilibrium F-adjoint process which yields to some local learning rule for deep feed-forward networks setting. Experimental results on MNIST and Fashion-MNIST datasets, demonstrate that the proposed approach provide a significant improvements on the standard back-propagation training procedure.