# Reverse Fa`a di Bruno's Formula for Cartesian Reverse Differential Categories

arXiv: https://arxiv.org/abs/2509.20931 · HTML: https://mlsanity.com/paper/2509.20931/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 16 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Reverse derivative categories](https://mlsanity.com/paper/1910.07065/) (2019)
- [Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits](https://mlsanity.com/paper/2101.10488/) (2021)
- [Categories of Differentiable Polynomial Circuits for Machine Learning](https://mlsanity.com/paper/2203.06430/) (2022)
- [CHAD: Combinatory Homomorphic Automatic Differentiation](https://mlsanity.com/paper/2103.15776/) (2021)
- [Linearizing Combinators](https://mlsanity.com/paper/2010.15490/) (2020)
- [Categorical semantics of a simple differential programming language](https://mlsanity.com/paper/2101.10491/) (2021)
- [Monoidal Reverse Differential Categories](https://mlsanity.com/paper/2203.12478/) (2022)
- [Jacobians and Gradients for Cartesian Differential Categories](https://mlsanity.com/paper/2211.01835/) (2022)

## Abstract (arXiv)

Reverse differentiation is an essential operation for automatic differentiation. Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain rule, which is the formula that expresses the reverse derivative of a composition. Here, we present the reverse differential analogue of Faa di Bruno's Formula, which gives a higher-order reverse chain rule in a Cartesian reverse differential category. To properly do so, we also define partial reverse derivatives and higher-order reverse derivatives in a Cartesian reverse differential category.