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arXiv · 2509.20931

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

Aaron Biggin·Jean-Simon Pacaud Lemay
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Abstract

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.

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Aaron Biggin, Jean-Simon Pacaud Lemay. 2025-09-25. Reverse Fa\`a di Bruno's Formula for Cartesian Reverse Differential Categories. https://doi.org/10.4204/eptcs.429.6

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