Reverse Faà di Bruno's Formula for Cartesian Reverse Differential Categories

2025-09-25

AI Summary

This paper establishes a rigorous categorical characterization of the higher-order reverse chain rule—i.e., the reverse differential analogue of the Faà di Bruno formula—within Cartesian reverse differential categories. To this end, it introduces the first axiomatic definitions of partial reverse derivatives and arbitrary-order higher-order reverse derivatives, and systematically develops their composition rules. Methodologically, it leverages the algebraic structure of reverse differential categories to formalize, via categorical tools, the recursive structure of higher-order reverse derivatives of composite functions. The main contributions are: (1) the first fully categorical formulation of the reverse Faà di Bruno formula; (2) the extension of reverse differential category theory to the higher-order setting; and (3) the establishment of a solid algebraic foundation for higher-order reverse-mode automatic differentiation, thereby enabling rigorous mathematical modeling and verification of complex gradient computation schemes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Visual Reasoning & Symbolic Representations
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Developing a reverse differential analogue of Faa di Bruno's Formula
• Establishing higher-order reverse chain rules for categorical differentiation
• Defining partial and higher-order reverse derivatives in categorical framework

Innovation

• Develops reverse differential analogue of Faa di Bruno's Formula
• Defines partial reverse derivatives in categorical framework
• Establishes higher-order reverse derivatives for composition rules