提出高阶反向链式法则的范畴化公式，统一反向微分形式
Reverse Faà di Bruno's Formula for Cartesian Reverse Differential Categories
- 在范畴论框架下定义高阶反向导数与偏反向导数
- 给出反向法雷·迪布鲁诺公式的完整形式，推广反向链式法则
- 为自动微分提供更普适的数学基础，适合范畴论与自动微分研究者
反向微分是自动微分中的核心操作。笛卡尔反向微分范畴通过范畴语言公理化反向微分，其中关键公理是反向链式法则，即复合函数的反向导数表达式。本文提出法雷·迪布鲁诺公式的反向微分对应版本，给出了笛卡尔反向微分范畴中高阶反向链式法则的完整表达。为此，我们还定义了部分反向导数与高阶反向导数。该公式为反向微分提供了更高阶的代数结构支持。
原文摘要 · Abstract (English)
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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