Reverse derivative categories
Reverse derivative categories
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反向导数类别
DOI：
10.4230/lipics.csl.2020.18
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发表时间：
2019
期刊：
影响因子：
20.3
通讯作者：
D. Pronk
中科院分区：
文献类型：
--
作者：
Robin Cockett;G. Cruttwell;J. Gallagher;J. Lemay;Benjamin MacAdam;G. Plotkin;D. Pronk
逆导数是机器学习和自动微分中的一种基本运算。本文给出了一个具有逆导数运算的范畴的直接公理化，其形式类似于笛卡尔微分范畴给出的正向导数的形式。有趣的是，一个有反向导数的范畴也有一个正向导数，但反之亦然。事实上，我们清楚地表明了什么是正向导数的缺失：在它的线性映射的子范畴上，反向导数等价于具有匕首结构的正向导数。此外，我们还证明了这些线性映射构成了一个具有匕首双积的可加丰富范畴。
The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.