We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.
翻译:我们强化了具有等级特征的单亚化分类字符串图的微积分,以捕捉封闭的单亚化(和卡通氏封闭)结构。使用这一新语法,我们为(应用的)简单打字式的羊羔微积分法制定了自动区分算法[Pearlmutter和Siskind,2008年],我们第一次证明了它是否正确。为了高效但有原则地实施AD 算法,我们定义了等级图的正确和完整表达方式,作为我们称之为高网络的等级高调分类。