I have another paper draft for submission to ICFP 2009. This one is called Beautiful differentiation, The paper is a culmination of the several posts I’ve written on derivatives and automatic differentiation (AD). I’m happy with how the derivation keeps getting simpler. Now I’ve boiled extremely general higher-order AD down to a Functor and Applicative morphism.

I’d love to get some readings and feedback. I’m a bit over the page the limit, so I’ll have to do some trimming before submitting.

The abstract:

Automatic differentiation (AD) is a precise, efficient, and convenient method for computing derivatives of functions. Its implementation can be quite simple even when extended to compute all of the higher-order derivatives as well. The higher-dimensional case has also been tackled, though with extra complexity. This paper develops an implementation of higher-dimensional, higher-order differentiation in the extremely general and elegant setting of calculus on manifolds and derives that implementation from a simple and precise specification.

In order to motivate and discover the implementation, the paper poses the question “What does AD mean, independently of implementation?” An answer arises in the form of naturality of sampling a function and its derivative. Automatic differentiation flows out of this naturality condition, together with the chain rule. Graduating from first-order to higher-order AD corresponds to sampling all derivatives instead of just one. Next, the notion of a derivative is generalized via the notions of vector space and linear maps. The specification of AD adapts to this elegant and very general setting, which even simplifies the development.

You can get the paper and see current errata here.

The submission deadline is March 2, so comments before then are most helpful to me.

Enjoy, and thanks!