Functional linear maps
Two earlier posts described a simple and general notion of derivative that unifies the many concrete notions taught in traditional calculus courses. All of those variations turn out to be concrete representations of the single abstract notion of a linear map. Correspondingly, the various forms of mulitplication in chain rules all turn out to be implementations of composition of linear maps. For simplicity, I suggested a direct implementation of linear maps as functions. Unfortunately, that direct representation thwarts efficiency, since functions, unlike data structures, do not cache by default.
This post presents a data representation of linear maps that makes crucial use of (a) linearity and (b) the recently added language feature indexed type families (“associated types”).
For a while now, I’ve wondered if a library for linear maps could replace and generalize matrix libraries. After all, matrices represent of a restricted class of linear maps. Unlike conventional matrix libraries, however, the linear map library described in this post captures matrix/linear-map dimensions via static typing. The composition function defined below statically enforces the conformability property required of matrix multiplication (which implements linear map composition). Likewise, conformance for addition of linear maps is also enforced simply and statically. Moreover, with sufficiently sophisticated coaxing of the Haskell compiler, of the sort Don Stewart does, perhaps a library like this one could also have terrific performance. (It doesn’t yet.)
You can read and try out the code for this post in the module Data.LinearMap in version 0.2.0 or later of the vector-space package.
That module also contains an implementation of linear map composition, as well as Functor-like and Applicative-like operations.
Andy Gill has been helping me get to the bottom of some some severe performance problems, apparently involving huge amounts of redundant dictionary creation.
Edits:
- 2008-06-04: Brief explanation of the associated data type declaration.