[Haskell-cafe] Automated Differentiation of Matrices (hmatrix)

[Edward Kmett]

hmatrix and ad don't (currently) mix.

The problem is that hmatrix uses a packed structure that can't hold any of
the AD mode variants we have as an Element. =(

I've been working with Alex Lang to explore in ad 4.0 ways that we can
support monomorphic AD modes and still present largely the same API. We've
seen a number of performance gains off of this refactoring already, but it
doesn't go far enough to address what you need.

A goal a bit farther out is to support AD on vector/matrix operations, but
it is a much bigger refactoring than the one currently in the pipeline. =/

To support automatic differentiation on vector-based operations in a form
that works with standard BLAS-like storage like the packed matrix rep used
in hmatrix we need to convert from a 'matrix of AD variables' to an 'AD
mode over of matrices'. This is similar to the difference between a matrix
of complex numbers and a real matrix plus an imaginary matrix.

This is a long term goal, but not one you're likely to see support for out
of 'ad' in the short term.

I can't build AD on hmatrix itself due in part to licensing restrictions
and differing underlying storage requirements, so there are a lot of little
issues in making that latter vision a reality.

-Edward


On Tue, Apr 9, 2013 at 10:46 AM, Dominic Steinitz <dominic at steinitz.org>wrote:

> Hi Cafe,
>
> Suppose I want to find the grad of a function then it's easy I just
> use http://hackage.haskell.org/package/ad-3.4:
>
> import Numeric.AD
> import Data.Foldable (Foldable)
> import Data.Traversable (Traversable)
>
> data MyMatrix a = MyMatrix (a, a)
>   deriving (Show, Functor, Foldable, Traversable)
>
> f :: Floating a => MyMatrix a -> a
> f (MyMatrix (x, y)) = exp $ negate $ (x^2 + y^2) / 2.0
>
> main :: IO ()
> main = do
>   putStrLn $ show $ f $ MyMatrix (0.0, 0.0)
>   putStrLn $ show $ grad f $ MyMatrix (0.0, 0.0)
>
> But now suppose I am doing some matrix calculations
> http://hackage.haskell.org/package/hmatrix-0.14.1.0 and I want to find
> the grad of a function of a matrix:
>
> import Numeric.AD
> import Numeric.LinearAlgebra
> import Data.Foldable (Foldable)
> import Data.Traversable (Traversable)
>
> g :: (Element a, Floating a) => Matrix a -> a
> g m = exp $ negate $ (x^2 + y^2) / 2.0
>   where r = (toLists m)!!0
>         x = r!!0
>         y = r!!1
>
> main :: IO ()
> main = do
>   putStrLn $ show $ g $ (1 >< 2) ([0.0, 0.0] :: [Double])
>   putStrLn $ show $ grad g $ (1 >< 2) ([0.0, 0.0] :: [Double])
>
> Then I am in trouble:
>
> /Users/dom/Dropbox/Private/Whales/MyAD.hs:24:21:
>     No instance for (Traversable Matrix) arising from a use of `grad'
>     Possible fix: add an instance declaration for (Traversable Matrix)
>     In the expression: grad g
>     In the second argument of `($)', namely
>       `grad g $ (1 >< 2) ([0.0, 0.0] :: [Double])'
>     In the second argument of `($)', namely
>       `show $ grad g $ (1 >< 2) ([0.0, 0.0] :: [Double])'
>
> /Users/dom/Dropbox/Private/Whales/MyAD.hs:24:26:
>     Could not deduce (Element
>                         (ad-3.4:Numeric.AD.Internal.Types.AD s Double))
>       arising from a use of `g'
>     from the context (Numeric.AD.Internal.Classes.Mode s)
>       bound by a type expected by the context:
>                  Numeric.AD.Internal.Classes.Mode s =>
>                  Matrix (ad-3.4:Numeric.AD.Internal.Types.AD s Double)
>                  -> ad-3.4:Numeric.AD.Internal.Types.AD s Double
>       at /Users/dom/Dropbox/Private/Whales/MyAD.hs:24:21-26
>     Possible fix:
>       add an instance declaration for
>       (Element (ad-3.4:Numeric.AD.Internal.Types.AD s Double))
>     In the first argument of `grad', namely `g'
>     In the expression: grad g
>     In the second argument of `($)', namely
>       `grad g $ (1 >< 2) ([0.0, 0.0] :: [Double])'
>
> What are my options here? Clearly I can convert my matrix into a list
> (which is traversable), find the grad and convert it back into a
> matrix but given I am doing numerical calculations and speed is an
> important factor, this seems undesirable.
>
> I think I would have the same problem with:
>
> http://hackage.haskell.org/package/repa
> http://hackage.haskell.org/package/yarr-1.3.1
>
> although I haven'¯t checked.
>
> Thanks, Dominic.
>
>
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