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

[Dominic Steinitz]

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.