ad-1.5: src/Numeric/AD/Internal/Dense.hs
{-# LANGUAGE Rank2Types, TypeFamilies, FlexibleContexts, UndecidableInstances, TemplateHaskell, DeriveDataTypeable, BangPatterns #-}
-- {-# OPTIONS_HADDOCK hide, prune #-}
-----------------------------------------------------------------------------
-- |
-- Module : Numeric.AD.Internal.Dense
-- Copyright : (c) Edward Kmett 2010
-- License : BSD3
-- Maintainer : ekmett@gmail.com
-- Stability : experimental
-- Portability : GHC only
--
-- Dense Forward AD. Useful when the result involves the majority of the input
-- elements. Do not use for 'Numeric.AD.Mode.Mixed.hessian' and beyond, since
-- they only contain a small number of unique @n@th derivatives --
-- @(n + k - 1) `choose` k@ for functions of @k@ inputs rather than the
-- @k^n@ that would be generated by using 'Dense', not to mention the redundant
-- intermediate derivatives that would be
-- calculated over and over during that process!
--
-- Assumes all instances of 'f' have the same number of elements.
--
-- NB: We don't need the full power of 'Traversable' here, we could get
-- by with a notion of zippable that can plug in 0's for the missing
-- entries. This might allow for gradients where @f@ has exponentials like @((->) a)@
-----------------------------------------------------------------------------
module Numeric.AD.Internal.Dense
( Dense(..)
, ds
, ds'
, vars
, apply
) where
import Language.Haskell.TH
import Data.Typeable ()
import Data.Traversable (Traversable, mapAccumL)
import Data.Data ()
import Numeric.AD.Internal.Types
import Numeric.AD.Internal.Combinators
import Numeric.AD.Internal.Classes
import Numeric.AD.Internal.Identity
data Dense f a
= Lift !a
| Dense !a (f a)
| Zero
instance Show a => Show (Dense f a) where
showsPrec d (Lift a) = showsPrec d a
showsPrec d (Dense a _) = showsPrec d a
showsPrec _ Zero = showString "0"
ds :: f a -> AD (Dense f) a -> f a
ds _ (AD (Dense _ da)) = da
ds z _ = z
{-# INLINE ds #-}
ds' :: Num a => f a -> AD (Dense f) a -> (a, f a)
ds' _ (AD (Dense a da)) = (a, da)
ds' z (AD (Lift a)) = (a, z)
ds' z (AD Zero) = (0, z)
{-# INLINE ds' #-}
-- Bind variables and count inputs
vars :: (Traversable f, Num a) => f a -> f (AD (Dense f) a)
vars as = snd $ mapAccumL outer (0 :: Int) as
where
outer !i a = (i + 1, AD $ Dense a $ snd $ mapAccumL (inner i) 0 as)
inner !i !j _ = (j + 1, if i == j then 1 else 0)
{-# INLINE vars #-}
apply :: (Traversable f, Num a) => (f (AD (Dense f) a) -> b) -> f a -> b
apply f as = f (vars as)
{-# INLINE apply #-}
instance Primal (Dense f) where
primal Zero = 0
primal (Lift a) = a
primal (Dense a _) = a
instance (Traversable f, Lifted (Dense f)) => Mode (Dense f) where
lift = Lift
zero = Zero
Zero <+> a = a
a <+> Zero = a
Lift a <+> Lift b = Lift (a + b)
Lift a <+> Dense b db = Dense (a + b) db
Dense a da <+> Lift b = Dense (a + b) da
Dense a da <+> Dense b db = Dense (a + b) $ zipWithT (+) da db
Zero <**> y = lift (0 ** primal y)
_ <**> Zero = lift 1
x <**> Lift y = lift1 (**y) (\z -> (y *^ z ** Id (y-1))) x
x <**> y = lift2_ (**) (\z xi yi -> (yi *! z /! xi, z *! log1 xi)) x y
_ *^ Zero = Zero
a *^ Lift b = Lift (a * b)
a *^ Dense b db = Dense (a * b) $ fmap (a*) db
Zero ^* _ = Zero
Lift a ^* b = Lift (a * b)
Dense a da ^* b = Dense (a * b) $ fmap (*b) da
Zero ^/ _ = Zero
Lift a ^/ b = Lift (a / b)
Dense a da ^/ b = Dense (a / b) $ fmap (/b) da
instance (Traversable f, Lifted (Dense f)) => Jacobian (Dense f) where
type D (Dense f) = Id
unary f _ Zero = Lift (f 0)
unary f _ (Lift b) = Lift (f b)
unary f (Id dadb) (Dense b db) = Dense (f b) (fmap (dadb *) db)
lift1 f _ Zero = Lift (f 0)
lift1 f _ (Lift b) = Lift (f b)
lift1 f df (Dense b db) = Dense (f b) (fmap (dadb *) db)
where
Id dadb = df (Id b)
lift1_ f _ Zero = Lift (f 0)
lift1_ f _ (Lift b) = Lift (f b)
lift1_ f df (Dense b db) = Dense a (fmap (dadb *) db)
where
a = f b
Id dadb = df (Id a) (Id b)
binary f _ _ Zero Zero = Lift (f 0 0)
binary f _ _ Zero (Lift c) = Lift (f 0 c)
binary f _ _ (Lift b) Zero = Lift (f b 0)
binary f _ _ (Lift b) (Lift c) = Lift (f b c)
binary f _ (Id dadc) Zero (Dense c dc) = Dense (f 0 c) $ fmap (* dadc) dc
binary f _ (Id dadc) (Lift b) (Dense c dc) = Dense (f b c) $ fmap (* dadc) dc
binary f (Id dadb) _ (Dense b db) Zero = Dense (f b 0) $ fmap (dadb *) db
binary f (Id dadb) _ (Dense b db) (Lift c) = Dense (f b c) $ fmap (dadb *) db
binary f (Id dadb) (Id dadc) (Dense b db) (Dense c dc) = Dense (f b c) $ zipWithT productRule db dc
where productRule dbi dci = dadb * dbi + dci * dadc
lift2 f _ Zero Zero = Lift (f 0 0)
lift2 f _ Zero (Lift c) = Lift (f 0 c)
lift2 f _ (Lift b) Zero = Lift (f b 0)
lift2 f _ (Lift b) (Lift c) = Lift (f b c)
lift2 f df Zero (Dense c dc) = Dense (f 0 c) $ fmap (*dadc) dc where dadc = runId (snd (df (Id 0) (Id c)))
lift2 f df (Lift b) (Dense c dc) = Dense (f b c) $ fmap (*dadc) dc where dadc = runId (snd (df (Id b) (Id c)))
lift2 f df (Dense b db) Zero = Dense (f b 0) $ fmap (dadb*) db where dadb = runId (fst (df (Id b) (Id 0)))
lift2 f df (Dense b db) (Lift c) = Dense (f b c) $ fmap (dadb*) db where dadb = runId (fst (df (Id b) (Id c)))
lift2 f df (Dense b db) (Dense c dc) = Dense (f b c) da
where
(Id dadb, Id dadc) = df (Id b) (Id c)
da = zipWithT productRule db dc
productRule dbi dci = dadb * dbi + dci * dadc
lift2_ f _ Zero Zero = Lift (f 0 0)
lift2_ f _ Zero (Lift c) = Lift (f 0 c)
lift2_ f _ (Lift b) Zero = Lift (f b 0)
lift2_ f _ (Lift b) (Lift c) = Lift (f b c)
lift2_ f df Zero (Dense c dc)
= Dense a $ fmap (*dadc) dc
where
a = f 0 c
(_, Id dadc) = df (Id a) (Id 0) (Id c)
lift2_ f df (Lift b) (Dense c dc)
= Dense a $ fmap (*dadc) dc
where
a = f b c
(_, Id dadc) = df (Id a) (Id b) (Id c)
lift2_ f df (Dense b db) Zero
= Dense a $ fmap (dadb*) db
where
a = f b 0
(Id dadb, _) = df (Id a) (Id b) (Id 0)
lift2_ f df (Dense b db) (Lift c)
= Dense a $ fmap (dadb*) db
where
a = f b c
(Id dadb, _) = df (Id a) (Id b) (Id c)
lift2_ f df (Dense b db) (Dense c dc)
= Dense a $ zipWithT productRule db dc
where
a = f b c
(Id dadb, Id dadc) = df (Id a) (Id b) (Id c)
productRule dbi dci = dadb * dbi + dci * dadc
let f = varT (mkName "f") in
deriveLifted
(classP ''Traversable [f]:)
(conT ''Dense `appT` f)