Copyright	(c) Edward Kmett 2010-2015
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	GHC only
Safe Haskell	None
Language	Haskell2010

Numeric.AD.Rank1.Sparse

Contents

Sparse Gradients
Variadic Gradients
Higher-Order Gradients
Variadic Higher-Order Gradients
Sparse Jacobians (synonyms)
Sparse Hessians

Description

Higher order derivatives via a "dual number tower".

Synopsis

data Sparse a
auto :: Mode t => Scalar t -> t
grad :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f a
grad' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f a)
gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> f b
gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> (a, f b)
class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o
vgrad :: Grad i o o' a => i -> o
grads :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> Cofree f a
class Num a => Grads i o a | i -> a o, o -> a i
vgrads :: Grads i o a => i -> o
jacobian :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f a)
jacobian' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f a)
jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (f b)
jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f b)
jacobians :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (Cofree f a)
hessian :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f (f a)
hessian' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f (a, f a))
hessianF :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f (f a))
hessianF' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f (a, f a))

Documentation

data Sparse a Source #

We only store partials in sorted order, so the map contained in a partial will only contain partials with equal or greater keys to that of the map in which it was found. This should be key for efficiently computing sparse hessians. there are only n + k - 1 choose k distinct nth partial derivatives of a function with k inputs.

Instances

(Num a, Bounded a) => Bounded (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods minBound :: Sparse a # maxBound :: Sparse a #
(Num a, Enum a) => Enum (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods succ :: Sparse a -> Sparse a # pred :: Sparse a -> Sparse a # toEnum :: Int -> Sparse a # fromEnum :: Sparse a -> Int # enumFrom :: Sparse a -> [Sparse a] # enumFromThen :: Sparse a -> Sparse a -> [Sparse a] # enumFromTo :: Sparse a -> Sparse a -> [Sparse a] # enumFromThenTo :: Sparse a -> Sparse a -> Sparse a -> [Sparse a] #
(Num a, Eq a) => Eq (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods (==) :: Sparse a -> Sparse a -> Bool # (/=) :: Sparse a -> Sparse a -> Bool #
Floating a => Floating (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pi :: Sparse a # exp :: Sparse a -> Sparse a # log :: Sparse a -> Sparse a # sqrt :: Sparse a -> Sparse a # (**) :: Sparse a -> Sparse a -> Sparse a # logBase :: Sparse a -> Sparse a -> Sparse a # sin :: Sparse a -> Sparse a # cos :: Sparse a -> Sparse a # tan :: Sparse a -> Sparse a # asin :: Sparse a -> Sparse a # acos :: Sparse a -> Sparse a # atan :: Sparse a -> Sparse a # sinh :: Sparse a -> Sparse a # cosh :: Sparse a -> Sparse a # tanh :: Sparse a -> Sparse a # asinh :: Sparse a -> Sparse a # acosh :: Sparse a -> Sparse a # atanh :: Sparse a -> Sparse a # log1p :: Sparse a -> Sparse a # expm1 :: Sparse a -> Sparse a # log1pexp :: Sparse a -> Sparse a # log1mexp :: Sparse a -> Sparse a #
Fractional a => Fractional (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods (/) :: Sparse a -> Sparse a -> Sparse a # recip :: Sparse a -> Sparse a # fromRational :: Rational -> Sparse a #
Data a => Data (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Sparse a -> c (Sparse a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Sparse a) # toConstr :: Sparse a -> Constr # dataTypeOf :: Sparse a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Sparse a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Sparse a)) # gmapT :: (forall b. Data b => b -> b) -> Sparse a -> Sparse a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Sparse a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Sparse a -> r # gmapQ :: (forall d. Data d => d -> u) -> Sparse a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Sparse a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Sparse a -> m (Sparse a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Sparse a -> m (Sparse a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Sparse a -> m (Sparse a) #
Num a => Num (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods (+) :: Sparse a -> Sparse a -> Sparse a # (-) :: Sparse a -> Sparse a -> Sparse a # (*) :: Sparse a -> Sparse a -> Sparse a # negate :: Sparse a -> Sparse a # abs :: Sparse a -> Sparse a # signum :: Sparse a -> Sparse a # fromInteger :: Integer -> Sparse a #
(Num a, Ord a) => Ord (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods compare :: Sparse a -> Sparse a -> Ordering # (<) :: Sparse a -> Sparse a -> Bool # (<=) :: Sparse a -> Sparse a -> Bool # (>) :: Sparse a -> Sparse a -> Bool # (>=) :: Sparse a -> Sparse a -> Bool # max :: Sparse a -> Sparse a -> Sparse a # min :: Sparse a -> Sparse a -> Sparse a #
Real a => Real (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods toRational :: Sparse a -> Rational #
RealFloat a => RealFloat (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods floatRadix :: Sparse a -> Integer # floatDigits :: Sparse a -> Int # floatRange :: Sparse a -> (Int, Int) # decodeFloat :: Sparse a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Sparse a # exponent :: Sparse a -> Int # significand :: Sparse a -> Sparse a # scaleFloat :: Int -> Sparse a -> Sparse a # isNaN :: Sparse a -> Bool # isInfinite :: Sparse a -> Bool # isDenormalized :: Sparse a -> Bool # isNegativeZero :: Sparse a -> Bool # isIEEE :: Sparse a -> Bool # atan2 :: Sparse a -> Sparse a -> Sparse a #
RealFrac a => RealFrac (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods properFraction :: Integral b => Sparse a -> (b, Sparse a) # truncate :: Integral b => Sparse a -> b # round :: Integral b => Sparse a -> b # ceiling :: Integral b => Sparse a -> b # floor :: Integral b => Sparse a -> b #
Show a => Show (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods showsPrec :: Int -> Sparse a -> ShowS # show :: Sparse a -> String # showList :: [Sparse a] -> ShowS #
Erf a => Erf (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods erf :: Sparse a -> Sparse a # erfc :: Sparse a -> Sparse a # erfcx :: Sparse a -> Sparse a # normcdf :: Sparse a -> Sparse a #
InvErf a => InvErf (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods inverf :: Sparse a -> Sparse a # inverfc :: Sparse a -> Sparse a # invnormcdf :: Sparse a -> Sparse a #
Num a => Mode (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Associated Types type Scalar (Sparse a) :: Type Source # Methods isKnownConstant :: Sparse a -> Bool Source # isKnownZero :: Sparse a -> Bool Source # auto :: Scalar (Sparse a) -> Sparse a Source # (^) :: Scalar (Sparse a) -> Sparse a -> Sparse a Source # (^) :: Sparse a -> Scalar (Sparse a) -> Sparse a Source # (^/) :: Sparse a -> Scalar (Sparse a) -> Sparse a Source # zero :: Sparse a Source #
Num a => Jacobian (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse Associated Types type D (Sparse a) :: Type Source # Methods unary :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> D (Sparse a) -> Sparse a -> Sparse a Source # lift1 :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a)) -> Sparse a -> Sparse a Source # lift1_ :: (Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> D (Sparse a)) -> Sparse a -> Sparse a Source # binary :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> D (Sparse a) -> D (Sparse a) -> Sparse a -> Sparse a -> Sparse a Source # lift2 :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> (D (Sparse a), D (Sparse a))) -> Sparse a -> Sparse a -> Sparse a Source # lift2_ :: (Scalar (Sparse a) -> Scalar (Sparse a) -> Scalar (Sparse a)) -> (D (Sparse a) -> D (Sparse a) -> D (Sparse a) -> (D (Sparse a), D (Sparse a))) -> Sparse a -> Sparse a -> Sparse a Source #
Num a => Grad (Sparse a) [a] (a, [a]) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: Sparse a -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> [a] Source # unpack' :: ([a] -> (a, [a])) -> (a, [a]) Source #
Num a => Grads (Sparse a) (Cofree [] a) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: Sparse a -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #
Grads i o a => Grads (Sparse a -> i) (a -> o) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> a -> o Source #
Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> a -> o Source # unpack' :: ([a] -> (a, [a])) -> a -> o' Source #
type Scalar (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse type Scalar (Sparse a) = a
type D (Sparse a) Source #
Instance details Defined in Numeric.AD.Internal.Sparse type D (Sparse a) = Sparse a

auto :: Mode t => Scalar t -> t Source #

Embed a constant

Sparse Gradients

grad :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f a Source #

grad' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f a) Source #

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> f b Source #

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> (a, f b) Source #

Variadic Gradients

Variadic combinators for variadic mixed-mode automatic differentiation.

Unfortunately, variadicity comes at the expense of being able to use quantification to avoid sensitivity confusion, so be careful when counting the number of auto calls you use when taking the gradient of a function that takes gradients!

class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o Source #

Minimal complete definition

pack, unpack, unpack'

Instances

Num a => Grad (Sparse a) [a] (a, [a]) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: Sparse a -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> [a] Source # unpack' :: ([a] -> (a, [a])) -> (a, [a]) Source #
Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods pack :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpack :: ([a] -> [a]) -> a -> o Source # unpack' :: ([a] -> (a, [a])) -> a -> o' Source #

vgrad :: Grad i o o' a => i -> o Source #

Higher-Order Gradients

grads :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> Cofree f a Source #

Variadic Higher-Order Gradients

class Num a => Grads i o a | i -> a o, o -> a i Source #

Minimal complete definition

packs, unpacks

Instances

Num a => Grads (Sparse a) (Cofree [] a) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: Sparse a -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> Cofree [] a Source #
Grads i o a => Grads (Sparse a -> i) (a -> o) a Source #
Instance details Defined in Numeric.AD.Internal.Sparse Methods packs :: (Sparse a -> i) -> [Sparse a] -> Sparse a Source # unpacks :: ([a] -> Cofree [] a) -> a -> o Source #

vgrads :: Grads i o a => i -> o Source #

Sparse Jacobians (synonyms)

jacobian :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f a) Source #

jacobian' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f a) Source #

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (f b) Source #

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f b) Source #

jacobians :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (Cofree f a) Source #

Sparse Hessians

hessian :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f (f a) Source #

hessian' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f (a, f a)) Source #

hessianF :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f (f a)) Source #

hessianF' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f (a, f a)) Source #