Downhill.Grad

Documentation

class (AdditiveGroup s, VectorSpace v, VectorSpace dv, Scalar v ~ s, Scalar dv ~ s) => Dual s v dv where

Dual of a vector v is a linear map v -> Scalar v.

Methods

evalGrad :: dv -> v -> s

Instances

Dual Double Double Double

Defined in Downhill.Grad

Methods

evalGrad :: Double -> Double -> Double

Dual Float Float Float

Defined in Downhill.Grad

Methods

evalGrad :: Float -> Float -> Float

Dual Integer Integer Integer

Defined in Downhill.Grad

Methods

evalGrad :: Integer -> Integer -> Integer

(Dual s a da, Dual s b db) => Dual s (a, b) (da, db)

Defined in Downhill.Grad

Methods

evalGrad :: (da, db) -> (a, b) -> s

(Dual s a da, Dual s b db, Dual s c dc) => Dual s (a, b, c) (da, db, dc)

Defined in Downhill.Grad

Methods

evalGrad :: (da, db, dc) -> (a, b, c) -> s

Num a => Dual (AsNum a) (AsNum a) (AsNum a)

Defined in Downhill.BVar.Num

Methods

evalGrad :: AsNum a -> AsNum a -> AsNum a

class (Dual (Scalar g) (MtVector g) (MtCovector g), VectorSpace g) => MetricTensor g where

MetricTensor converts gradients to vectors.

It is really inverse of a metric tensor, because it maps cotangent space into tangent space. Gradient descent doesn't need metric tensor, it needs inverse.

Minimal complete definition

evalMetric

Associated Types

type MtVector g :: Type

type MtCovector g :: Type

Methods

evalMetric :: g -> MtCovector g -> MtVector g

m must be symmetric:

evalGrad x (evalMetric m y) = evalGrad y (evalMetric m x)

innerProduct :: g -> MtCovector g -> MtCovector g -> Scalar g

innerProduct m x y = evalGrad x (evalMetric m y)

sqrNorm :: g -> MtCovector g -> Scalar g

sqrNorm m x = innerProduct m x x

Instances

MetricTensor Double

Defined in Downhill.Grad

Associated Types

type MtVector Double

type MtCovector Double

Methods

evalMetric :: Double -> MtCovector Double -> MtVector Double

innerProduct :: Double -> MtCovector Double -> MtCovector Double -> Scalar Double

sqrNorm :: Double -> MtCovector Double -> Scalar Double

MetricTensor Float

Defined in Downhill.Grad

Associated Types

type MtVector Float

type MtCovector Float

Methods

evalMetric :: Float -> MtCovector Float -> MtVector Float

innerProduct :: Float -> MtCovector Float -> MtCovector Float -> Scalar Float

sqrNorm :: Float -> MtCovector Float -> Scalar Float

MetricTensor Integer

Defined in Downhill.Grad

Associated Types

type MtVector Integer

type MtCovector Integer

Methods

evalMetric :: Integer -> MtCovector Integer -> MtVector Integer

innerProduct :: Integer -> MtCovector Integer -> MtCovector Integer -> Scalar Integer

sqrNorm :: Integer -> MtCovector Integer -> Scalar Integer

Num a => MetricTensor (AsNum a)

Defined in Downhill.BVar.Num

Associated Types

type MtVector (AsNum a)

type MtCovector (AsNum a)

Methods

evalMetric :: AsNum a -> MtCovector (AsNum a) -> MtVector (AsNum a)

innerProduct :: AsNum a -> MtCovector (AsNum a) -> MtCovector (AsNum a) -> Scalar (AsNum a)

sqrNorm :: AsNum a -> MtCovector (AsNum a) -> Scalar (AsNum a)

(MetricTensor ma, MetricTensor mb, Scalar ma ~ Scalar mb) => MetricTensor (ma, mb)

Defined in Downhill.Grad

Associated Types

type MtVector (ma, mb)

type MtCovector (ma, mb)

Methods

evalMetric :: (ma, mb) -> MtCovector (ma, mb) -> MtVector (ma, mb)

innerProduct :: (ma, mb) -> MtCovector (ma, mb) -> MtCovector (ma, mb) -> Scalar (ma, mb)

sqrNorm :: (ma, mb) -> MtCovector (ma, mb) -> Scalar (ma, mb)

(MetricTensor ma, MetricTensor mb, MetricTensor mc, Scalar ma ~ Scalar mb, Scalar ma ~ Scalar mc) => MetricTensor (ma, mb, mc)

Defined in Downhill.Grad

Associated Types

type MtVector (ma, mb, mc)

type MtCovector (ma, mb, mc)

Methods

evalMetric :: (ma, mb, mc) -> MtCovector (ma, mb, mc) -> MtVector (ma, mb, mc)

innerProduct :: (ma, mb, mc) -> MtCovector (ma, mb, mc) -> MtCovector (ma, mb, mc) -> Scalar (ma, mb, mc)

sqrNorm :: (ma, mb, mc) -> MtCovector (ma, mb, mc) -> Scalar (ma, mb, mc)

class (Dual (MScalar p) (Tang p) (Grad p), MetricTensor (Metric p), MtVector (Metric p) ~ Tang p, MtCovector (Metric p) ~ Grad p, BasicVector (Tang p), BasicVector (Grad p)) => HasGrad p

HasGrad is a collection of types and constraints that are useful in many places. It helps to keep type signatures short.

Associated Types

type MScalar p :: Type

Scalar of Tang p and Grad p.

type Tang p :: Type

Tangent vector of manifold p. If p is AffineSpace, Tang p should be Diff p. If p is VectorSpace, Tang p might be the same as p itself.

type Grad p :: Type

Dual of tangent space of p.

type Metric p :: Type

A MetricTensor.

Instances

HasGrad Double

Defined in Downhill.Grad

Associated Types

type MScalar Double

type Tang Double

type Grad Double

type Metric Double

HasGrad Float

Defined in Downhill.Grad

Associated Types

type MScalar Float

type Tang Float

type Grad Float

type Metric Float

HasGrad Integer

Defined in Downhill.Grad

Associated Types

type MScalar Integer

type Tang Integer

type Grad Integer

type Metric Integer

Num a => HasGrad (AsNum a)

Defined in Downhill.BVar.Num

Associated Types

type MScalar (AsNum a)

type Tang (AsNum a)

type Grad (AsNum a)

type Metric (AsNum a)

(HasGrad a, HasGrad b, MScalar b ~ MScalar a) => HasGrad (a, b)

Defined in Downhill.Grad

Associated Types

type MScalar (a, b)

type Tang (a, b)

type Grad (a, b)

type Metric (a, b)

(HasGrad a, HasGrad b, HasGrad c, MScalar b ~ MScalar a, MScalar c ~ MScalar a) => HasGrad (a, b, c)

Defined in Downhill.Grad

Associated Types

type MScalar (a, b, c)

type Tang (a, b, c)

type Grad (a, b, c)

type Metric (a, b, c)

HasGrad a => HasGrad (TraversableVar f a)

Defined in Downhill.BVar.Traversable

Associated Types

type MScalar (TraversableVar f a)

type Tang (TraversableVar f a)

type Grad (TraversableVar f a)

type Metric (TraversableVar f a)

type GradBuilder v = VecBuilder (Grad v)

type HasFullGrad p = (HasGrad p, FullVector (Grad p))

type HasGradAffine p = (AffineSpace p, HasGrad p, HasGrad (Tang p), Tang p ~ Diff p, Tang (Tang p) ~ Tang p, Grad (Tang p) ~ Grad p)