Goal.Geometry.Manifold

The core mathematical definitions used by the rest of Goal. The central object is a Point on a Manifold. A Manifold is an object with a Dimension, and a Point represents an element of the Manifold in a particular coordinate system, represented by a chart.

Manifolds

class KnownNat (Dimension x) => Manifold x

A geometric object with a certain Dimension.

Associated Types

type Dimension x :: Nat

Instances

Manifold x => Manifold [x]

Defined in Goal.Geometry.Manifold

Associated Types

type Dimension [x] :: Nat

KnownNat k => Manifold (Euclidean k)

Defined in Goal.Geometry.Manifold

Associated Types

type Dimension (Euclidean k) :: Nat

(Manifold x, Manifold y) => Manifold (x, y)

Defined in Goal.Geometry.Manifold

Associated Types

type Dimension (x, y) :: Nat

(KnownNat k, Manifold x) => Manifold (Replicated k x)

Defined in Goal.Geometry.Manifold

Associated Types

type Dimension (Replicated k x) :: Nat

(Manifold x, Manifold y) => Manifold (Tensor y x)

Defined in Goal.Geometry.Map.Linear

Associated Types

type Dimension (Tensor y x) :: Nat

(Manifold z, Manifold (f y x)) => Manifold (Affine f y z x)

Defined in Goal.Geometry.Map.Linear

Associated Types

type Dimension (Affine f y z x) :: Nat

(Manifold (Affine f z z y), Manifold (NeuralNetwork gys g y x)) => Manifold (NeuralNetwork ('(g, y) ': gys) f z x)

Defined in Goal.Geometry.Map.NeuralNetwork

Associated Types

type Dimension (NeuralNetwork ('(g, y) ': gys) f z x) :: Nat

Manifold (Affine f z z x) => Manifold (NeuralNetwork ('[] :: [(Type -> Type -> Type, Type)]) f z x)

Defined in Goal.Geometry.Map.NeuralNetwork

Associated Types

type Dimension (NeuralNetwork '[] f z x) :: Nat

(1 <= (r * c), Manifold x, Manifold y, KnownNat r, KnownNat c, KnownNat rd, KnownNat (Div (Dimension x) (r * c)), KnownNat (Div (Dimension y) (r * c))) => Manifold (Convolutional rd r c y x)

Defined in Goal.Geometry.Map.Linear.Convolutional

Associated Types

type Dimension (Convolutional rd r c y x) :: Nat

dimension :: Manifold x => Proxy x -> Int

The Dimension of the given Manifold.

Combinators

data Replicated (k :: Nat) m

A Sum type for repetitions of the same Manifold.

Instances

(KnownNat k, Manifold x, Transition c d x) => Transition c d (Replicated k x)

Defined in Goal.Geometry.Manifold

Methods

transition :: (c # Replicated k x) -> d # Replicated k x

(KnownNat k, Manifold x) => Manifold (Replicated k x)

Defined in Goal.Geometry.Manifold

Associated Types

type Dimension (Replicated k x) :: Nat

(DuallyFlat x, KnownNat k) => DuallyFlat (Replicated k x)

Defined in Goal.Geometry.Differential

Methods

dualPotential :: (PotentialCoordinates (Replicated k x) #* Replicated k x) -> Double

(Legendre x, KnownNat k) => Legendre (Replicated k x)

Defined in Goal.Geometry.Differential

Methods

potential :: (PotentialCoordinates (Replicated k x) # Replicated k x) -> Double

type Dimension (Replicated k x)

Defined in Goal.Geometry.Manifold

type Dimension (Replicated k x) = k * Dimension x

type PotentialCoordinates (Replicated k x)

Defined in Goal.Geometry.Differential

type PotentialCoordinates (Replicated k x) = PotentialCoordinates x

type R k x = Replicated k x

An abbreviation for Replicated.

Points

newtype Point c x

A Point on a Manifold. The phantom type m represents the Manifold, and the phantom type c represents the coordinate system, or chart, in which the Point is represented.

Constructors

Point

Fields

• coordinates :: Vector (Dimension x) Double

Instances

Eq (Point c x)

Defined in Goal.Geometry.Manifold

Methods

(==) :: Point c x -> Point c x -> Bool

(/=) :: Point c x -> Point c x -> Bool

(Manifold x, KnownNat (Dimension x)) => Floating (Point c x)

Defined in Goal.Geometry.Manifold

Methods

pi :: Point c x

exp :: Point c x -> Point c x

log :: Point c x -> Point c x

sqrt :: Point c x -> Point c x

(**) :: Point c x -> Point c x -> Point c x

logBase :: Point c x -> Point c x -> Point c x

sin :: Point c x -> Point c x

cos :: Point c x -> Point c x

tan :: Point c x -> Point c x

asin :: Point c x -> Point c x

acos :: Point c x -> Point c x

atan :: Point c x -> Point c x

sinh :: Point c x -> Point c x

cosh :: Point c x -> Point c x

tanh :: Point c x -> Point c x

asinh :: Point c x -> Point c x

acosh :: Point c x -> Point c x

atanh :: Point c x -> Point c x

log1p :: Point c x -> Point c x

expm1 :: Point c x -> Point c x

log1pexp :: Point c x -> Point c x

log1mexp :: Point c x -> Point c x

(Manifold x, KnownNat (Dimension x)) => Fractional (Point c x)

Defined in Goal.Geometry.Manifold

Methods

(/) :: Point c x -> Point c x -> Point c x

recip :: Point c x -> Point c x

fromRational :: Rational -> Point c x

(Manifold x, KnownNat (Dimension x)) => Num (c # x)

Defined in Goal.Geometry.Manifold

Methods

(+) :: (c # x) -> (c # x) -> c # x

(-) :: (c # x) -> (c # x) -> c # x

(*) :: (c # x) -> (c # x) -> c # x

negate :: (c # x) -> c # x

abs :: (c # x) -> c # x

signum :: (c # x) -> c # x

fromInteger :: Integer -> c # x

Ord (Point c x)

Defined in Goal.Geometry.Manifold

Methods

compare :: Point c x -> Point c x -> Ordering

(<) :: Point c x -> Point c x -> Bool

(<=) :: Point c x -> Point c x -> Bool

(>) :: Point c x -> Point c x -> Bool

(>=) :: Point c x -> Point c x -> Bool

max :: Point c x -> Point c x -> Point c x

min :: Point c x -> Point c x -> Point c x

Show (Point c x)

Defined in Goal.Geometry.Manifold

Methods

showsPrec :: Int -> Point c x -> ShowS

show :: Point c x -> String

showList :: [Point c x] -> ShowS

KnownNat (Dimension x) => Storable (Point c x)

Defined in Goal.Geometry.Manifold

Methods

sizeOf :: Point c x -> Int

alignment :: Point c x -> Int

peekElemOff :: Ptr (Point c x) -> Int -> IO (Point c x)

pokeElemOff :: Ptr (Point c x) -> Int -> Point c x -> IO ()

peekByteOff :: Ptr b -> Int -> IO (Point c x)

pokeByteOff :: Ptr b -> Int -> Point c x -> IO ()

peek :: Ptr (Point c x) -> IO (Point c x)

poke :: Ptr (Point c x) -> Point c x -> IO ()

NFData (Point c x)

Defined in Goal.Geometry.Manifold

Methods

rnf :: Point c x -> ()

type (#) c x = Point c x infix 3

An infix version of Point, where x is assumed to be of type Double.

breakPoint :: Dimension x ~ Dimension y => (c # x) -> Point d y

Throws away the type-level information about the chart and manifold of the given Point.

listCoordinates :: (c # x) -> [Double]

Returns the coordinates of the point in list form.

boxCoordinates :: (c # x) -> Vector (Dimension x) Double

Returns the coordinates of the point as a boxed vector.

Constructors

singleton :: Dimension x ~ 1 => Double -> c # x

Constructs a Point with Dimension 1.

fromTuple :: (IndexedListLiterals ds (Dimension x) Double, KnownNat (Dimension x)) => ds -> c # x

Constructs a Point from a tuple.

fromBoxed :: Vector (Dimension x) Double -> c # x

Constructs a point with coordinates given by a boxed vector.

class (Manifold (First z), Manifold (Second z), Manifold z, Dimension z ~ (Dimension (First z) + Dimension (Second z))) => Product z where

A Product Manifold is one that is produced out of the sumproductconcatenation of two source Manifolds.

Associated Types

type First z :: Type

The First Manifold.

type Second z :: Type

The 'Second Manifold.

Methods

join :: (c # First z) -> (c # Second z) -> c # z

Combine Points from the First and Second Manifold into a Point on the Product Manifold.

split :: (c # z) -> (c # First z, c # Second z)

Split a Point on the Product Manifold into Points from the First and Second Manifold.

Instances

(Manifold x, Manifold y) => Product (x, y)

Defined in Goal.Geometry.Manifold

Associated Types

type First (x, y)

type Second (x, y)

Methods

join :: (c # First (x, y)) -> (c # Second (x, y)) -> c # (x, y)

split :: (c # (x, y)) -> (c # First (x, y), c # Second (x, y))

(Manifold z, Manifold (f y x)) => Product (Affine f y z x)

Defined in Goal.Geometry.Map.Linear

Associated Types

type First (Affine f y z x)

type Second (Affine f y z x)

Methods

join :: (c # First (Affine f y z x)) -> (c # Second (Affine f y z x)) -> c # Affine f y z x

split :: (c # Affine f y z x) -> (c # First (Affine f y z x), c # Second (Affine f y z x))

(Manifold (Affine f z z y), Manifold (NeuralNetwork gys g y x)) => Product (NeuralNetwork ('(g, y) ': gys) f z x)

Defined in Goal.Geometry.Map.NeuralNetwork

Associated Types

type First (NeuralNetwork ('(g, y) ': gys) f z x)

type Second (NeuralNetwork ('(g, y) ': gys) f z x)

Methods

join :: (c # First (NeuralNetwork ('(g, y) ': gys) f z x)) -> (c # Second (NeuralNetwork ('(g, y) ': gys) f z x)) -> c # NeuralNetwork ('(g, y) ': gys) f z x

split :: (c # NeuralNetwork ('(g, y) ': gys) f z x) -> (c # First (NeuralNetwork ('(g, y) ': gys) f z x), c # Second (NeuralNetwork ('(g, y) ': gys) f z x))

Reshaping Points

splitReplicated :: (KnownNat k, Manifold x) => (c # Replicated k x) -> Vector k (c # x)

Splits a Point on a Replicated Manifold into a Vector of of Points.

joinReplicated :: (KnownNat k, Manifold x) => Vector k (c # x) -> c # Replicated k x

Joins a Vector of of Points into a Point on a Replicated Manifold.

joinBoxedReplicated :: (KnownNat k, Manifold x) => Vector k (c # x) -> c # Replicated k x

Joins a Vector of of Points into a Point on a Replicated Manifold.

mapReplicated :: (Storable a, KnownNat k, Manifold x) => ((c # x) -> a) -> (c # Replicated k x) -> Vector k a

A combination of splitReplicated and fmap.

mapReplicatedPoint :: (KnownNat k, Manifold x, Manifold y) => ((c # x) -> Point d y) -> (c # Replicated k x) -> Point d (Replicated k y)

A combination of splitReplicated and fmap, where the value of the mapped function is also a point.

splitReplicatedProduct :: (KnownNat k, Product x) => (c # Replicated k x) -> (c # Replicated k (First x), c # Replicated k (Second x))

Splits a Replicated Product Manifold into a pair of Replicated Manifolds.

joinReplicatedProduct :: (KnownNat k, Product x) => (c # Replicated k (First x)) -> (c # Replicated k (Second x)) -> c # Replicated k x

joins a Replicated Product Manifold out of a pair of Replicated Manifolds.

Euclidean Manifolds

data Euclidean (n :: Nat)

n-dimensional Euclidean space.

Instances

Transition Polar Cartesian (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Polar # Euclidean 2) -> Cartesian # Euclidean 2

Transition Cartesian Polar (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Cartesian # Euclidean 2) -> Polar # Euclidean 2

KnownNat k => Riemannian Cartesian (Euclidean k)

Defined in Goal.Geometry.Differential

Methods

metric :: (Cartesian # Euclidean k) -> Cartesian #* Tensor (Euclidean k) (Euclidean k)

flat :: (Cartesian # Euclidean k) -> (Cartesian # Euclidean k) -> Cartesian #* Euclidean k

sharp :: (Cartesian # Euclidean k) -> (Cartesian #* Euclidean k) -> Cartesian # Euclidean k

KnownNat k => Manifold (Euclidean k)

Defined in Goal.Geometry.Manifold

Associated Types

type Dimension (Euclidean k) :: Nat

type Dimension (Euclidean k)

Defined in Goal.Geometry.Manifold

type Dimension (Euclidean k) = k

Charts

data Cartesian

Cartesian coordinates on Euclidean space.

Instances

Primal Cartesian

Defined in Goal.Geometry.Vector

Associated Types

type Dual Cartesian

Transition Polar Cartesian (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Polar # Euclidean 2) -> Cartesian # Euclidean 2

Transition Cartesian Polar (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Cartesian # Euclidean 2) -> Polar # Euclidean 2

KnownNat k => Riemannian Cartesian (Euclidean k)

Defined in Goal.Geometry.Differential

Methods

metric :: (Cartesian # Euclidean k) -> Cartesian #* Tensor (Euclidean k) (Euclidean k)

flat :: (Cartesian # Euclidean k) -> (Cartesian # Euclidean k) -> Cartesian #* Euclidean k

sharp :: (Cartesian # Euclidean k) -> (Cartesian #* Euclidean k) -> Cartesian # Euclidean k

type Dual Cartesian

Defined in Goal.Geometry.Vector

type Dual Cartesian = Cartesian

data Polar

Polar coordinates on Euclidean space.

Instances

Transition Polar Cartesian (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Polar # Euclidean 2) -> Cartesian # Euclidean 2

Transition Cartesian Polar (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Cartesian # Euclidean 2) -> Polar # Euclidean 2

Transition

class Transition c d x where

A transition involves taking a point represented by the chart c, and re-representing in terms of the chart d.

Methods

transition :: (c # x) -> d # x

Instances

Transition Polar Cartesian (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Polar # Euclidean 2) -> Cartesian # Euclidean 2

Transition Cartesian Polar (Euclidean 2)

Defined in Goal.Geometry.Manifold

Methods

transition :: (Cartesian # Euclidean 2) -> Polar # Euclidean 2

(KnownNat k, Manifold x, Transition c d x) => Transition c d (Replicated k x)

Defined in Goal.Geometry.Manifold

Methods

transition :: (c # Replicated k x) -> d # Replicated k x

(Manifold x, Manifold y, Transition c d x, Transition c d y) => Transition c d (x, y)

Defined in Goal.Geometry.Manifold

Methods

transition :: (c # (x, y)) -> d # (x, y)

transition2 :: (Transition cx dx x, Transition cy dy y) => ((dx # x) -> (dy # y) -> a) -> (cx # x) -> (cy # y) -> a

Generalizes a function of two points in given coordinate systems to a function on arbitrary coordinate systems.