manifolds-0.4.0.0: Data/Manifold/Types.hs
-- |
-- Module : Data.Manifold.Types
-- Copyright : (c) Justus Sagemüller 2015
-- License : GPL v3
--
-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
-- Stability : experimental
-- Portability : portable
--
-- Several commonly-used manifolds, represented in some simple way as Haskell
-- data types. All these are in the 'PseudoAffine' class.
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE LiberalTypeSynonyms #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE UnicodeSyntax #-}
module Data.Manifold.Types (
-- * Index / ASCII names
Real0, Real1, RealPlus, Real2, Real3
, Sphere0, Sphere1, Sphere2
, Projective1, Projective2
, Disk1, Disk2, Cone, OpenCone
-- * Linear manifolds
, ZeroDim(..)
, ℝ, ℝ⁰, ℝ¹, ℝ², ℝ³, ℝ⁴
-- * Hyperspheres
-- ** General form: Stiefel manifolds
, Stiefel1(..), stiefel1Project, stiefel1Embed
-- ** Specific examples
, HasUnitSphere(..)
, S⁰(..), S¹(..), S²(..)
-- * Projective spaces
, ℝP¹, ℝP²(..)
-- * Intervals\/disks\/cones
, D¹(..), D²(..)
, ℝay
, CD¹(..), Cℝay(..)
-- * Affine subspaces
-- ** Lines
, Line(..), lineAsPlaneIntersection
-- ** Hyperplanes
, Cutplane(..)
, fathomCutDistance, sideOfCut, cutPosBetween
-- * Linear mappings
, LinearMap, LocalLinear
) where
import Data.VectorSpace
import Data.VectorSpace.Free
import Data.AffineSpace
import Data.MemoTrie (HasTrie(..))
import Data.Basis
import Data.Fixed
import Data.Tagged
import qualified Data.Vector.Generic as Arr
import qualified Data.Vector
import qualified Data.Vector.Unboxed as UArr
import Data.List (maximumBy)
import Data.Ord (comparing)
import Data.Manifold.Types.Primitive
import Data.Manifold.Types.Stiefel
import Data.Manifold.PseudoAffine
import Data.Manifold.Cone
import Math.LinearMap.Category
import qualified Prelude
import Control.Category.Constrained.Prelude hiding ((^))
import Control.Arrow.Constrained
import Control.Monad.Constrained
import Data.Foldable.Constrained
import Data.Type.Coercion
#define deriveAffine(c,t) \
instance (c) => Semimanifold (t) where { \
type Needle (t) = Diff (t); \
fromInterior = id; \
toInterior = pure; \
translateP = Tagged (.+~^); \
(.+~^) = (.+^) }; \
instance (c) => PseudoAffine (t) where { \
a.-~.b = pure (a.-.b); }
newtype Stiefel1Needle v = Stiefel1Needle { getStiefel1Tangent :: UArr.Vector (Scalar v) }
newtype Stiefel1Basis v = Stiefel1Basis { getStiefel1Basis :: Int }
s1bTrie :: ∀ v b. FiniteFreeSpace v => (Stiefel1Basis v->b) -> Stiefel1Basis v:->:b
s1bTrie = \f -> St1BTrie $ fmap (f . Stiefel1Basis) allIs
where d = freeDimension ([]::[v])
allIs = Arr.fromList [0 .. d-2]
instance FiniteFreeSpace v => HasTrie (Stiefel1Basis v) where
data (Stiefel1Basis v :->: a) = St1BTrie ( Array a )
trie = s1bTrie; untrie (St1BTrie a) (Stiefel1Basis i) = a Arr.! i
enumerate (St1BTrie a) = Arr.ifoldr (\i x l -> (Stiefel1Basis i,x):l) [] a
type Array = Data.Vector.Vector
instance (FiniteFreeSpace v, UArr.Unbox (Scalar v))
=> AdditiveGroup(Stiefel1Needle v) where
Stiefel1Needle v ^+^ Stiefel1Needle w = Stiefel1Needle $ uarrAdd v w
Stiefel1Needle v ^-^ Stiefel1Needle w = Stiefel1Needle $ uarrSubtract v w
zeroV = s1nZ; negateV (Stiefel1Needle v) = Stiefel1Needle $ UArr.map negate v
uarrAdd :: (Num n, UArr.Unbox n) => UArr.Vector n -> UArr.Vector n -> UArr.Vector n
uarrAdd = UArr.zipWith (+)
uarrSubtract :: (Num n, UArr.Unbox n) => UArr.Vector n -> UArr.Vector n -> UArr.Vector n
uarrSubtract = UArr.zipWith (-)
s1nZ :: ∀ v. (FiniteFreeSpace v, UArr.Unbox (Scalar v)) => Stiefel1Needle v
s1nZ = Stiefel1Needle . UArr.fromList $ replicate (d-1) 0
where d = freeDimension ([]::[v])
instance (FiniteFreeSpace v, UArr.Unbox (Scalar v)) => VectorSpace (Stiefel1Needle v) where
type Scalar (Stiefel1Needle v) = Scalar v
μ *^ Stiefel1Needle v = Stiefel1Needle $ uarrScale μ v
uarrScale :: (Num n, UArr.Unbox n) => n -> UArr.Vector n -> UArr.Vector n
uarrScale μ = UArr.map (*μ)
instance (FiniteFreeSpace v, UArr.Unbox (Scalar v)) => HasBasis (Stiefel1Needle v) where
type Basis (Stiefel1Needle v) = Stiefel1Basis v
basisValue = s1bV
decompose (Stiefel1Needle v) = zipWith ((,).Stiefel1Basis) [0..] $ UArr.toList v
decompose' (Stiefel1Needle v) (Stiefel1Basis i) = v UArr.! i
s1bV :: ∀ v b. (FiniteFreeSpace v, UArr.Unbox (Scalar v))
=> Stiefel1Basis v -> Stiefel1Needle v
s1bV = \(Stiefel1Basis i) -> Stiefel1Needle
$ UArr.fromList [ if k==i then 1 else 0 | k<-[0..d-2] ]
where d = freeDimension ([]::[v])
instance (FiniteFreeSpace v, UArr.Unbox (Scalar v))
=> FiniteFreeSpace (Stiefel1Needle v) where
freeDimension = s1nD
toFullUnboxVect = getStiefel1Tangent
unsafeFromFullUnboxVect = Stiefel1Needle
s1nD :: ∀ v p . FiniteFreeSpace v => p (Stiefel1Needle v) -> Int
s1nD _ = freeDimension ([]::[v]) - 1
instance (FiniteFreeSpace v, UArr.Unbox (Scalar v)) => AffineSpace (Stiefel1Needle v) where
type Diff (Stiefel1Needle v) = Stiefel1Needle v
(.+^) = (^+^)
(.-.) = (^-^)
deriveAffine((FiniteFreeSpace v, UArr.Unbox (Scalar v)), Stiefel1Needle v)
instance ∀ v . (LSpace v, FiniteFreeSpace v, UArr.Unbox (Scalar v))
=> TensorSpace (Stiefel1Needle v) where
type TensorProduct (Stiefel1Needle v) w = Array w
scalarSpaceWitness = case scalarSpaceWitness :: ScalarSpaceWitness v of
ScalarSpaceWitness -> ScalarSpaceWitness
zeroTensor = Tensor $ Arr.replicate (freeDimension ([]::[v]) - 1) zeroV
toFlatTensor = LinearFunction $ Tensor . Arr.convert . getStiefel1Tangent
fromFlatTensor = LinearFunction $ Stiefel1Needle . Arr.convert . getTensorProduct
addTensors (Tensor a) (Tensor b) = Tensor $ Arr.zipWith (^+^) a b
scaleTensor = bilinearFunction $ \μ (Tensor a) -> Tensor $ Arr.map (μ*^) a
negateTensor = LinearFunction $ \(Tensor a) -> Tensor $ Arr.map negateV a
tensorProduct = bilinearFunction $ \(Stiefel1Needle n) w
-> Tensor $ Arr.map (*^w) $ Arr.convert n
transposeTensor = LinearFunction $ \(Tensor a) -> Arr.foldl' (^+^) zeroV
$ Arr.imap ( \i w -> (getLinearFunction tensorProduct w) $ Stiefel1Needle
$ UArr.generate d (\j -> if i==j then 1 else 0) ) a
where d = freeDimension ([]::[v]) - 1
fmapTensor = bilinearFunction $ \f (Tensor a) -> Tensor $ Arr.map (f$) a
fzipTensorWith = bilinearFunction $ \f (Tensor a, Tensor b)
-> Tensor $ Arr.zipWith (curry $ arr f) a b
coerceFmapTensorProduct _ Coercion = Coercion
asTensor :: Coercion (LinearMap s a b) (Tensor s (DualVector a) b)
asTensor = Coercion
asLinearMap :: Coercion (Tensor s (DualVector a) b) (LinearMap s a b)
asLinearMap = Coercion
infixr 0 +$>
(+$>) :: (LinearSpace a, TensorSpace b, Scalar a ~ s, Scalar b ~ s)
=> LinearMap s a b -> a -> b
(+$>) = getLinearFunction . getLinearFunction applyLinear
instance ∀ v . (LSpace v, FiniteFreeSpace v, UArr.Unbox (Scalar v))
=> LinearSpace (Stiefel1Needle v) where
type DualVector (Stiefel1Needle v) = Stiefel1Needle v
linearId = LinearMap . Arr.generate d $ \i -> Stiefel1Needle . Arr.generate d $
\j -> if i==j then 1 else 0
where d = freeDimension ([]::[v]) - 1
tensorId = ti dualSpaceWitness
where ti :: ∀ w . (LinearSpace w, Scalar w ~ Scalar v)
=> DualSpaceWitness w -> (Stiefel1Needle v ⊗ w) +> (Stiefel1Needle v ⊗ w)
ti DualSpaceWitness = LinearMap . Arr.generate d
$ \i -> fmap (LinearFunction $ \w -> Tensor . Arr.generate d $
\j -> if i==j then w else zeroV) $ asTensor $ id
d = freeDimension ([]::[v]) - 1
dualSpaceWitness = case dualSpaceWitness :: DualSpaceWitness v of
DualSpaceWitness -> DualSpaceWitness
coerceDoubleDual = Coercion
contractTensorMap = LinearFunction $ \(LinearMap m)
-> Arr.ifoldl' (\acc i (Tensor t) -> acc ^+^ t Arr.! i) zeroV m
contractMapTensor = LinearFunction $ \(Tensor m)
-> Arr.ifoldl' (\acc i (LinearMap t) -> acc ^+^ t Arr.! i) zeroV m
contractLinearMapAgainst = bilinearFunction $ \(LinearMap m) f
-> Arr.ifoldl' (\acc i w -> case f $ w of
Stiefel1Needle n -> n UArr.! i ) 0 m
applyDualVector = bilinearFunction $ \(Stiefel1Needle v) (Stiefel1Needle w)
-> UArr.sum $ UArr.zipWith (*) v w
applyLinear = bilinearFunction $ \(LinearMap m) (Stiefel1Needle v)
-> Arr.ifoldl' (\acc i w -> acc ^+^ v UArr.! i *^ w) zeroV m
applyTensorFunctional = bilinearFunction $ \(LinearMap f) (Tensor t)
-> Arr.ifoldl' (\acc i u -> acc + u <.>^ t Arr.! i) 0 f
applyTensorLinMap = bilinearFunction $ \(LinearMap f) (Tensor t)
-> Arr.ifoldl' (\w i u -> w ^+^ ((asLinearMap $ f Arr.! i) +$> u)) zeroV t
composeLinear = bilinearFunction $ \f (LinearMap g)
-> LinearMap $ Arr.map (getLinearFunction applyLinear f$) g
instance ∀ k v .
( WithField k LinearManifold v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v)
, RealFloat k, UArr.Unbox k ) => Semimanifold (Stiefel1 v) where
type Needle (Stiefel1 v) = Stiefel1Needle v
fromInterior = id
toInterior = pure
translateP = Tagged (.+~^)
(.+~^) = tpst dualSpaceWitness
where tpst :: DualSpaceWitness v -> Stiefel1 v -> Stiefel1Needle v -> Stiefel1 v
tpst DualSpaceWitness (Stiefel1 s) (Stiefel1Needle n)
= Stiefel1 . unsafeFromFullUnboxVect . uarrScale (signum s'i)
$ if| ν==0 -> s' -- ν'≡0 is a special case of this, so if not ν=0
-- we can otherwise assume ν'>0.
| ν<=2 -> let m = uarrScale ιmν spro
`uarrAdd` uarrScale ((1-abs ιmν)/ν') n
ιmν = 1-ν
in insi ιmν m
| otherwise -> let m = uarrScale ιmν spro
`uarrAdd` uarrScale ((abs ιmν-1)/ν') n
ιmν = ν-3
in insi ιmν m
where d = UArr.length s'
s'= toFullUnboxVect s
ν' = l2norm n
quop = signum s'i / ν'
ν = ν' `mod'` 4
im = UArr.maxIndex $ UArr.map abs s'
s'i = s' UArr.! im
spro = let v = deli s' in uarrScale (recip s'i) v
deli v = Arr.take im v Arr.++ Arr.drop (im+1) v
insi ti v = Arr.generate d $ \i -> if | i<im -> v Arr.! i
| i>im -> v Arr.! (i-1)
| otherwise -> ti
instance ∀ k v .
( WithField k LinearManifold v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v)
, RealFloat k, UArr.Unbox k ) => PseudoAffine (Stiefel1 v) where
(.-~.) = dpst dualSpaceWitness
where dpst :: DualSpaceWitness v -> Stiefel1 v -> Stiefel1 v -> Maybe (Stiefel1Needle v)
dpst DualSpaceWitness (Stiefel1 s) (Stiefel1 t)
= pure . Stiefel1Needle $ case s' UArr.! im of
0 -> uarrScale (recip $ l2norm delis) delis
s'i | v <- uarrScale (recip s'i) delis `uarrSubtract` tpro
, absv <- l2norm v
, absv > 0
-> let μ = (signum (t'i/s'i) - recip(absv + 1)) / absv
in uarrScale μ v
| t'i/s'i > 0 -> samePoint
| otherwise -> antipode
where d = UArr.length t'
s'= toFullUnboxVect s; t' = toFullUnboxVect t
im = UArr.maxIndex $ UArr.map abs t'
t'i = t' UArr.! im
tpro = let v = deli t' in uarrScale (recip t'i) v
delis = deli s'
deli v = Arr.take im v Arr.++ Arr.drop (im+1) v
samePoint = UArr.replicate (d-1) 0
antipode = (d-1) `UArr.fromListN` (2 : repeat 0)
-- instance ( WithField ℝ HilbertManifold x ) => ConeSemimfd (Stiefel1 x) where
-- type CℝayInterior (Stiefel1 x) = x
l2norm :: (Floating s, UArr.Unbox s) => UArr.Vector s -> s
l2norm = sqrt . UArr.sum . UArr.map (^2)
data Line x = Line { lineHandle :: x
, lineDirection :: Stiefel1 (Needle' x) }
-- | Oriented hyperplanes, naïvely generalised to 'PseudoAffine' manifolds:
-- @'Cutplane' p w@ represents the set of all points 'q' such that
-- @(q.-~.p) ^\<.\> w ≡ 0@.
--
-- In vector spaces this is indeed a hyperplane; for general manifolds it should
-- behave locally as a plane, globally as an (/n/−1)-dimensional submanifold.
data Cutplane x = Cutplane { sawHandle :: x
, cutNormal :: Stiefel1 (Needle x) }
sideOfCut :: (WithField ℝ PseudoAffine x, LinearSpace (Needle x))
=> Cutplane x -> x -> Maybe S⁰
sideOfCut (Cutplane sh (Stiefel1 cn)) p
= decideSide . (cn<.>^) =<< p.-~.sh
where decideSide 0 = mzero
decideSide μ | μ > 0 = pure PositiveHalfSphere
| otherwise = pure NegativeHalfSphere
fathomCutDistance :: ∀ x . (WithField ℝ PseudoAffine x, LinearSpace (Needle x))
=> Cutplane x -- ^ Hyperplane to measure the distance from.
-> Metric' x -- ^ Metric to use for measuring that distance.
-- This can only be accurate if the metric
-- is valid both around the cut-plane's 'sawHandle', and
-- around the points you measure.
-- (Strictly speaking, we would need /parallel transport/
-- to ensure this).
-> x -- ^ Point to measure the distance to.
-> Maybe ℝ -- ^ A signed number, giving the distance from plane
-- to point with indication on which side the point lies.
-- 'Nothing' if the point isn't reachable from the plane.
fathomCutDistance = fcd dualSpaceWitness
where fcd (DualSpaceWitness :: DualSpaceWitness (Needle x))
(Cutplane sh (Stiefel1 cn)) met
= \x -> fmap fathom $ x .-~. sh
where fathom v = (cn <.>^ v) / scaleDist
scaleDist = met|$|cn
cutPosBetween :: WithField ℝ Manifold x => Cutplane x -> (x,x) -> Maybe D¹
cutPosBetween (Cutplane h (Stiefel1 cn)) (x₀,x₁)
| Just [d₀,d₁] <- map (cn<.>^) <$> sequenceA [x₀.-~.h, x₁.-~.h]
, d₀*d₁ < 0 = pure . D¹ $ 2 * d₀ / (d₀ - d₁) - 1
| otherwise = empty
lineAsPlaneIntersection :: ∀ x .
(WithField ℝ Manifold x, FiniteDimensional (Needle' x))
=> Line x -> [Cutplane x]
lineAsPlaneIntersection = lapi dualSpaceWitness
where lapi (DualSpaceWitness :: DualSpaceWitness (Needle x)) (Line h (Stiefel1 dir))
= [ Cutplane h . Stiefel1
$ candidate ^-^ worstCandidate ^* (overlap/worstOvlp)
| (i, (candidate, overlap)) <- zip [0..] $ zip candidates overlaps
, i /= worstId ]
where candidates = enumerateSubBasis entireBasis
overlaps = (<.>^dir) <$> candidates
(worstId, worstOvlp) = maximumBy (comparing $ abs . snd) $ zip [0..] overlaps
worstCandidate = candidates !! worstId