manifolds-0.1.3.0: 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 RecordWildCards #-}
module Data.Manifold.Types (
-- * Index / ASCII names
Real0, Real1, RealPlus, Real2, Real3
, Sphere0, Sphere1, Sphere2
, Projective1, Projective2
, Disk1, Disk2, Cone, OpenCone
-- * Linear manifolds
, ZeroDim(..)
, ℝ⁰, ℝ, ℝ², ℝ³
-- * 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(..)
-- * Misc
-- * Cut-planes
, Cutplane(..)
, fathomCutDistance, sideOfCut
) where
import Data.VectorSpace
import Data.AffineSpace
import Data.MemoTrie (HasTrie(..))
import Data.Basis
import Data.Fixed
import Data.Void
import Data.Tagged
import Data.Monoid
import Data.Semigroup
import qualified Numeric.LinearAlgebra.HMatrix as HMat
import qualified Data.Vector.Generic as Arr
import qualified Data.Vector
import Data.Manifold.Types.Primitive
import Data.Manifold.PseudoAffine
import Data.LinearMap.HerMetric
import Data.VectorSpace.FiniteDimensional
import qualified Prelude
import Control.Category.Constrained.Prelude hiding ((^))
import Control.Arrow.Constrained
import Control.Monad.Constrained
import Data.Foldable.Constrained
#define deriveAffine(c,t) \
instance (c) => Semimanifold (t) where { \
type Needle (t) = Diff (t); \
(.+~^) = (.+^) }; \
instance (c) => PseudoAffine (t) where { \
a.-~.b = pure (a.-.b); }
-- | The /n/-th Stiefel manifold is the space of all possible configurations of
-- /n/ orthonormal vectors. In the case /n/ = 1, simply the subspace of normalised
-- vectors, i.e. equivalent to the 'UnitSphere'. Even so, it strictly speaking
-- requires the containing space to be at least metric (if not Hilbert); we would
-- however like to be able to use this concept also in spaces with no inner product,
-- therefore we define this space not as normalised vectors, but rather as all
-- vectors modulo scaling by positive factors.
newtype Stiefel1 v = Stiefel1 { getStiefel1N :: DualSpace v }
newtype Stiefel1Needle v = Stiefel1Needle { getStiefel1Tangent :: HMat.Vector (Scalar v) }
newtype Stiefel1Basis v = Stiefel1Basis { getStiefel1Basis :: Int }
s1bTrie :: forall v b. FiniteDimensional v => (Stiefel1Basis v->b) -> Stiefel1Basis v:->:b
s1bTrie = \f -> St1BTrie $ fmap (f . Stiefel1Basis) allIs
where (Tagged d) = dimension :: Tagged v Int
allIs = Arr.fromList [0 .. d-2]
instance FiniteDimensional 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(SmoothScalar(Scalar v),FiniteDimensional v)=>AdditiveGroup(Stiefel1Needle v) where
Stiefel1Needle v ^+^ Stiefel1Needle w = Stiefel1Needle $ v + w
zeroV = s1nZ; negateV (Stiefel1Needle v) = Stiefel1Needle $ negate v
s1nZ :: forall v. FiniteDimensional v => Stiefel1Needle v
s1nZ=Stiefel1Needle .HMat.fromList$replicate(d-1)0 where(Tagged d)=dimension::Tagged v Int
instance (SmoothScalar(Scalar v),FiniteDimensional v)=>VectorSpace(Stiefel1Needle v) where
type Scalar (Stiefel1Needle v) = Scalar v
μ *^ Stiefel1Needle v = Stiefel1Needle $ HMat.scale μ v
instance (SmoothScalar (Scalar v), FiniteDimensional v)=>HasBasis (Stiefel1Needle v) where
type Basis (Stiefel1Needle v) = Stiefel1Basis v
basisValue = s1bV
decompose (Stiefel1Needle v) = zipWith ((,).Stiefel1Basis) [0..] $ HMat.toList v
decompose' (Stiefel1Needle v) (Stiefel1Basis i) = v HMat.! i
s1bV :: forall v b. FiniteDimensional v => Stiefel1Basis v -> Stiefel1Needle v
s1bV = \(Stiefel1Basis i) -> Stiefel1Needle
$ HMat.fromList [ if k==i then 1 else 0 | k<-[0..d-2] ]
where (Tagged d) = dimension :: Tagged v Int
instance (SmoothScalar (Scalar v), FiniteDimensional v)
=> FiniteDimensional (Stiefel1Needle v) where
dimension = s1nD
basisIndex = Tagged $ \(Stiefel1Basis i) -> i
indexBasis = Tagged Stiefel1Basis
fromPackedVector = Stiefel1Needle
asPackedVector = getStiefel1Tangent
s1nD :: forall v. FiniteDimensional v => Tagged (Stiefel1Needle v) Int
s1nD = Tagged (d - 1) where (Tagged d) = dimension :: Tagged v Int
instance (SmoothScalar (Scalar v), FiniteDimensional v)
=> AffineSpace (Stiefel1Needle v) where
type Diff (Stiefel1Needle v) = Stiefel1Needle v
(.+^) = (^+^)
(.-.) = (^-^)
deriveAffine((SmoothScalar (Scalar v), FiniteDimensional v), Stiefel1Needle v)
instance (MetricScalar (Scalar v), FiniteDimensional v)
=> HasMetric' (Stiefel1Needle v) where
type DualSpace (Stiefel1Needle v) = Stiefel1Needle v
Stiefel1Needle v <.>^ Stiefel1Needle w = HMat.dot v w
functional = s1nF
doubleDual = id; doubleDual' = id
s1nF :: forall v. FiniteDimensional v => (Stiefel1Needle v->Scalar v)->Stiefel1Needle v
s1nF = \f -> Stiefel1Needle $ HMat.fromList [f $ basisValue b | b <- cb]
where (Tagged cb) = completeBasis :: Tagged (Stiefel1Needle v) [Stiefel1Basis v]
instance (WithField k LinearManifold v, Real k) => Semimanifold (Stiefel1 v) where
type Needle (Stiefel1 v) = Stiefel1Needle v
Stiefel1 s .+~^ Stiefel1Needle n = Stiefel1 . fromPackedVector . HMat.scale (signum s'i)
$ if| ν==0 -> s' -- ν'≡0 is a special case of this, so we can otherwise assume ν'>0.
-- -- | ν<=1 -> let -- κ = (-1 − 1/(ν−1)) / ν'
-- -- m ∝ spro + κ · n
-- -- ∝ (1−ν) · spro + (1−ν) · κ · n
-- -- = (1−ν) · spro + (-(1−ν) − -1)/ν' · n
-- m = HMat.scale (1-ν) spro + HMat.scale (ν/ν') n
-- in insi (1-ν) m
| ν<=2 -> let -- κ = (1/(ν−1) − 1) / ν'
-- m ∝ - spro + κ · n
-- ∝ (1−ν) · spro + (ν−1) · κ · n
-- = (1−ν) · spro + (1 − (ν−1))/ν' · n
m = HMat.scale ιmν spro + HMat.scale ((1-abs ιmν)/ν') n
ιmν = 1-ν
in insi ιmν m
| otherwise -> let m = HMat.scale ιmν spro + HMat.scale ((abs ιmν-1)/ν') n
ιmν = ν-3
in insi ιmν m
where d = HMat.size s'
s'= asPackedVector s
ν' = l2norm n
quop = signum s'i / ν'
ν = ν' `mod'` 4
im = HMat.maxIndex $ HMat.cmap abs s'
s'i = s' HMat.! im
spro = let v = deli s' in HMat.scale (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 (WithField k LinearManifold v, Real k) => PseudoAffine (Stiefel1 v) where
Stiefel1 s .-~. Stiefel1 t = pure . Stiefel1Needle $ case s' HMat.! im of
0 -> HMat.scale (recip $ l2norm delis) delis
s'i | v <- HMat.scale (recip s'i) delis - tpro
, absv <- l2norm v
, absv > 0
-> let μ -- = (1 − recip (|v| + 1)) / |v| for sgn sᵢ = sgn tᵢ
= (signum (t'i/s'i) - recip(absv + 1)) / absv
in HMat.scale μ v
| t'i/s'i > 0 -> samePoint
| otherwise -> antipode
where d = HMat.size t'
s'= asPackedVector s; t' = asPackedVector t
im = HMat.maxIndex $ HMat.cmap abs t'
t'i = t' HMat.! im
tpro = let v = deli t' in HMat.scale (recip t'i) v
delis = deli s'
deli v = Arr.take im v Arr.++ Arr.drop (im+1) v
samePoint = (d-1) HMat.|> repeat 0
antipode = (d-1) HMat.|> (2 : repeat 0)
l2norm :: MetricScalar s => HMat.Vector s -> s
l2norm = realToFrac . HMat.norm_2
stiefel1Project :: LinearManifold v =>
DualSpace v -- ^ Must be nonzero.
-> Stiefel1 v
stiefel1Project = Stiefel1
stiefel1Embed :: HilbertSpace v => Stiefel1 v -> v
stiefel1Embed (Stiefel1 n) = normalized n
class (PseudoAffine v, InnerSpace v, NaturallyEmbedded (UnitSphere v) (DualSpace v))
=> HasUnitSphere v where
type UnitSphere v :: *
stiefel :: UnitSphere v -> Stiefel1 v
stiefel = Stiefel1 . embed
unstiefel :: Stiefel1 v -> UnitSphere v
unstiefel = coEmbed . getStiefel1N
instance HasUnitSphere ℝ where type UnitSphere ℝ = S⁰
instance HasUnitSphere ℝ² where type UnitSphere ℝ² = S¹
instance HasUnitSphere ℝ³ where type UnitSphere ℝ³ = S²
instance (HasUnitSphere v, v ~ DualSpace v) => NaturallyEmbedded (Stiefel1 v) v where
embed = embed . unstiefel
coEmbed = stiefel . coEmbed
-- | 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 Manifold x => Cutplane x -> x -> Option 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 :: WithField ℝ Manifold x
=> Cutplane x -- ^ Hyperplane to measure the distance from.
-> HerMetric'(Needle 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.
-> Option 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 (Cutplane sh (Stiefel1 cn)) met = \x -> fmap fathom $ x .-~. sh
where fathom v = (cn <.>^ v) / scaleDist
scaleDist = metric' met cn