manifolds-0.4.4.0: Data/Function/Differentiable.hs
-- |
-- Module : Data.Function.Differentiable
-- Copyright : (c) Justus Sagemüller 2015
-- License : GPL v3
--
-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
-- Stability : experimental
-- Portability : portable
--
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE LiberalTypeSynonyms #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UnicodeSyntax #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE RecordWildCards #-}
{-# LANGUAGE CPP #-}
module Data.Function.Differentiable (
-- * Everywhere differentiable functions
Differentiable
-- * Region-wise defined diff'able functions
, RWDiffable
-- ** Operators for piecewise definition
-- $definitionRegionOps
, (?->), (?>), (?<), (?|:), backupRegions
-- * Regions within a manifold
, Region
, smoothIndicator
-- * Evaluation of differentiable functions
, discretisePathIn
, discretisePathSegs
, continuityRanges
, regionOfContinuityAround
, analyseLocalBehaviour
, intervalImages
) where
import Data.List
import Data.Maybe
import Data.Semigroup
import Data.Embedding
import Data.MemoTrie (HasTrie)
import Data.VectorSpace
import Math.LinearMap.Category
import Data.AffineSpace
import Data.Function.Differentiable.Data
import Data.Function.Affine
import Data.Basis
import Data.Tagged
import Data.Manifold.Types.Primitive
import Data.Manifold.PseudoAffine
import Data.Manifold.Atlas
import qualified Prelude
import qualified Control.Applicative as Hask
import Control.Category.Constrained.Prelude hiding ((^))
import Control.Arrow.Constrained
import Control.Monad.Constrained
import Data.Foldable.Constrained
type RealDimension s
= ( RealFloat' s, SimpleSpace s, Show s, Atlas s, HasTrie (ChartIndex s)
, s ~ Needle s, s ~ Interior s, s ~ Scalar s, s ~ DualVector s )
discretisePathIn :: (WithField ℝ Manifold y, SimpleSpace (Needle y))
=> Int -- ^ Limit the number of steps taken in either direction. Note this will not cap the resolution but /length/ of the discretised path.
-> ℝInterval -- ^ Parameter interval of interest.
-> (RieMetric ℝ, RieMetric y) -- ^ Inaccuracy allowance /ε/.
-> (Differentiable ℝ ℝ y) -- ^ Path specification.
-> [(ℝ,y)] -- ^ Trail of points along the path, such that a linear interpolation deviates nowhere by more as /ε/.
discretisePathIn nLim (xl, xr) (mx,my) (Differentiable f)
= reverse (tail . take nLim $ traceFwd xl xm (-1))
++ take nLim (traceFwd xr xm 1)
where traceFwd xlim x₀ dir
| signum (x₀-xlim) == signum dir = [(xlim, fxlim)]
| otherwise = (x₀, fx₀) : traceFwd xlim (x₀+xstep) dir
where (fx₀, jf, δx²) = f x₀
εx = my fx₀ `relaxNorm` [jf $ normalLength $ mx x₀]
χ = δx² εx |$| 1
xstep = dir * min (abs x₀+1) (recip χ)
(fxlim, _, _) = f xlim
xm = (xr + xl) / 2
type ℝInterval = (ℝ,ℝ)
continuityRanges :: WithField ℝ Manifold y
=> Int -- ^ Max number of exploration steps per region
-> RieMetric ℝ -- ^ Needed resolution of boundaries
-> RWDiffable ℝ ℝ y -- ^ Function to investigate
-> ([ℝInterval], [ℝInterval]) -- ^ Subintervals on which the function is guaranteed continuous.
continuityRanges nLim δbf (RWDiffable f)
| (GlobalRegion, _) <- f xc
= ([], [(-huge,huge)])
| otherwise = glueMid (go xc (-1)) (go xc 1)
where go x₀ dir
| yq₀ <= abs ((jq₀$1) * step₀)
= go (x₀ + step₀/2) dir
| RealSubray PositiveHalfSphere xl' <- rangeHere
= let stepl' = dir/(δbf xl'|$| 2)
in if dir>0
then if definedHere then [(max (xl'+stepl') x₀, huge)]
else []
else if definedHere && x₀ > xl'+stepl'
then (xl'+stepl',x₀) : go (xl'-stepl') dir
else go (xl'-stepl') dir
| RealSubray NegativeHalfSphere xr' <- rangeHere
= let stepr' = dir/(δbf xr'|$| 2)
in if dir<0
then if definedHere then [(-huge, min (xr'-stepr') x₀)]
else []
else if definedHere && x₀ < xr'-stepr'
then (x₀,xr'-stepr') : go (xr'+stepr') dir
else go (xr'+stepr') dir
| otherwise = exit nLim dir x₀
where (rangeHere, fq₀) = f x₀
(PreRegion (Differentiable r₀)) = genericisePreRegion rangeHere
(yq₀, jq₀, δyq₀) = r₀ x₀
step₀ = dir/(δbf x₀|$| 1)
exit 0 _ xq
| not definedHere = []
| xq < xc = [(xq,x₀)]
| otherwise = [(x₀,xq)]
exit nLim' dir' xq
| yq₁<0 || as_devεδ δyq yq₁<abs stepp
= exit (nLim'-1) (dir'/2) xq
| yq₂<0
, as_devεδ δyq (-yq₂)>=abs stepp
, resoHere stepp<1 = (if definedHere
then ((min x₀ xq₁, max x₀ xq₁):)
else id) $ go xq₂ dir
| otherwise = exit (nLim'-1) dir xq₁
where (yq, jq, δyq) = r₀ xq
xq₁ = xq + stepp
xq₂ = xq₁ + stepp
yq₁ = yq + f'x*stepp
yq₂ = yq₁ + f'x*stepp
f'x = jq $ 1
stepp | f'x*dir < 0 = -0.9 * abs dir' * yq/f'x
| otherwise = dir' * as_devεδ δyq yq -- TODO: memoise in `exit` recursion
resoHere = normSq $ δbf xq
resoStep = dir/sqrt(resoHere 1)
definedHere = case fq₀ of
Just _ -> True
Nothing -> False
glueMid ((l,le):ls) ((re,r):rs) | le==re = (ls, (l,r):rs)
glueMid l r = (l,r)
huge = exp $ fromIntegral nLim
xc = 0
discretisePathSegs :: (WithField ℝ Manifold y, SimpleSpace (Needle y))
=> Int -- ^ Maximum number of path segments and/or points per segment.
-> ( RieMetric ℝ
, RieMetric y ) -- ^ Inaccuracy allowance /δ/ for arguments
-- (mostly relevant for resolution of discontinuity boundaries –
-- consider it a “safety margin from singularities”),
-- and /ε/ for results in the target space.
-> RWDiffable ℝ ℝ y -- ^ Path specification. It is recommended that this
-- function be limited to a compact interval (e.g. with
-- '?>', '?<' and '?->'). For many functions the discretisation
-- will even work on an infinite interval: the point density
-- is exponentially decreased towards the infinities. But
-- this is still pretty bad for performance.
-> ([[(ℝ,y)]], [[(ℝ,y)]]) -- ^ Discretised paths: continuous segments in either direction
discretisePathSegs nLim (mx,my) f@(RWDiffable ff)
= ( map discretise ivsL, map discretise ivsR )
where (ivsL, ivsR) = continuityRanges nLim mx f
discretise rng@(l,r) = discretisePathIn nLim rng (mx,my) fr
where (_, Just fr) = ff $ (l+r)/2
analyseLocalBehaviour ::
RWDiffable ℝ ℝ ℝ
-> ℝ -- ^ /x/₀ value.
-> Maybe ( (ℝ,ℝ)
, ℝ->Maybe ℝ ) -- ^ /f/ /x/₀, derivative (i.e. Taylor-1-coefficient),
-- and reverse propagation of /O/ (/δ/²) bound.
analyseLocalBehaviour (RWDiffable f) x₀ = case f x₀ of
(r, Just (Differentiable fd))
| inRegion r x₀ -> return $
let (fx, j, δf) = fd x₀
epsprop ε
| ε>0 = case (δf $ spanNorm [recip ε])|$| 1 of
0 -> empty
δ' -> return $ recip δ'
| otherwise = pure 0
in ((fx, j $ 1), epsprop)
_ -> empty
where -- This check shouldn't really be necessary,
-- because the initial value lies by definition
inRegion GlobalRegion _ = True -- in its domain.
inRegion (PreRegion (Differentiable rf)) x
| (yr,_,_) <- rf x = yr>0
inRegion (RealSubray PositiveHalfSphere xl) x = x>xl
inRegion (RealSubray NegativeHalfSphere xr) x = x<xr
-- | Represent a 'Region' by a smooth function which is positive within the region,
-- and crosses zero at the boundary.
smoothIndicator :: LocallyScalable ℝ q => Region ℝ q -> Differentiable ℝ q ℝ
smoothIndicator (Region _ r₀) = let (PreRegion r) = genericisePreRegion r₀
in r
regionOfContinuityAround :: RWDiffable ℝ q x -> q -> Region ℝ q
regionOfContinuityAround (RWDiffable f) q = Region q . fst . f $ q
intervalImages ::
Int -- ^ Max number of exploration steps per region
-> (RieMetric ℝ, RieMetric ℝ) -- ^ Needed resolution in (x,y) direction
-> RWDiffable ℝ ℝ ℝ -- ^ Function to investigate
-> ( [(ℝInterval,ℝInterval)]
, [(ℝInterval,ℝInterval)] ) -- ^ (XInterval, YInterval) rectangles in which
-- the function graph lies.
intervalImages nLim (mx,my) f@(RWDiffable fd)
= (map (id&&&ivimg) domsL, map (id&&&ivimg) domsR)
where (domsL, domsR) = continuityRanges nLim mx f
ivimg (xl,xr) = go xl 1 i₀ ∪ go xr (-1) i₀
where (_, Just fdd@(Differentiable fddd))
= second (fmap genericiseDifferentiable) $ fd xc
xc = (xl+xr)/2
i₀ = minimum&&&maximum $ [fdd$xl, fdd$xc, fdd$xr]
go x dir (a,b)
| dir>0 && x>xc = (a,b)
| dir<0 && x<xc = (a,b)
| χ == 0 = (y + (x-xl)*y', y + (x-xr)*y')
| y < a+resoHere = go (x + dir/χ) dir (y,b)
| y > b-resoHere = go (x + dir/χ) dir (a,y)
| otherwise = go (x + safeStep stepOut₀) dir (a,b)
where (y, j, δε) = fddd x
y' = j $ 1
εx = my y
resoHere = normalLength εx
χ = δε εx|$| 1
safeStep s₀
| as_devεδ δε (safetyMarg s₀) > abs s₀ = s₀
| otherwise = safeStep (s₀*0.5)
stepOut₀ | y'*dir>0 = 0.5 * (b-y)/y'
| otherwise = -0.5 * (y-a)/y'
safetyMarg stp = minimum [y-a, y+stp*y'-a, b-y, b-y-stp*y']
infixl 3 ∪
(a,b) ∪ (c,d) = (min a c, max b d)
hugeℝVal :: ℝ
hugeℝVal = 1e+100
unsafe_dev_ε_δ :: ∀ a . RealDimension a
=> String -> (a -> a) -> LinDevPropag a a
unsafe_dev_ε_δ = case ( linearManifoldWitness :: LinearManifoldWitness a
, closedScalarWitness :: ClosedScalarWitness a ) of
(LinearManifoldWitness _, ClosedScalarWitness) -> \errHint f d
-> let ε'² = normSq d 1
in if ε'²>0
then let δ = f . sqrt $ recip ε'²
in if δ > 0
then spanNorm [recip δ]
else error $ "ε-δ propagator function for "
++errHint++", with ε="
++show(sqrt $ recip ε'²)
++ " gives non-positive δ="++show δ++"."
else mempty
dev_ε_δ :: ∀ a . RealDimension a
=> (a -> a) -> Metric a -> Maybe (Metric a)
dev_ε_δ = case ( linearManifoldWitness :: LinearManifoldWitness a
, closedScalarWitness :: ClosedScalarWitness a ) of
(LinearManifoldWitness _, ClosedScalarWitness) -> \f d
-> let ε'² = normSq d 1
in if ε'²>0
then let δ = f . sqrt $ recip ε'²
in if δ > 0
then pure (spanNorm [recip δ])
else empty
else pure mempty
as_devεδ :: ∀ a . RealDimension a => LinDevPropag a a -> a -> a
as_devεδ = asdevεδ linearManifoldWitness closedScalarWitness where
asdevεδ :: LinearManifoldWitness a -> ClosedScalarWitness a -> LinDevPropag a a -> a -> a
asdevεδ (LinearManifoldWitness _) ClosedScalarWitness
ldp ε | ε>0
, δ'² <- normSq (ldp $ spanNorm [recip ε]) 1
, δ'² > 0
= sqrt $ recip δ'²
| otherwise = 0
genericiseDifferentiable :: (LocallyScalable s d, LocallyScalable s c)
=> Differentiable s d c -> Differentiable s d c
genericiseDifferentiable (AffinDiffable _ af)
= Differentiable $ \x -> let (y₀, ϕ) = evalAffine af x
in (y₀, ϕ, const mempty)
genericiseDifferentiable f = f
instance RealFrac' s => Category (Differentiable s) where
type Object (Differentiable s) o = LocallyScalable s o
id = Differentiable $ \x -> (x, id, const mempty)
Differentiable f . Differentiable g = Differentiable $
\x -> let (y, g', devg) = g x
(z, f', devf) = f y
devfg δz = let δy = transformNorm f' δz
εy = devf δz
in transformNorm g' εy <> devg δy <> devg εy
in (z, f' . g', devfg)
AffinDiffable ef f . AffinDiffable eg g = AffinDiffable (ef . eg) (f . g)
f . g = genericiseDifferentiable f . genericiseDifferentiable g
-- instance (RealDimension s) => EnhancedCat (Differentiable s) (Affine s) where
-- arr (Affine co ao sl) = actuallyAffineEndo (ao .-^ lapply sl co) sl
instance (RealDimension s) => EnhancedCat (->) (Differentiable s) where
arr (Differentiable f) x = let (y,_,_) = f x in y
arr (AffinDiffable _ f) x = f $ x
instance (RealFrac' s) => Cartesian (Differentiable s) where
type UnitObject (Differentiable s) = ZeroDim s
swap = Differentiable $ \(x,y) -> ((y,x), swap, const mempty)
attachUnit = Differentiable $ \x -> ((x, Origin), attachUnit, const mempty)
detachUnit = Differentiable $ \(x, Origin) -> (x, detachUnit, const mempty)
regroup = Differentiable $ \(x,(y,z)) -> (((x,y),z), regroup, const mempty)
regroup' = Differentiable $ \((x,y),z) -> ((x,(y,z)), regroup', const mempty)
instance (RealFrac' s) => Morphism (Differentiable s) where
Differentiable f *** Differentiable g = Differentiable h
where h (x,y) = ((fx, gy), f'***g', devfg)
where (fx, f', devf) = f x
(gy, g', devg) = g y
devfg δs = transformNorm fst δx
<> transformNorm snd δy
where δx = devf $ transformNorm (id&&&zeroV) δs
δy = devg $ transformNorm (zeroV&&&id) δs
AffinDiffable IsDiffableEndo f *** AffinDiffable IsDiffableEndo g
= AffinDiffable IsDiffableEndo $ f *** g
AffinDiffable _ f *** AffinDiffable _ g = AffinDiffable NotDiffableEndo $ f *** g
f *** g = genericiseDifferentiable f *** genericiseDifferentiable g
instance (RealFrac' s) => PreArrow (Differentiable s) where
terminal = Differentiable $ \_ -> (Origin, zeroV, const mempty)
fst = Differentiable $ \(x,_) -> (x, fst, const mempty)
snd = Differentiable $ \(_,y) -> (y, snd, const mempty)
Differentiable f &&& Differentiable g = Differentiable h
where h x = ((fx, gx), f'&&&g', devfg)
where (fx, f', devf) = f x
(gx, g', devg) = g x
devfg δs = (devf $ transformNorm (id&&&zeroV) δs)
<> (devg $ transformNorm (zeroV&&&id) δs)
f &&& g = genericiseDifferentiable f &&& genericiseDifferentiable g
instance (RealFrac' s) => WellPointed (Differentiable s) where
unit = Tagged Origin
globalElement x = Differentiable $ \Origin -> (x, zeroV, const mempty)
const x = Differentiable $ \_ -> (x, zeroV, const mempty)
type DfblFuncValue s = GenericAgent (Differentiable s)
instance (RealFrac' s) => HasAgent (Differentiable s) where
alg = genericAlg
($~) = genericAgentMap
instance ∀ s . (RealFrac' s) => CartesianAgent (Differentiable s) where
alg1to2 = genericAlg1to2
alg2to1 = a2t1
where a2t1 :: ∀ α β γ . (LocallyScalable s α, LocallyScalable s β)
=> (∀ q . LocallyScalable s q
=> DfblFuncValue s q α -> DfblFuncValue s q β -> DfblFuncValue s q γ )
-> Differentiable s (α,β) γ
a2t1 = case ( dualSpaceWitness :: DualSpaceWitness (Needle α)
, dualSpaceWitness :: DualSpaceWitness (Needle β) ) of
(DualSpaceWitness, DualSpaceWitness) -> genericAlg2to1
alg2to2 = a2t1
where a2t1 :: ∀ α β γ δ . ( LocallyScalable s α, LocallyScalable s β
, LocallyScalable s γ, LocallyScalable s δ )
=> (∀ q . LocallyScalable s q
=> DfblFuncValue s q α -> DfblFuncValue s q β
-> (DfblFuncValue s q γ, DfblFuncValue s q δ) )
-> Differentiable s (α,β) (γ,δ)
a2t1 = case ( dualSpaceWitness :: DualSpaceWitness (Needle α)
, dualSpaceWitness :: DualSpaceWitness (Needle β)
, dualSpaceWitness :: DualSpaceWitness (Needle γ)
, dualSpaceWitness :: DualSpaceWitness (Needle δ) ) of
(DualSpaceWitness, DualSpaceWitness, DualSpaceWitness, DualSpaceWitness)
-> genericAlg2to2
instance (RealFrac' s)
=> PointAgent (DfblFuncValue s) (Differentiable s) a x where
point = genericPoint
actuallyLinearEndo :: (Object (Affine s) x, Object (LinearMap s) x)
=> (x+>x) -> Differentiable s x x
actuallyLinearEndo = AffinDiffable IsDiffableEndo . arr
actuallyAffineEndo :: (Object (Affine s) x, Object (LinearMap s) x)
=> x -> (x+>Needle x) -> Differentiable s x x
actuallyAffineEndo y₀ f = AffinDiffable IsDiffableEndo $ fromOffsetSlope y₀ f
actuallyLinear :: ( Object (Affine s) x, Object (Affine s) y
, Object (LinearMap s) x, Object (LinearMap s) y )
=> (x+>y) -> Differentiable s x y
actuallyLinear = AffinDiffable NotDiffableEndo . arr
actuallyAffine :: ( Object (Affine s) x, Object (Affine s) y
, Object (LinearMap s) x, Object (LinearMap s) (Needle y) )
=> y -> (x+>Needle y) -> Differentiable s x y
actuallyAffine y₀ f = AffinDiffable NotDiffableEndo $ fromOffsetSlope y₀ f
-- affinPoint :: (WithField s LinearManifold c, WithField s LinearManifold d)
-- => c -> DfblFuncValue s d c
-- affinPoint p = GenericAgent (AffinDiffable (const p))
dfblFnValsFunc :: ( LocallyScalable s c, LocallyScalable s c', LocallyScalable s d
, v ~ Needle c, v' ~ Needle c'
, ε ~ Norm v, ε ~ Norm v'
, RealFrac' s )
=> (c' -> (c, v'+>v, ε->ε)) -> DfblFuncValue s d c' -> DfblFuncValue s d c
dfblFnValsFunc f = (Differentiable f $~)
dfblFnValsCombine :: forall d c c' c'' v v' v'' ε ε' ε'' s.
( LocallyScalable s c, LocallyScalable s c', LocallyScalable s c''
, LocallyScalable s d
, v ~ Needle c, v' ~ Needle c', v'' ~ Needle c''
, ε ~ Norm v , ε' ~ Norm v' , ε'' ~ Norm v'', ε~ε', ε~ε''
, RealFrac' s )
=> ( c' -> c'' -> (c, (v',v'')+>v, ε -> (ε',ε'')) )
-> DfblFuncValue s d c' -> DfblFuncValue s d c'' -> DfblFuncValue s d c
dfblFnValsCombine cmb (GenericAgent (Differentiable f))
(GenericAgent (Differentiable g))
= GenericAgent . Differentiable $
\d -> let (c', jf, devf) = f d
(c'', jg, devg) = g d
(c, jh, devh) = cmb c' c''
jhl = jh . (id&&&zeroV); jhr = jh . (zeroV&&&id)
in ( c
, jh <<< jf&&&jg
, \εc -> let εc' = transformNorm jhl εc
εc'' = transformNorm jhr εc
(δc',δc'') = devh εc
in devf εc' <> devg εc''
<> transformNorm jf δc'
<> transformNorm jg δc''
)
dfblFnValsCombine cmb (GenericAgent fa) (GenericAgent ga)
= dfblFnValsCombine cmb (GenericAgent $ genericiseDifferentiable fa)
(GenericAgent $ genericiseDifferentiable ga)
instance ∀ v s a . (LinearSpace v, Scalar v ~ s, LocallyScalable s a, RealFloat' s)
=> AdditiveGroup (DfblFuncValue s a v) where
zeroV = case ( linearManifoldWitness :: LinearManifoldWitness v
, dualSpaceWitness :: DualSpaceWitness v ) of
(LinearManifoldWitness _, DualSpaceWitness) -> point zeroV
(^+^) = case ( linearManifoldWitness :: LinearManifoldWitness v
, dualSpaceWitness :: DualSpaceWitness v ) of
(LinearManifoldWitness _, DualSpaceWitness)
-> curry $ \case
(GenericAgent (AffinDiffable ef f), GenericAgent (AffinDiffable eg g))
-> GenericAgent $ AffinDiffable (ef<>eg) (f^+^g)
(α,β) -> dfblFnValsCombine (\a b -> (a^+^b, arr addV, const mempty)) α β
negateV = case ( linearManifoldWitness :: LinearManifoldWitness v
, dualSpaceWitness :: DualSpaceWitness v ) of
(LinearManifoldWitness _, DualSpaceWitness) -> \case
(GenericAgent (AffinDiffable ef f))
-> GenericAgent $ AffinDiffable ef (negateV f)
α -> dfblFnValsFunc (\a -> (negateV a, negateV id, const mempty)) α
instance ∀ n a . (RealDimension n, LocallyScalable n a)
=> Num (DfblFuncValue n a n) where
fromInteger = case ( linearManifoldWitness :: LinearManifoldWitness n
, closedScalarWitness :: ClosedScalarWitness n ) of
(LinearManifoldWitness _, ClosedScalarWitness) -> point . fromInteger
(+) = case closedScalarWitness :: ClosedScalarWitness n of
ClosedScalarWitness -> (^+^)
(*) = case ( linearManifoldWitness :: LinearManifoldWitness n
, closedScalarWitness :: ClosedScalarWitness n ) of
(LinearManifoldWitness _, ClosedScalarWitness) -> dfblFnValsCombine $
\a b -> ( a*b
, arr $ addV <<< (scale $ a)***(scale $ b)
, unsafe_dev_ε_δ(show a++"*"++show b) (sqrt :: n->n)
>>> \d¹₂ -> (d¹₂,d¹₂)
-- ε δa δb = (a+δa)·(b+δb) - (a·b + (a·δa + b·δb))
-- = δa·δb
-- so choose δa = δb = √ε
)
negate = case closedScalarWitness :: ClosedScalarWitness n of
ClosedScalarWitness -> negateV
abs = mkabs linearManifoldWitness closedScalarWitness
where mkabs :: LinearManifoldWitness n -> ClosedScalarWitness n
-> DfblFuncValue n a n -> DfblFuncValue n a n
mkabs (LinearManifoldWitness _) ClosedScalarWitness = dfblFnValsFunc dfblAbs
where dfblAbs a
| a>0 = (a, id, unsafe_dev_ε_δ("abs "++show a) $ \ε -> a + ε/2)
| a<0 = (-a, negateV id, unsafe_dev_ε_δ("abs "++show a) $ \ε -> ε/2 - a)
| otherwise = (0, zeroV, scaleNorm (sqrt 0.5))
signum = mksgn linearManifoldWitness closedScalarWitness
where mksgn :: LinearManifoldWitness n -> ClosedScalarWitness n
-> DfblFuncValue n a n -> DfblFuncValue n a n
mksgn (LinearManifoldWitness _) ClosedScalarWitness = dfblFnValsFunc dfblSgn
where dfblSgn a
| a>0 = (1, zeroV, unsafe_dev_ε_δ("signum "++show a) $ const a)
| a<0 = (-1, zeroV, unsafe_dev_ε_δ("signum "++show a) $ \_ -> -a)
| otherwise = (0, zeroV, const $ spanNorm [1])
-- VectorSpace instance is more problematic than you'd think: multiplication
-- requires the allowed-deviation backpropagators to be split as square
-- roots, but the square root of a nontrivial-vector-space metric requires
-- an eigenbasis transform, which we have not implemented yet.
--
-- instance (WithField s LinearManifold v, LocallyScalable s a, Floating s)
-- => VectorSpace (DfblFuncValue s a v) where
-- type Scalar (DfblFuncValue s a v) = DfblFuncValue s a (Scalar v)
-- (*^) = dfblFnValsCombine $ \μ v -> (μ*^v, lScl, \ε -> (ε ^* sqrt 2, ε ^* sqrt 2))
-- where lScl = linear $ uncurry (*^)
-- | Important special operator needed to compute intersection of 'Region's.
minDblfuncs :: ∀ s m . (LocallyScalable s m, RealDimension s)
=> Differentiable s m s -> Differentiable s m s -> Differentiable s m s
minDblfuncs (Differentiable f) (Differentiable g)
= Differentiable $ h linearManifoldWitness closedScalarWitness
where h :: LinearManifoldWitness s -> ClosedScalarWitness s
-> m -> (s, Needle m+>Needle s, LinDevPropag m s)
h (LinearManifoldWitness _) ClosedScalarWitness x
| fx < gx = ( fx, jf
, \d -> devf d <> devg d
<> transformNorm δj
(spanNorm [recip $ recip(d|$|1) + gx - fx]) )
| fx > gx = ( gx, jg
, \d -> devf d <> devg d
<> transformNorm δj
(spanNorm [recip $ recip(d|$|1) + fx - gx]) )
| otherwise = ( fx, (jf^+^jg)^/2
, \d -> devf d <> devg d
<> transformNorm δj d )
where (fx, jf, devf) = f x
(gx, jg, devg) = g x
δj = jf ^-^ jg
postEndo :: ∀ c a b . (HasAgent c, Object c a, Object c b)
=> c a a -> GenericAgent c b a -> GenericAgent c b a
postEndo = genericAgentMap
genericisePreRegion :: ∀ s m . (RealDimension s, LocallyScalable s m)
=> PreRegion s m -> PreRegion s m
genericisePreRegion GlobalRegion = case ( linearManifoldWitness :: LinearManifoldWitness s
, closedScalarWitness :: ClosedScalarWitness s ) of
(LinearManifoldWitness _, ClosedScalarWitness) -> PreRegion $ const 1
genericisePreRegion (RealSubray PositiveHalfSphere xl) = preRegionToInfFrom' xl
genericisePreRegion (RealSubray NegativeHalfSphere xr) = preRegionFromMinInfTo' xr
genericisePreRegion r = r
-- | Set-intersection of regions would not be guaranteed to yield a connected result
-- or even have the reference point of one region contained in the other. This
-- combinator assumes (unchecked) that the references are in a connected
-- sub-intersection, which is used as the result.
unsafePreRegionIntersect :: (RealDimension s, LocallyScalable s a)
=> PreRegion s a -> PreRegion s a -> PreRegion s a
unsafePreRegionIntersect GlobalRegion r = r
unsafePreRegionIntersect r GlobalRegion = r
unsafePreRegionIntersect (RealSubray PositiveHalfSphere xl) (RealSubray PositiveHalfSphere xl')
= RealSubray PositiveHalfSphere $ max xl xl'
unsafePreRegionIntersect (RealSubray NegativeHalfSphere xr) (RealSubray NegativeHalfSphere xr')
= RealSubray NegativeHalfSphere $ min xr xr'
unsafePreRegionIntersect (PreRegion ra) (PreRegion rb) = PreRegion $ minDblfuncs ra rb
unsafePreRegionIntersect ra rb
= unsafePreRegionIntersect (genericisePreRegion ra) (genericisePreRegion rb)
-- | Cartesian product of two regions.
regionProd :: (RealDimension s, LocallyScalable s a, LocallyScalable s b)
=> Region s a -> Region s b -> Region s (a,b)
regionProd (Region a₀ ra) (Region b₀ rb) = Region (a₀,b₀) (preRegionProd ra rb)
-- | Cartesian product of two pre-regions.
preRegionProd :: ∀ s a b . (RealDimension s, LocallyScalable s a, LocallyScalable s b)
=> PreRegion s a -> PreRegion s b -> PreRegion s (a,b)
preRegionProd = prp linearManifoldWitness closedScalarWitness
where prp :: LinearManifoldWitness s -> ClosedScalarWitness s
-> PreRegion s a -> PreRegion s b -> PreRegion s (a,b)
prp _ _ GlobalRegion GlobalRegion = GlobalRegion
prp (LinearManifoldWitness _) ClosedScalarWitness GlobalRegion (PreRegion rb)
= PreRegion $ rb . snd
prp (LinearManifoldWitness _) ClosedScalarWitness (PreRegion ra) GlobalRegion
= PreRegion $ ra . fst
prp (LinearManifoldWitness _) ClosedScalarWitness (PreRegion ra) (PreRegion rb)
= PreRegion $ minDblfuncs (ra.fst) (rb.snd)
prp _ _ ra rb = preRegionProd (genericisePreRegion ra) (genericisePreRegion rb)
positivePreRegion, negativePreRegion :: (RealDimension s) => PreRegion s s
positivePreRegion = RealSubray PositiveHalfSphere 0
negativePreRegion = RealSubray NegativeHalfSphere 0
positivePreRegion', negativePreRegion' :: ∀ s . (RealDimension s) => PreRegion s s
positivePreRegion' = PreRegion . Differentiable
$ prr linearManifoldWitness closedScalarWitness
where prr :: LinearManifoldWitness s -> ClosedScalarWitness s
-> s -> (s, Needle s+>Needle s, LinDevPropag s s)
prr (LinearManifoldWitness _) ClosedScalarWitness
x = ( 1 - 1/xp1
, (1/xp1²) *^ id
, unsafe_dev_ε_δ("positivePreRegion@"++show x) δ )
-- ε = (1 − 1/(1+x)) + (-δ · 1/(x+1)²) − (1 − 1/(1+x−δ))
-- = 1/(1+x−δ) − 1/(1+x) − δ · 1/(x+1)²
--
-- ε·(1+x−δ) = 1 − (1+x−δ)/(1+x) − δ·(1+x-δ)/(x+1)²
-- ε·(1+x) − ε·δ = 1 − 1/(1+x) − x/(1+x) + δ/(1+x)
-- − δ/(x+1)² − δ⋅x/(x+1)² + δ²/(x+1)²
-- = 1 − (1+x)/(1+x) + ((x+1) − 1)⋅δ/(x+1)²
-- − δ⋅x/(x+1)² + δ²/(x+1)²
-- = 1 − 1 + x⋅δ/(x+1)² − δ⋅x/(x+1)² + δ²/(x+1)²
-- = δ²/(x+1)²
--
-- ε·(x+1)⋅(x+1)² − ε·δ⋅(x+1)² = δ²
-- 0 = δ² + ε·(x+1)²·δ − ε·(x+1)³
--
-- δ = let μ = ε·(x+1)²/2 -- Exact form
-- in -μ + √(μ² + ε·(x+1)³) -- (not overflow save)
--
-- Safe approximation for large x:
-- ε = 1/(1+x−δ) − 1/(1+x) − δ · 1/(x+1)²
-- ≤ 1/(1+x−δ) − 1/(1+x)
--
-- ε⋅(1+x−δ)⋅(1+x) ≤ 1+x − (1+x−δ) = δ
--
-- δ ≥ ε + ε⋅x − ε⋅δ + ε⋅x + ε⋅x² − ε⋅δ⋅x
--
-- δ⋅(1 + ε + ε⋅x) ≥ ε + ε⋅x + ε⋅x + ε⋅x² ≥ ε⋅x²
--
-- δ ≥ ε⋅x²/(1 + ε + ε⋅x)
-- = ε⋅x/(1/x + ε/x + ε)
where δ ε | x<100 = let μ = ε*xp1²/2
in sqrt(μ^2 + ε * xp1² * xp1) - μ
| otherwise = ε * x / ((1+ε)/x + ε)
xp1 = (x+1)
xp1² = xp1 ^ 2
negativePreRegion' = npr (linearManifoldWitness :: LinearManifoldWitness s)
(closedScalarWitness :: ClosedScalarWitness s)
where npr (LinearManifoldWitness BoundarylessWitness)
(ClosedScalarWitness :: ClosedScalarWitness s)
= PreRegion $ ppr . ngt
where PreRegion ppr = positivePreRegion' :: PreRegion s s
ngt = actuallyLinearEndo $ negateV id
preRegionToInfFrom, preRegionFromMinInfTo :: RealDimension s => s -> PreRegion s s
preRegionToInfFrom = RealSubray PositiveHalfSphere
preRegionFromMinInfTo = RealSubray NegativeHalfSphere
preRegionToInfFrom', preRegionFromMinInfTo' :: ∀ s . RealDimension s => s -> PreRegion s s
preRegionToInfFrom' = prif (linearManifoldWitness :: LinearManifoldWitness s)
(closedScalarWitness :: ClosedScalarWitness s)
where prif (LinearManifoldWitness BoundarylessWitness)
(ClosedScalarWitness :: ClosedScalarWitness s)
xs = PreRegion $ ppr . trl
where PreRegion ppr = positivePreRegion' :: PreRegion s s
trl = actuallyAffineEndo (-xs) id
preRegionFromMinInfTo' = prif (linearManifoldWitness :: LinearManifoldWitness s)
(closedScalarWitness :: ClosedScalarWitness s)
where prif (LinearManifoldWitness BoundarylessWitness)
(ClosedScalarWitness :: ClosedScalarWitness s)
xe = PreRegion $ ppr . flp
where PreRegion ppr = positivePreRegion' :: PreRegion s s
flp = actuallyAffineEndo xe (negateV id)
intervalPreRegion :: ∀ s . RealDimension s => (s,s) -> PreRegion s s
intervalPreRegion (lb,rb) = PreRegion . Differentiable
$ prr linearManifoldWitness closedScalarWitness
where m = lb + radius; radius = (rb - lb)/2
prr :: LinearManifoldWitness s -> ClosedScalarWitness s
-> s -> (s, Needle s+>Needle s, LinDevPropag s s)
prr (LinearManifoldWitness _) ClosedScalarWitness
x = ( 1 - ((x-m)/radius)^2
, (2*(m-x)/radius^2) *^ id
, unsafe_dev_ε_δ("intervalPreRegion@"++show x) $ (*radius) . sqrt )
instance (RealDimension s) => Category (RWDiffable s) where
type Object (RWDiffable s) o = (LocallyScalable s o, Manifold o, SimpleSpace (Needle o))
id = RWDiffable $ \x -> (GlobalRegion, pure id)
RWDiffable f . RWDiffable g = RWDiffable h where
h x₀ = case g x₀ of
( rg, Just gr'@(AffinDiffable IsDiffableEndo gr) )
-> let (y₀, ϕg) = evalAffine gr x₀
in case f y₀ of
(GlobalRegion, Just (AffinDiffable fe fr))
-> (rg, Just (AffinDiffable fe (fr.gr)))
(GlobalRegion, fhr)
-> (rg, fmap (. gr') fhr)
(RealSubray diry yl, fhr)
-> let hhr = fmap (. gr') fhr
in case ϕg $ 1 of
y' | y'>0 -> ( unsafePreRegionIntersect rg
$ RealSubray diry (x₀ + (yl-y₀)/y')
-- y'⋅(xl−x₀) + y₀ ≝ yl
, hhr )
| y'<0 -> ( unsafePreRegionIntersect rg
$ RealSubray (otherHalfSphere diry)
(x₀ + (yl-y₀)/y')
, hhr )
| otherwise -> (rg, hhr)
(PreRegion ry, fhr)
-> ( PreRegion $ ry . gr', fmap (. gr') fhr )
( rg, Just gr'@(AffinDiffable _ gr) )
-> error "( rg, Just gr'@(AffinDiffable gr) )"
(GlobalRegion, Just gr@(Differentiable grd))
-> let (y₀,_,_) = grd x₀
in case f y₀ of
(GlobalRegion, Nothing)
-> (GlobalRegion, notDefinedHere)
(GlobalRegion, Just fr)
-> (GlobalRegion, pure (fr . gr))
(r, Nothing) | PreRegion ry <- genericisePreRegion r
-> ( PreRegion $ ry . gr, notDefinedHere )
(r, (Just fr)) | PreRegion ry <- genericisePreRegion r
-> ( PreRegion $ ry . gr, pure (fr . gr) )
(rg@(RealSubray _ _), Just gr@(Differentiable grd))
-> let (y₀,_,_) = grd x₀
in case f y₀ of
(GlobalRegion, Nothing)
-> (rg, notDefinedHere)
(GlobalRegion, Just fr)
-> (rg, pure (fr . gr))
(rf, Nothing)
| PreRegion rx <- genericisePreRegion rg
, PreRegion ry <- genericisePreRegion rf
-> ( PreRegion $ minDblfuncs (ry . gr) rx
, notDefinedHere )
(rf, Just fr)
| PreRegion rx <- genericisePreRegion rg
, PreRegion ry <- genericisePreRegion rf
-> ( PreRegion $ minDblfuncs (ry . gr) rx
, pure (fr . gr) )
(PreRegion rx, Just gr@(Differentiable grd))
-> let (y₀,_,_) = grd x₀
in case f y₀ of
(GlobalRegion, Nothing)
-> (PreRegion rx, notDefinedHere)
(GlobalRegion, Just fr)
-> (PreRegion rx, pure (fr . gr))
(r, Nothing) | PreRegion ry <- genericisePreRegion r
-> ( PreRegion $ minDblfuncs (ry . gr) rx
, notDefinedHere )
(r, Just fr) | PreRegion ry <- genericisePreRegion r
-> ( PreRegion $ minDblfuncs (ry . gr) rx
, pure (fr . gr) )
(r, Nothing)
-> (r, notDefinedHere)
globalDiffable' :: Differentiable s a b -> RWDiffable s a b
globalDiffable' f = RWDiffable $ const (GlobalRegion, pure f)
instance (RealDimension s) => EnhancedCat (RWDiffable s) (Differentiable s) where
arr = globalDiffable'
instance (RealDimension s) => Cartesian (RWDiffable s) where
type UnitObject (RWDiffable s) = ZeroDim s
swap = globalDiffable' swap
attachUnit = globalDiffable' attachUnit
detachUnit = globalDiffable' detachUnit
regroup = globalDiffable' regroup
regroup' = globalDiffable' regroup'
instance (RealDimension s) => Morphism (RWDiffable s) where
RWDiffable f *** RWDiffable g = RWDiffable h
where h (x,y) = (preRegionProd rfx rgy, liftA2 (***) dff dfg)
where (rfx, dff) = f x
(rgy, dfg) = g y
instance (RealDimension s) => PreArrow (RWDiffable s) where
RWDiffable f &&& RWDiffable g = RWDiffable h
where h x = (unsafePreRegionIntersect rfx rgx, liftA2 (&&&) dff dfg)
where (rfx, dff) = f x
(rgx, dfg) = g x
terminal = globalDiffable' terminal
fst = globalDiffable' fst
snd = globalDiffable' snd
instance (RealDimension s) => WellPointed (RWDiffable s) where
unit = Tagged Origin
globalElement x = RWDiffable $ \Origin -> (GlobalRegion, pure (globalElement x))
const x = RWDiffable $ \_ -> (GlobalRegion, pure (const x))
data RWDfblFuncValue s d c where
ConstRWDFV :: c -> RWDfblFuncValue s d c
RWDFV_IdVar :: RWDfblFuncValue s c c
GenericRWDFV :: RWDiffable s d c -> RWDfblFuncValue s d c
genericiseRWDFV :: ( RealDimension s
, LocallyScalable s c, SimpleSpace (Needle c)
, LocallyScalable s d, SimpleSpace (Needle d)
, Manifold d, Manifold c )
=> RWDfblFuncValue s d c -> RWDfblFuncValue s d c
genericiseRWDFV (ConstRWDFV c) = GenericRWDFV $ const c
genericiseRWDFV RWDFV_IdVar = GenericRWDFV id
genericiseRWDFV v = v
instance RealDimension s => HasAgent (RWDiffable s) where
type AgentVal (RWDiffable s) d c = RWDfblFuncValue s d c
alg fq = case fq RWDFV_IdVar of
GenericRWDFV f -> f
ConstRWDFV c -> const c
RWDFV_IdVar -> id
($~) = postCompRW
instance RealDimension s => CartesianAgent (RWDiffable s) where
alg1to2 fgq = case fgq RWDFV_IdVar of
(GenericRWDFV f, GenericRWDFV g) -> f &&& g
alg2to1 fq = case fq (GenericRWDFV fst) (GenericRWDFV snd) of
GenericRWDFV f -> f
alg2to2 fgq = case fgq (GenericRWDFV fst) (GenericRWDFV snd) of
(GenericRWDFV f, GenericRWDFV g) -> f &&& g
instance (RealDimension s)
=> PointAgent (RWDfblFuncValue s) (RWDiffable s) a x where
point = ConstRWDFV
grwDfblFnValsFunc
:: ( RealDimension s
, LocallyScalable s c, LocallyScalable s c', LocallyScalable s d
, Manifold d, Manifold c, Manifold c'
, v ~ Needle c, v' ~ Needle c'
, SimpleSpace v, SimpleSpace (Needle d)
, ε ~ Norm v, ε ~ Norm v' )
=> (c' -> (c, v'+>v, ε->ε)) -> RWDfblFuncValue s d c' -> RWDfblFuncValue s d c
grwDfblFnValsFunc f = (RWDiffable (\_ -> (GlobalRegion, pure (Differentiable f))) $~)
grwDfblFnValsCombine :: forall d c c' c'' v v' v'' ε ε' ε'' s.
( LocallyScalable s c, LocallyScalable s c', LocallyScalable s c''
, LocallyScalable s d, RealDimension s
, Manifold d, Manifold c', Manifold c''
, v ~ Needle c, v' ~ Needle c', v'' ~ Needle c''
, SimpleSpace v, SimpleSpace (Needle d)
, ε ~ Norm v , ε' ~ Norm v' , ε'' ~ Norm v'', ε~ε', ε~ε'' )
=> ( c' -> c'' -> (c, (v',v'')+>v, ε -> (ε',ε'')) )
-> RWDfblFuncValue s d c' -> RWDfblFuncValue s d c'' -> RWDfblFuncValue s d c
grwDfblFnValsCombine cmb (GenericRWDFV (RWDiffable fpcs))
(GenericRWDFV (RWDiffable gpcs))
= GenericRWDFV . RWDiffable $
\d₀ -> let (rc', fmay) = fpcs d₀
(rc'',gmay) = gpcs d₀
in (unsafePreRegionIntersect rc' rc'',) $
case (genericiseDifferentiable<$>fmay, genericiseDifferentiable<$>gmay) of
(Just(Differentiable f), Just(Differentiable g)) ->
pure . Differentiable $ \d
-> let (c', jf, devf) = f d
(c'',jg, devg) = g d
(c, jh, devh) = cmb c' c''
jhl = jh . (id&&&zeroV); jhr = jh . (zeroV&&&id)
in ( c
, jh <<< jf&&&jg
, \εc -> let εc' = transformNorm jhl εc
εc'' = transformNorm jhr εc
(δc',δc'') = devh εc
in devf εc' <> devg εc''
<> transformNorm jf δc'
<> transformNorm jg δc''
)
_ -> notDefinedHere
grwDfblFnValsCombine cmb fv gv
= grwDfblFnValsCombine cmb (genericiseRWDFV fv) (genericiseRWDFV gv)
rwDfbl_plus :: ∀ s a v .
( WithField s Manifold a
, LinearSpace v, Scalar v ~ s
, RealDimension s )
=> RWDiffable s a v -> RWDiffable s a v -> RWDiffable s a v
rwDfbl_plus (RWDiffable f) (RWDiffable g) = RWDiffable
$ h linearManifoldWitness dualSpaceWitness
where h :: LinearManifoldWitness v -> DualSpaceWitness v
-> a -> (PreRegion s a, Maybe (Differentiable s a v))
h (LinearManifoldWitness _) DualSpaceWitness
x₀ = (rh, liftA2 fgplus ff gf)
where (rf, ff) = f x₀
(rg, gf) = g x₀
rh = unsafePreRegionIntersect rf rg
fgplus :: Differentiable s a v -> Differentiable s a v -> Differentiable s a v
fgplus (Differentiable fd) (Differentiable gd) = Differentiable hd
where hd x = (fx^+^gx, jf^+^jg, \ε -> δf(scaleNorm 2 ε)
<> δg(scaleNorm 2 ε))
where (fx, jf, δf) = fd x
(gx, jg, δg) = gd x
fgplus (Differentiable fd) (AffinDiffable _ ga)
= Differentiable hd
where hd x = (fx^+^gx, jf^+^ϕg, δf)
where (fx, jf, δf) = fd x
(gx, ϕg) = evalAffine ga x
fgplus (AffinDiffable _ fa) (Differentiable gd)
= Differentiable hd
where hd x = (fx^+^gx, ϕf^+^jg, δg)
where (gx, jg, δg) = gd x
(fx, ϕf) = evalAffine fa x
fgplus (AffinDiffable fe fa) (AffinDiffable ge ga)
= AffinDiffable (fe<>ge) (fa^+^ga)
rwDfbl_negateV :: ∀ s a v .
( WithField s Manifold a
, LinearSpace v, Scalar v ~ s
, RealDimension s )
=> RWDiffable s a v -> RWDiffable s a v
rwDfbl_negateV (RWDiffable f) = RWDiffable $ h linearManifoldWitness dualSpaceWitness
where h :: LinearManifoldWitness v -> DualSpaceWitness v
-> a -> (PreRegion s a, Maybe (Differentiable s a v))
h (LinearManifoldWitness _) DualSpaceWitness
x₀ = (rf, fmap fneg ff)
where (rf, ff) = f x₀
fneg :: Differentiable s a v -> Differentiable s a v
fneg (Differentiable fd) = Differentiable hd
where hd x = (negateV fx, negateV jf, δf)
where (fx, jf, δf) = fd x
fneg (AffinDiffable ef af) = AffinDiffable ef $ negateV af
postCompRW :: ( RealDimension s
, LocallyScalable s a, LocallyScalable s b, LocallyScalable s c
, Manifold a, Manifold b, Manifold c
, SimpleSpace (Needle a), SimpleSpace (Needle b), SimpleSpace (Needle c) )
=> RWDiffable s b c -> RWDfblFuncValue s a b -> RWDfblFuncValue s a c
postCompRW (RWDiffable f) (ConstRWDFV x) = case f x of
(_, Just fd) -> ConstRWDFV $ fd $ x
postCompRW f RWDFV_IdVar = GenericRWDFV f
postCompRW f (GenericRWDFV g) = GenericRWDFV $ f . g
instance ∀ s a v . ( WithField s Manifold a, SimpleSpace (Needle a)
, Atlas v, HasTrie (ChartIndex v), SimpleSpace v, Scalar v ~ s
, RealDimension s )
=> AdditiveGroup (RWDfblFuncValue s a v) where
zeroV = case ( linearManifoldWitness :: LinearManifoldWitness v
, dualSpaceWitness :: DualSpaceWitness v ) of
(LinearManifoldWitness BoundarylessWitness, DualSpaceWitness) -> point zeroV
(^+^) = case ( linearManifoldWitness :: LinearManifoldWitness v
, dualSpaceWitness :: DualSpaceWitness v ) of
(LinearManifoldWitness BoundarylessWitness, DualSpaceWitness)
-> curry $ \case
(ConstRWDFV c₁, ConstRWDFV c₂) -> ConstRWDFV (c₁^+^c₂)
(ConstRWDFV c₁, RWDFV_IdVar) -> GenericRWDFV $
globalDiffable' (actuallyAffineEndo c₁ id)
(RWDFV_IdVar, ConstRWDFV c₂) -> GenericRWDFV $
globalDiffable' (actuallyAffineEndo c₂ id)
(ConstRWDFV c₁, GenericRWDFV g) -> GenericRWDFV $
globalDiffable' (actuallyAffineEndo c₁ id) . g
(GenericRWDFV f, ConstRWDFV c₂) -> GenericRWDFV $
globalDiffable' (actuallyAffineEndo c₂ id) . f
(fa, ga) | GenericRWDFV f <- genericiseRWDFV fa
, GenericRWDFV g <- genericiseRWDFV ga
-> GenericRWDFV $ rwDfbl_plus f g
negateV = case ( linearManifoldWitness :: LinearManifoldWitness v
, dualSpaceWitness :: DualSpaceWitness v ) of
(LinearManifoldWitness BoundarylessWitness, DualSpaceWitness) -> \case
(ConstRWDFV c) -> ConstRWDFV (negateV c)
RWDFV_IdVar -> GenericRWDFV $ globalDiffable' (actuallyLinearEndo $ negateV id)
(GenericRWDFV f) -> GenericRWDFV $ rwDfbl_negateV f
dualCoCoProduct :: ∀ v w s .
( SimpleSpace v, HilbertSpace v
, SimpleSpace w, Scalar v ~ s, Scalar w ~ s )
=> LinearMap s w v -> LinearMap s w v -> Norm w
dualCoCoProduct = dccp (dualSpaceWitness::DualSpaceWitness w)
where dccp DualSpaceWitness s t = Norm $ (tSpread*sSpread) *^ t²Ps²M
where t' = adjoint $ t :: LinearMap s v (DualVector w)
s' = adjoint $ s :: LinearMap s v (DualVector w)
tSpread = sum . map recip_t²PLUSs² $ snd (decomposeLinMap t') []
sSpread = sum . map recip_t²PLUSs² $ snd (decomposeLinMap s') []
t²PLUSs²@(Norm t²Ps²M)
= transformNorm t euclideanNorm <> transformNorm s euclideanNorm :: Norm w
recip_t²PLUSs² = normSq (dualNorm t²PLUSs²) :: DualVector w -> s
instance ( RealDimension n, WithField n Manifold a
, LocallyScalable n a, SimpleSpace (Needle a))
=> Num (RWDfblFuncValue n a n) where
fromInteger i = point $ fromInteger i
(+) = (^+^)
ConstRWDFV c₁ * ConstRWDFV c₂ = ConstRWDFV (c₁*c₂)
ConstRWDFV c₁ * RWDFV_IdVar = GenericRWDFV $
globalDiffable' (actuallyLinearEndo . arr $ scale $ c₁)
RWDFV_IdVar * ConstRWDFV c₂ = GenericRWDFV $
globalDiffable' (actuallyLinearEndo . arr $ scale $ c₂)
ConstRWDFV c₁ * GenericRWDFV g = GenericRWDFV $
globalDiffable' (actuallyLinearEndo . arr $ scale $ c₁) . g
GenericRWDFV f * ConstRWDFV c₂ = GenericRWDFV $
globalDiffable' (actuallyLinearEndo . arr $ scale $ c₂) . f
f*g = genericiseRWDFV f ⋅ genericiseRWDFV g
where (⋅) :: ∀ n a . (RealDimension n, LocallyScalable n a, SimpleSpace (Needle a))
=> RWDfblFuncValue n a n -> RWDfblFuncValue n a n -> RWDfblFuncValue n a n
GenericRWDFV (RWDiffable fpcs) ⋅ GenericRWDFV (RWDiffable gpcs)
= GenericRWDFV . RWDiffable $
\d₀ -> let (rc₁, fmay) = fpcs d₀
(rc₂,gmay) = gpcs d₀
in (unsafePreRegionIntersect rc₁ rc₂, mulDi <$> fmay <*> gmay)
where mulDi :: Differentiable n a n -> Differentiable n a n -> Differentiable n a n
mulDi f@(AffinDiffable ef af) g@(AffinDiffable eg ag) = case ef<>eg of
IsDiffableEndo ->
{- let f' = lapply slf 1; g' = lapply slg 1
in case f'*g' of
0 -> AffinDiffableEndo $ const (aof*aog)
f'g' -> -} Differentiable $
\d -> let (fd,ϕf) = evalAffine af d
(gd,ϕg) = evalAffine ag d
jf = ϕf $ 1; jg = ϕg $ 1
invf'g' = recip $ jf*jg
in ( fd*gd
, arr $ scale $ fd*jg + gd*jf
, unsafe_dev_ε_δ "*" $ sqrt . (*invf'g') )
_ -> mulDi (genericiseDifferentiable f) (genericiseDifferentiable g)
mulDi (Differentiable f) (Differentiable g)
= Differentiable $
\d -> let (c₁, jf, devf) = f d
(c₂, jg, devg) = g d
c = c₁*c₂; c₁² = c₁^2; c₂² = c₂^2
h' = c₁*^jg ^+^ c₂*^jf
in ( c
, h'
, \εc -> let rε = εc|$|1
c₁worst = sqrt $ c₁² + recip(1 + c₂²*rε^2)
c₂worst = sqrt $ c₂² + recip(1 + c₁²*rε^2)
in scaleNorm (2*rε) (dualCoCoProduct jf jg)
<> devf (scaleNorm (2*c₂worst) εc)
<> devg (scaleNorm (2*c₁worst) εc)
-- TODO: add formal proof for this (or, if necessary, the correct form)
)
mulDi f g = mulDi (genericiseDifferentiable f) (genericiseDifferentiable g)
negate = negateV
abs = (RWDiffable absPW $~)
where absPW a₀
| a₀<0 = (negativePreRegion, pure desc)
| otherwise = (positivePreRegion, pure asc)
desc = actuallyLinearEndo $ negateV id
asc = actuallyLinearEndo id
signum = (RWDiffable sgnPW $~)
where sgnPW a₀
| a₀<0 = (negativePreRegion, pure (const $ -1))
| otherwise = (positivePreRegion, pure (const 1))
instance ( RealDimension n, WithField n Manifold a
, LocallyScalable n a, SimpleSpace (Needle a))
=> Fractional (RWDfblFuncValue n a n) where
fromRational i = point $ fromRational i
recip = postCompRW . RWDiffable $ \a₀ -> if a₀<0
then (negativePreRegion, pure (Differentiable negp))
else (positivePreRegion, pure (Differentiable posp))
where negp x = (x'¹, (- x'¹^2) *^ id, unsafe_dev_ε_δ("1/"++show x) δ)
-- ε = 1/x − δ/x² − 1/(x+δ)
-- ε·x + ε·δ = 1 + δ/x − δ/x − δ²/x² − 1
-- = -δ²/x²
-- 0 = δ² + ε·x²·δ + ε·x³
-- δ = let mph = -ε·x²/2 in mph + sqrt (mph² − ε·x³)
where δ ε = let mph = -ε*x^2/2
δ₀ = mph + sqrt (mph^2 - ε*x^3)
in if δ₀ > 0
then δ₀
else - x -- numerical underflow of εx³ vs mph
-- ≡ ε*x^3 / (2*mph) (Taylor-expansion of the root)
x'¹ = recip x
posp x = (x'¹, (- x'¹^2) *^ id, unsafe_dev_ε_δ("1/"++show x) δ)
where δ ε = let mph = ε*x^2/2
δ₀ = sqrt (mph^2 + ε*x^3) - mph
in if δ₀>0 then δ₀ else x
x'¹ = recip x
instance ∀ n a . ( RealDimension n, WithField n Manifold a
, LocallyScalable n a, SimpleSpace (Needle a) )
=> Floating (RWDfblFuncValue n a n) where
pi = point pi
exp = grwDfblFnValsFunc
$ \x -> let ex = exp x
in if ex*2 == ex -- numerical trouble...
then if x<0 then ( 0, zeroV, unsafe_dev_ε_δ("exp "++show x) $ \ε -> log ε - x )
else ( ex, ex*^id
, unsafe_dev_ε_δ("exp "++show x) $ \_ -> 1e-300 :: n )
else ( ex, ex *^ id, unsafe_dev_ε_δ("exp "++show x)
$ \ε -> case acosh(ε/(2*ex) + 1) of
δ | δ==δ -> δ
| otherwise -> log ε - x )
-- ε = e^(x+δ) − eˣ − eˣ·δ
-- = eˣ·(e^δ − 1 − δ)
-- ≤ eˣ · (e^δ − 1 + e^(-δ) − 1)
-- = eˣ · 2·(cosh(δ) − 1)
-- cosh(δ) ≥ ε/(2·eˣ) + 1
-- δ ≥ acosh(ε/(2·eˣ) + 1)
log = postCompRW . RWDiffable $ \x -> if x>0
then (positivePreRegion, pure (Differentiable lnPosR))
else (negativePreRegion, notDefinedHere)
where lnPosR x = ( log x, recip x *^ id, unsafe_dev_ε_δ("log "++show x) $ \ε -> x * sqrt(1 - exp(-ε)) )
-- ε = ln x + (-δ)/x − ln(x−δ)
-- = ln (x / ((x−δ) · exp(δ/x)))
-- x/e^ε = (x−δ) · exp(δ/x)
-- let γ = δ/x ∈ [0,1[
-- exp(-ε) = (1−γ) · e^γ
-- ≥ (1−γ) · (1+γ)
-- = 1 − γ²
-- γ ≥ sqrt(1 − exp(-ε))
-- δ ≥ x · sqrt(1 − exp(-ε))
sqrt = postCompRW . RWDiffable $ \x -> if x>0
then (positivePreRegion, pure (Differentiable sqrtPosR))
else (negativePreRegion, notDefinedHere)
where sqrtPosR x = ( sx, id ^/ (2*sx), unsafe_dev_ε_δ("sqrt "++show x) $
\ε -> 2 * (s2 * sqrt sx^3 * sqrt ε + signum (ε*2-sx) * sx * ε) )
where sx = sqrt x; s2 = sqrt 2
-- Exact inverse of O(δ²) remainder.
sin = grwDfblFnValsFunc sinDfb
where sinDfb x = ( sx, cx *^ id, unsafe_dev_ε_δ("sin "++show x) δ )
where sx = sin x; cx = cos x
sx² = sx^2; cx² = cx^2
sx' = abs sx; cx' = abs cx
sx'³ = sx'*sx²; cx⁴ = cx²*cx²
δ ε = (ε*(1.8 + ε^2/(cx' + (2+40*cx⁴)/ε)) + σ₃³*sx'³)**(1/3) - σ₃*sx'
+ σ₂*sqrt ε/(σ₂+cx²)
-- Carefully fine-tuned to give everywhere a good and safe bound.
-- The third root makes it pretty slow too, but since tight
-- deviation bounds can dramatically reduce the number of evaluations
-- needed in the first place, this is probably worthwhile.
σ₂ = 1.4; σ₃ = 1.75; σ₃³ = σ₃^3
-- Safety margins for overlap between quadratic and cubic model
-- (these aren't naturally compatible to be used both together)
cos = sin . (globalDiffable' (actuallyAffineEndo (pi/2) id) $~)
sinh x = (exp x - exp (-x))/2
{- = grwDfblFnValsFunc sinhDfb
where sinhDfb x = ( sx, cx *^ idL, unsafe_dev_ε_δ δ )
where sx = sinh x; cx = cosh x
δ ε = undefined -}
-- ε = sinh x + δ · cosh x − sinh(x+δ)
-- = ½ · ( eˣ − e⁻ˣ + δ · (eˣ + e⁻ˣ) − exp(x+δ) + exp(-x−δ) )
-- = ½·e⁻ˣ · ( e²ˣ − 1 + δ · (e²ˣ + 1) − e²ˣ·e^δ + e^-δ )
-- = ½ · ( eˣ − e⁻ˣ + δ · (eˣ + e⁻ˣ) − exp(x+δ) + exp(-x−δ) )
cosh x = (exp x + exp (-x))/2
tanh = grwDfblFnValsFunc tanhDfb
where tanhDfb x = ( tnhx, id ^/ (cosh x^2), unsafe_dev_ε_δ("tan "++show x) δ )
where tnhx = tanh x
c = (tnhx*2/pi)^2
p = 1 + abs x/(2*pi)
δ ε = p * (sqrt ε + ε * c)
-- copied from 'atan' definition. Empirically works safely, in fact
-- with quite a big margin. TODO: find a tighter definition.
atan = grwDfblFnValsFunc atanDfb
where atanDfb x = ( atnx, id ^/ (1+x^2), unsafe_dev_ε_δ("atan "++show x) δ )
where atnx = atan x
c = (atnx*2/pi)^2
p = 1 + abs x/(2*pi)
δ ε = p * (sqrt ε + ε * c)
-- Semi-empirically obtained: with the epsEst helper,
-- it is observed that this function is (for xc≥0) a lower bound
-- to the arctangent. The growth of the p coefficient makes sense
-- and holds for arbitrarily large xc, because those move us linearly
-- away from the only place where the function is not virtually constant
-- (around 0).
asin = postCompRW . RWDiffable $ \x -> if
| x < (-1) -> (preRegionFromMinInfTo (-1), notDefinedHere)
| x > 1 -> (preRegionToInfFrom 1, notDefinedHere)
| otherwise -> (intervalPreRegion (-1,1), pure (Differentiable asinDefdR))
where asinDefdR x = ( asinx, asin'x *^ id, unsafe_dev_ε_δ("asin "++show x) δ )
where asinx = asin x; asin'x = recip (sqrt $ 1 - x^2)
c = 1 - x^2
δ ε = sqrt ε * c
-- Empirical, with epsEst upper bound.
acos = postCompRW . RWDiffable $ \x -> if
| x < (-1) -> (preRegionFromMinInfTo (-1), notDefinedHere)
| x > 1 -> (preRegionToInfFrom 1, notDefinedHere)
| otherwise -> (intervalPreRegion (-1,1), pure (Differentiable acosDefdR))
where acosDefdR x = ( acosx, acos'x *^ id, unsafe_dev_ε_δ("acos "++show x) δ )
where acosx = acos x; acos'x = - recip (sqrt $ 1 - x^2)
c = 1 - x^2
δ ε = sqrt ε * c -- Like for asin – it's just a translation/reflection.
asinh = grwDfblFnValsFunc asinhDfb
where asinhDfb x = ( asinhx, id ^/ sqrt(1+x^2), unsafe_dev_ε_δ("asinh "++show x) δ )
where asinhx = asinh x
δ ε = abs x * sqrt((1 - exp(-ε))*0.8 + ε^2/(3*abs x + 1)) + sqrt(ε/(abs x+0.5))
-- Empirical, modified from log function (the area hyperbolic sine
-- resembles two logarithmic lobes), with epsEst-checked lower bound.
acosh = postCompRW . RWDiffable $ \x -> if x>1
then (preRegionToInfFrom 1, pure (Differentiable acoshDfb))
else (preRegionFromMinInfTo 1, notDefinedHere)
where acoshDfb x = ( acosh x, id ^/ sqrt(x^2 - 1), unsafe_dev_ε_δ("acosh "++show x) δ )
where δ ε = (2 - 1/sqrt x) * (s2 * sqrt sx^3 * sqrt(ε/s2) + signum (ε*s2-sx) * sx * ε/s2)
sx = sqrt(x-1)
s2 = sqrt 2
-- Empirical, modified from sqrt function – the area hyperbolic cosine
-- strongly resembles \x -> sqrt(2 · (x-1)).
atanh = postCompRW . RWDiffable $ \x -> if
| x < (-1) -> (preRegionFromMinInfTo (-1), notDefinedHere)
| x > 1 -> (preRegionToInfFrom 1, notDefinedHere)
| otherwise -> (intervalPreRegion (-1,1), pure (Differentiable atnhDefdR))
where atnhDefdR x = ( atanh x, recip(1-x^2) *^ id, unsafe_dev_ε_δ("atanh "++show x) $ \ε -> sqrt(tanh ε)*(1-abs x) )
-- Empirical, with epsEst upper bound.
-- $definitionRegionOps
-- Because the agents of 'RWDiffable' aren't really values in /Hask/, you can't use
-- the standard comparison operators on them, nor the built-in syntax of guards
-- or if-statements.
--
-- However, because this category allows functions to be undefined in some region,
-- such decisions can be faked quite well: '?->' restricts a function to
-- some region, by simply marking it undefined outside, and '?|:' replaces these
-- regions with values from another function.
--
-- Example: define a function that is compactly supported on the interval ]-1,1[,
-- i.e. exactly zero everywhere outside.
--
-- @
-- Graphics.Dynamic.Plot.R2> plotWindow [fnPlot (\\x -> -1 '?<' x '?<' 1 '?->' cos (x*pi/2)^2 '?|:' 0)]
-- @
--
-- <<images/examples/DiffableFunction-plots/Hann-window.png>>
--
-- Note that it may not be necessary to restrict explicitly: for instance if a
-- square root appears somewhere in an expression, then the expression is automatically
-- restricted so that the root has a positive argument!
--
-- @
-- Graphics.Dynamic.Plot.R2> plotWindow [fnPlot (\\x -> sqrt x '?|:' -sqrt (-x))]
-- @
--
-- <<images/examples/DiffableFunction-plots/safe-sqrt.png>>
infixr 4 ?->
-- | Require the LHS to be defined before considering the RHS as result.
-- This works analogously to the standard `Control.Applicative.Applicative` method
--
-- @
-- ('Control.Applicative.*>') :: Maybe a -> Maybe b -> Maybe b
-- Just _ 'Control.Applicative.*>' a = a
-- _ 'Control.Applicative.*>' a = Nothing
-- @
(?->) :: ( RealDimension n, LocallyScalable n a, LocallyScalable n b, LocallyScalable n c
, Manifold b, Manifold c
, SimpleSpace (Needle b), SimpleSpace (Needle c) )
=> RWDfblFuncValue n c a -> RWDfblFuncValue n c b -> RWDfblFuncValue n c b
ConstRWDFV _ ?-> f = f
RWDFV_IdVar ?-> f = f
GenericRWDFV (RWDiffable r) ?-> ConstRWDFV c = GenericRWDFV (RWDiffable s)
where s x₀ = case r x₀ of
(rd, Just q) -> (rd, return $ const c)
(rd, Nothing) -> (rd, empty)
GenericRWDFV (RWDiffable f) ?-> GenericRWDFV (RWDiffable g) = GenericRWDFV (RWDiffable h)
where h x₀ = case f x₀ of
(rf, Just _) | (rg, q) <- g x₀
-> (unsafePreRegionIntersect rf rg, q)
(rf, Nothing) -> (rf, empty)
c ?-> f = c ?-> genericiseRWDFV f
positiveRegionalId :: RealDimension n => RWDiffable n n n
positiveRegionalId = RWDiffable $ \x₀ ->
if x₀ > 0 then (positivePreRegion, pure . AffinDiffable IsDiffableEndo $ id)
else (negativePreRegion, notDefinedHere)
infixl 5 ?> , ?<
-- | Return the RHS, if it is less than the LHS.
-- (Really the purpose is just to compare the values, but returning one of them
-- allows chaining of comparison operators like in Python.)
-- Note that less-than comparison is <http://www.paultaylor.eu/ASD/ equivalent>
-- to less-or-equal comparison, because there is no such thing as equality.
(?>) :: (RealDimension n, LocallyScalable n a, Manifold a, SimpleSpace (Needle a))
=> RWDfblFuncValue n a n -> RWDfblFuncValue n a n -> RWDfblFuncValue n a n
a ?> b = (positiveRegionalId $~ a-b) ?-> b
-- | Return the RHS, if it is greater than the LHS.
(?<) :: (RealDimension n, LocallyScalable n a, Manifold a, SimpleSpace (Needle a))
=> RWDfblFuncValue n a n -> RWDfblFuncValue n a n -> RWDfblFuncValue n a n
ConstRWDFV a ?< RWDFV_IdVar = GenericRWDFV . RWDiffable $
\x₀ -> if a < x₀ then ( preRegionToInfFrom a
, pure . AffinDiffable IsDiffableEndo $ id)
else (preRegionFromMinInfTo a, notDefinedHere)
RWDFV_IdVar ?< ConstRWDFV a = GenericRWDFV . RWDiffable $
\x₀ -> if x₀ < a then ( preRegionFromMinInfTo a
, pure . AffinDiffable IsDiffableEndo $ const a)
else (preRegionToInfFrom a, notDefinedHere)
a ?< b = (positiveRegionalId $~ b-a) ?-> b
infixl 3 ?|:
-- | Try the LHS, if it is undefined use the RHS. This works analogously to
-- the standard `Control.Applicative.Alternative` method
--
-- @
-- ('Control.Applicative.<|>') :: Maybe a -> Maybe a -> Maybe a
-- Just x 'Control.Applicative.<|>' _ = Just x
-- _ 'Control.Applicative.<|>' a = a
-- @
--
-- Basically a weaker and agent-ised version of 'backupRegions'.
(?|:) :: ( RealDimension n, LocallyScalable n a, LocallyScalable n b
, Manifold a, Manifold b
, SimpleSpace (Needle a), SimpleSpace (Needle b) )
=> RWDfblFuncValue n a b -> RWDfblFuncValue n a b -> RWDfblFuncValue n a b
ConstRWDFV c ?|: _ = ConstRWDFV c
RWDFV_IdVar ?|: _ = RWDFV_IdVar
GenericRWDFV (RWDiffable f) ?|: ConstRWDFV c = GenericRWDFV (RWDiffable h)
where h x₀ = case f x₀ of
(rd, Just q) -> (rd, Just q)
(rd, Nothing) -> (rd, Just $ const c)
GenericRWDFV (RWDiffable f) ?|: GenericRWDFV (RWDiffable g) = GenericRWDFV (RWDiffable h)
where h x₀ = case f x₀ of
(rf, Just q) -> (rf, pure q)
(rf, Nothing) | (rg, q) <- g x₀
-> (unsafePreRegionIntersect rf rg, q)
c ?|: f = c ?|: genericiseRWDFV f
-- | Replace the regions in which the first function is undefined with values
-- from the second function.
backupRegions :: (RealDimension n, LocallyScalable n a, LocallyScalable n b)
=> RWDiffable n a b -> RWDiffable n a b -> RWDiffable n a b
backupRegions (RWDiffable f) (RWDiffable g) = RWDiffable h
where h x₀ = case f x₀ of
(rf, q@(Just _)) -> (rf, q)
(rf, Nothing) | (rg, q) <- g x₀
-> (unsafePreRegionIntersect rf rg, q)
-- | Like 'Data.VectorSpace.lerp', but gives a differentiable function
-- instead of a Hask one.
lerp_diffable :: ∀ m s . ( LinearSpace m, Scalar m ~ s, Atlas m
, HasTrie (ChartIndex m), RealDimension s )
=> m -> m -> Differentiable s s m
lerp_diffable = case ( linearManifoldWitness :: LinearManifoldWitness m
, dualSpaceWitness :: DualSpaceWitness m ) of
(LinearManifoldWitness BoundarylessWitness, DualSpaceWitness)
-> \a b -> actuallyAffine a . arr $ flipBilin scale $ b.-.a