Data.Manifold.PseudoAffine

This is the second prototype of a manifold class. It appears to give considerable advantages over Manifold, so that class will probably soon be replaced with the one we define here (though PseudoAffine does not follow the standard notion of a manifold very closely, it should work quite equivalently for pretty much all Haskell types that qualify as manifolds).

Manifolds are interesting as objects of various categories, from continuous to diffeomorphic. At the moment, we mainly focus on region-wise differentiable functions, which are a promising compromise between flexibility of definition and provability of analytic properties. In particular, they are well-suited for visualisation purposes.

Manifold class

class PseudoAffine x where

PseudoAffine is intended as an alternative class for Manifolds. The interface is almost identical to the better-known AffineSpace class, but unlike in the mathematical definition of affine spaces we don't require associativity of .+~^ with ^+^ – except in an asymptotic sense for small vectors.

That innocent-looking change makes the class applicable to vastly more general types: while an affine space is basically nothing but a vector space without particularly designated origin, a pseudo-affine space can have nontrivial topology on the global scale, and yet be used in practically the same way as an affine space. At least the usual spheres and tori make good instances, perhaps the class is in fact equivalent to parallelisable manifolds.

Associated Types

type PseudoDiff x :: *

Methods

(.-~.) :: x -> x -> Option (PseudoDiff x) infix 6

(.+~^) :: x -> PseudoDiff x -> x infixl 6

Instances

PseudoAffine Double

PseudoAffine Rational

PseudoAffine S²

PseudoAffine S¹

PseudoAffine (ZeroDim k)

(PseudoAffine a, PseudoAffine b) => PseudoAffine (a, b)

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => PseudoAffine (a, b, c)

Regions within a manifold

data Region s m

A pathwise connected subset of a manifold m, whose tangent space has scalar s.

Hierarchy of manifold-categories

data Differentiable s d c

The category of differentiable functions between manifolds over scalar s.

As you might guess, these offer automatic differentiation of sorts (basically, simple forward AD), but that's in itself is not really the killer feature here. More interestingly, we actually have the (à la Curry-Howard) proof built in: the function f has at x₀ derivative f'ₓ₀, if, for¹ ε>0, there exists δ such that |f x − (f x₀ + x⋅f'ₓ₀)| < ε for all |x − x₀| < δ.

Observe that, though this looks quite similar to the standard definition of differentiability, it is not equivalent thereto – in fact it does not prove any analytic properties at all. To make it equivalent, we need a lower bound on δ: simply δ gives us continuity, and for continuous differentiability, δ must grow at least like √ε for small ε. Neither of these conditions are enforced by the type system, but we do require them for any allowed values because these proofs are obviously tremendously useful – for instance, you can have a root-finding algorithm and actually be sure you get all solutions correctly, not just some that are (hopefully) the closest to some reference point you'd need to laborously define!

Unfortunately however, this also prevents doing any serious algebra etc. with the category, because even something as simple as division necessary introduces singularities where the derivatives must diverge. Not to speak of many trigonometric e.g. trigonometric functions that are undefined on whole regions. The PWDiffable and RWDiffable categories have explicit handling for those issues built in; you may simply use these categories even when you know the result will be smooth in your relevant domain (or must be, for e.g. physics reasons).

¹(The implementation does not deal with ε and δ as difference-bounding reals, but rather as metric tensors that define a boundary by prohibiting the overlap from exceeding one; this makes the concept actually work on general manifolds.)

Instances

VectorSpace s => Morphism (Differentiable s)

VectorSpace s => PreArrow (Differentiable s)

VectorSpace s => WellPointed (Differentiable s)

VectorSpace s => CartesianAgent (Differentiable s)

VectorSpace s => Category (Differentiable s)

VectorSpace s => Cartesian (Differentiable s)

VectorSpace s => HasAgent (Differentiable s)

RealDimension s => EnhancedCat (RWDiffable s) (Differentiable s)

RealDimension s => EnhancedCat (PWDiffable s) (Differentiable s)

type UnitObject (Differentiable s) = ZeroDim s

type PointObject (Differentiable s) x = ()

type Object (Differentiable s) o

type PairObjects (Differentiable s) a b = ()

type AgentVal (Differentiable s) a v = GenericAgent (Differentiable s) a v

data PWDiffable s d c

Category of functions that almost everywhere have an open region in which they are continuously differentiable, i.e. PieceWiseDiff'able.

Instances

RealDimension s => Morphism (PWDiffable s)

RealDimension s => PreArrow (PWDiffable s)

RealDimension s => WellPointed (PWDiffable s)

RealDimension s => CartesianAgent (PWDiffable s)

RealDimension s => Category (PWDiffable s)

RealDimension s => Cartesian (PWDiffable s)

RealDimension s => HasAgent (PWDiffable s)

RealDimension s => EnhancedCat (RWDiffable s) (PWDiffable s)

RealDimension s => EnhancedCat (PWDiffable s) (Differentiable s)

type UnitObject (PWDiffable s) = ZeroDim s

type PointObject (PWDiffable s) x = ()

type Object (PWDiffable s) o

type PairObjects (PWDiffable s) a b = ()

type AgentVal (PWDiffable s) a v = GenericAgent (PWDiffable s) a v

data RWDiffable s d c

Category of functions that, where defined, have an open region in which they are continuously differentiable. Hence RegionWiseDiff'able. Basically these are the partial version of PWDiffable.

Though the possibility of undefined regions is of course not too nice (we don't need Java to demonstrate this with its everywhere-looming null values...), this category will propably be the “workhorse” for most serious calculus applications, because it contains all the usual trig etc. functions and of course everything algebraic you can do in the reals.

The easiest way to define ordinary functions in this category is hence with its AgentValues, which have instances of the standard classes Num through Floating. For instance, the following defines the binary entropy as a differentiable function on the interval ]0,1[: (it will actually know where it's defined and where not! – and I don't mean you need to exhaustively isNaN-check all results...)

hb :: RWDiffable R R R
hb = alg (\p -> - p * logBase 2 p - (1-p) * logBase 2 (1-p) )

Instances

RealDimension s => Morphism (RWDiffable s)

RealDimension s => PreArrow (RWDiffable s)

RealDimension s => WellPointed (RWDiffable s)

RealDimension s => CartesianAgent (RWDiffable s)

RealDimension s => Category (RWDiffable s)

RealDimension s => Cartesian (RWDiffable s)

RealDimension s => HasAgent (RWDiffable s)

RealDimension s => EnhancedCat (RWDiffable s) (PWDiffable s)

RealDimension s => EnhancedCat (RWDiffable s) (Differentiable s)

type UnitObject (RWDiffable s) = ZeroDim s

type PointObject (RWDiffable s) x = ()

type Object (RWDiffable s) o

type PairObjects (RWDiffable s) a b = ()

type AgentVal (RWDiffable s) a v = GenericAgent (RWDiffable s) a v