Conal Elliott

Additive groups and vector spaces

We’ll need a bit of preliminary before jumping into basis types.

An additive group has addition operation, with an identity (zero) and an additive inverse:

class AdditiveGroup v where
  (^+^)   ∷ v → v → v
  zeroV   ∷ v
  negateV ∷ v → v

A vector space is an additive group with scalar multiplication, so it also has an associated type of scalars:

class AdditiveGroup v ⇒ VectorSpace v where
  type Scalar v ∷ *
  (*^) ∷ Scalar v → v → v

This associated type could instead by written as a functional dependency (fundep). In fact, the type family implementation in ghc 6.9 is not quite up to working with the code in this post, so the library contains a (6.9-friendly) version together with as a fundep (Data.VectorSpace) and the version given in this post (Data.AVectorSpace). Sometime after ghc-6.10 is released, I will retire the fundep version and rename AVectorSpace to VectorSpace. Similarly, the library temporarily contains Data.ABasis.

Basis types

Since Haskell doesn't have subtyping, we can't represent a basis type directly as a subset. Instead, for an arbitrary vector space v, represent the (canonical) basis as an associated type, Basis v, and a function that interprets a basis representation as a vector.

class VectorSpace v ⇒ HasBasis v where
  type Basis v ∷ *
  basisValue   ∷ Basis v → v

Another method extracts coordinates (coefficients) for a vector. Each coordinate is associated with a basis element.

  decompose    ∷ v → [(Basis v, Scalar v)]

We can also reverse the process, recomposing into a vector, by linearly combining the basis elements:

linearCombo ∷ VectorSpace v ⇒ [(v,Scalar v)] → v
linearCombo ps = sumV [s *^ v | (v,s) ← ps]

recompose ∷ HasBasis v ⇒ [(Basis v, Scalar v)] → v
recompose = linearCombo ∘ fmap (first basisValue)

The defining property is

recompose ∘ decompose ≡ id

Primitive bases

Any scalar field is also a vector space over itself. For instance,

instance AdditiveGroup Double where
  zeroV   = 0.0
  (^+^)   = (+)
  negateV = negate

instance VectorSpace Double where
  type Scalar Double = Double
  (*^) = (*)

The canonical basis of a one-dimensional space has only one element, namely 1, which can be represented with no information.

instance HasBasis Double where
  type Basis Double = ()
  basisValue ()     = 1
  decompose s       = [((),s)]

Composing bases

Pairs of additive groups form additive groups:

instance (AdditiveGroup u,AdditiveGroup v)
      ⇒ AdditiveGroup (u,v) where
  zeroV             = (zeroV,zeroV)
  (u,v) ^+^ (u',v') = (u^+^u',v^+^v')
  negateV (u,v)     = (negateV u,negateV v)

Pairs of vector spaces, over the same scalar field, form vector spaces:

instance (VectorSpace u,VectorSpace v, Scalar u ~ Scalar v)
      ⇒ VectorSpace (u,v) where
  type Scalar (u,v) = Scalar u
  s *^ (u,v) = (s*^u,s*^v)

Given vector spaces u and v, a basis representation for (u,v) will be one basis representation or the other, tagged with Left or Right:

instance (HasBasis u, HasBasis v, Scalar u ~ Scalar v)
      ⇒ HasBasis (u,v) where
  type Basis (u,v) = Basis u `Either` Basis v

The basis vectors themselves will be (ub,0) or (0,vb), where ub is a basis vector for u, and vb is a basis vector for v. As expected then, the dimensionality of the cross product is the sum of the dimensions.

  basisValue (Left  a) = (basisValue a, zeroV)
  basisValue (Right b) = (zeroV, basisValue b)

The decomposition of a vector (u,v) contains left-tagged decompositions of u and right-tagged decompositions of v.

  decompose (u,v) = decomp2 Left u ++ decomp2 Right v

where

decomp2 ∷ HasBasis w ⇒ (Basis w → b) → w → [(Scalar w, b)]
decomp2 inject = fmap (first inject) ∘ decompose

Triples etc, can be handled similarly. Instead, the library implementation reduces them to the pair case:

instance ( HasBasis u, s ~ Scalar u
         , HasBasis v, s ~ Scalar v
         , HasBasis w, s ~ Scalar w )
      ⇒ HasBasis (u,v,w) where
  type Basis (u,v,w) = Basis (u,(v,w))
  ⋯

Exercise: complete this instance definition (without peeking).

Bases in spaces

What about other vector spaces, particularly infinite dimensional ones? The result type for decompose is not convenient:

decompose ∷ v → [(Basis v, Scalar v)]

Moreover, its definition for pair types would have to be changed, e.g., to use interleaving instead of append. On the other hand, this type can be thought of as an association list, representing Basis v → Scalar v. Instead, we might use the function representation directly:

decompose ∷ v → (Basis v → Scalar v)

In that case, the definition of decompose on pairs is

decompose (u,v) = decompose u `either` decompose v

which beautifully mirrors the basis type definition:

type Basis (u,v) = Basis u `Either` Basis v

I guess we'd have to somehow extend the definition of recompose as well, or avoid it.

One example of an infinite dimensional vector space is a function over an infinite domain. The additive group and vector space instances follow a standard form for applicative functors applied to an additive group or vector space. In this case, the applicative functor is ((→) a).

instance AdditiveGroup v ⇒ AdditiveGroup (a → v) where
  zeroV   = pure   zeroV
  (^+^)   = liftA2 (^+^)
  negateV = fmap   negateV

instance VectorSpace v ⇒ VectorSpace (a → v) where
  type Scalar (a → v) = Scalar v
  (*^) s = fmap ((*^) s)

As a basis for a function space a→u, let's use the subset of functions that map one domain value to some basis vector for u and map all other domain values to zero. By linearly combining these functions, we can construct any function in a→u that is nonzero for finitely many domain values. If we stretch the notion of linear combinations beyond finite combinations, perhaps we can go further and cover at least the countably infinite domain types.

To represent these functions, it suffices to record the choice of domain element and the representation of the corresponding basis vector for u:

instance (Eq a, HasBasis u) ⇒ HasBasis (a → u) where
  type Basis (a → u) = (a, Basis u)

  basisValue (a,b) a' | a ≡ a'   = basisValue b
                      | otherwise = zeroV

The actual implementation is a bit more efficient, avoiding recomputation of basisValue b for each a'.

Decomposing a function into its coordinates is even simpler:

  decompose g (a,b) = decompose (g a) b

Some isomorphisms

This instance rule for functions will be applied repeatedly for curried functions. For instance,

Basis (a → b → u) ≡ (a, (b, Basis u))

The isomorphic uncurried form has an isomorphic basis:

Basis ((a,b) → u) ≡ ((a,b), Basis u)

Pairing in the range instead of domain gives rise to another pair of isomorphisms:

Basis (a → (b,c)) ≡ (a, Basis b `Either` Basis c)

Basis ((a → b, a → c)) ≡ (a, Basis b) `Either` (a, Basis c)

The rules for basis types look like logarithms, especially if we add a basis for ():

  type Basis () = Void

  type Basis (u,v) = Basis u `Either` Basis v

  type Basis (a → u) = (a, Basis u)

Compare with:

The rules are also essentially the same as the ones used for memo tries, but phrased in terms of logarithms instead of (explicit) exponents.