Test Linearity

[oxinabox]

Pushforwards and pullbacks are requred to be linear operators. Which formally is defined as φ(𝛼x) = 𝛼φ(x) and φ(a+b) = φ(a) + φ(b) for a,b,x differentials/vector space elements, and 𝛼 being a scalar.

A important collory of that is φ(0) = 0 where two 0s are the additive identity for the relivent vector spaces. Which we can test with iszero(φ(zero(d)), since we always have access to a differential d, and all differentials must define zero and iszero.

Similarly we can test φ(𝛼x) = 𝛼φ(x) via φ(2d) == 2φ(d) since all differnetial define multiplication with a scalar. and we have one provided d.

We can't easily test φ(a+b) = φ(a) + φ(b) though, as we only have one differential. We can test φ(d+d) = φ(d) + φ(d) but that is less interesting. If we are generating them using rand_tangent we can easily generate another. Though perhaps we should be insisting that things define the addition of there natural differentials with the structural differentials that rand_tangent will most often provide (though sometimes rand_tangent's structural differential will have extra nondifferentiable fields so we can't trust it, unless we insist the user does actually overload it). Or we could avoid direct additon and do (primal + rand_tangent(d)) + d) though this is getting off-topic.. Possibly just skip that in initial implementation of this

The linear pullback can be extracted from the rrule via (f,primal...) -> (ds...) -> last(rrule(f, primal...))(ds...) (or equiv: (f,primal) -> last(rrule(f, primal))) The linear pushforward can be extracted from frulevia(f,primal) -> (ds...) -> frule(ds..., f, primal...)`

[willtebbutt]

We can't easily test φ(a+b) = φ(a) + φ(b) though, as we only have one differential.

Could we just introduce two scalars p and q s.t. p + q = 1, and check that

φ(d) = φ(p * d) + φ(q * d)

I think that would cover the cases we care about.

[oxinabox]

Fair, it isn't comprehensive but yeah i think:

6φ(d) = φ(2d) + φ(4d)

would catch a lot.

Just like iszero(φ(zero(d))