Improve `rand` for tangent vectors on `SpecialOrthogonal(2)`

[olivierverdier]

If M is defined as [0 1; -1 0], then

SO2 = SpecialOrthogonal(2)
rand(SO2; vector_at=identity_element(SO2)) # always return ±M/sqrt(2)

This contaminates products or semidirect products with SO(2), for instance, tangent vector of SpecialEuclidean(2) are not random either.

[olivierverdier]

Okay, it's random, so the title is wrong, but it is not normally distributed.

[mateuszbaran]

Yes, we should change the distribution to something like M .* randn()

[kellertuer]

I think this is really and nice random – rand ( https://docs.julialang.org/en/v1/stdlib/Random/#Base.rand ) does not say, that the randomness might be samples from a finite set.

But you are right, something more normally distributed would be nicer here. but getting distributions (and hence a proper generic rand) to work, cf. https://github.com/JuliaManifolds/Manifolds.jl/issues/57 is something we have in mind for a really long time, but did not yet have the time to get to.

Here, Mateusz remark should fix this for now at least.

[olivierverdier]

I think this is really and nice random – rand ( https://docs.julialang.org/en/v1/stdlib/Random/#Base.rand ) does not say, that the randomness might be samples from a finite set.

I see, that explains it. I should have been warned by the documentation: Usually [...] a Gaussian-like distribution for non-compact manifolds and tangent vectors, although it is not guaranteed.

So random vectors on a circle group returning zero (#686) is also fine, since it samples from a finite set of one element.

I guess the conclusion is that rand on manifolds is not really dependable for now. I'll try to roll out my own random functions instead. You can disregard these issues about random vector generation, then.

[mateuszbaran]

Well, it should be dependable and return something more reasonable. I will patch both this and CircleGroup.

[kellertuer]

If you have nice random functions, PRs are very welcome. But there is also a reason that we had the idea to start ManifoldMeasures.jl a few years back – these days maybe again more like ManifoldsDistributions.jl – since that business is nontrivial. And then we (just) need someone to approach such a package.

We can not guarantee much by now, but as Mateusz write, we can surely improve them a bit. I personally need the rand functions very often just to get some point (not far beyond memory allocation).

[mateuszbaran]

This should fix both issues: https://github.com/JuliaManifolds/Manifolds.jl/pull/689