Poincaré models for Hyperbolic space

[kellertuer]

I started testing how different formats for the same manifold work, i.e. besides the arrays, which are assumed to be form the hyperbolic model implicitly there is now

• the Hyperboloid model
• the Poincaré ball model
• the Poincaré half plane model (maybe better to name it half space)?

Where all have types for their MPoints and TVectors. All concerning converts have been implemented. Still todo

• exp/log/inner for the plane model
• check which models do not directly work, especially those that originally came from the embedding (which only holds for the first model)
• check whether we have to extend the allocate_result functions
• add tests.
• convert tangent vectors

[codecov]

Codecov Report

Merging #219 into master will decrease coverage by 0.32%.
The diff coverage is 91.89%.

@@            Coverage Diff             @@
##           master     #219      +/-   ##
==========================================
- Coverage   95.50%   95.17%   -0.33%     
==========================================
  Files          65       69       +4     
  Lines        4089     4248     +159     
==========================================
+ Hits         3905     4043     +138     
- Misses        184      205      +21     

Impacted Files	Coverage Δ
src/Manifolds.jl	100.00% <ø> (ø)
src/groups/group.jl	76.96% <0.00%> (-0.82%)	⬇️
src/manifolds/Stiefel.jl	92.72% <ø> (ø)
src/manifolds/Hyperbolic.jl	89.09% <91.30%> (+3.72%)	⬆️
src/manifolds/HyperbolicHyperboloid.jl	92.85% <92.85%> (ø)
src/manifolds/HyperbolicPoincareBall.jl	95.00% <95.00%> (ø)
src/manifolds/HyperbolicPoincareHalfspace.jl	95.00% <95.00%> (ø)
src/statistics.jl	99.42% <100.00%> (ø)
src/tests/tests_general.jl	98.54% <100.00%> (ø)
src/manifolds/SymmetricPositiveDefinite.jl	93.10% <0.00%> (-6.90%)	⬇️
... and 16 more

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[kellertuer]

I noticed something interesting:

While to define your own allocate_result boils down to defining a function for allocate_result(M, log, p, q), i.e. arguments after the function name),
for allocate_result_type one has to define allocate_result_type(M, log, (p,q)), i.e. putting the arguments in a tuple. Would it be reasonable to unify that to the first? Makes defining those nicer. But actually it only affects log/inverse_retract since they are specific in the type, since from two MPoints we have to specify the fitting TVector.
Maybe that could be globally done by defining a map from MPoint to TVector for these types?

So maybe something like vector_type(::Type{T}) where {T} = T for the general case and for specific own types vector_type(::Type{HyperboloidPoint}) = HyperboloidTVector for example? this would collapse a lot of my current allocation type definitions.

[mateuszbaran]

That's a good idea. There may be some minor things to work out though. Type conversion would have to be more precise (forwarding the type argument of the wrapper) like vector_type(::Type{HyperboloidPoint{T}}) where {T} = HyperboloidTVector{T} (not tested).

[kellertuer]

So now the main thing missing might be either look for log/exp in the different 2 other models or derive conversions for tangent vectors accordingly (taking the derivative of the conversion functions). And it might be worth doing both.

[kellertuer]

despite two conversions being done as double conversions, one conversion is missing, namely how to transform Poincaré ball tangent vectors to Hyperboloid ones. Still have to look for that formula (or derive it from the push forward of the isometry myself).

[kellertuer]

All methods should now be implemented, all formula documented (derived the last one by hand just now); still have to check whether the remaining exp/log/retract methods are failing due to a typo in the formulae or whether that's numerical instability.

[kellertuer]

It seems, that conversion between both Poincaré models still has a bug.

[kellertuer]

There is so much converting going on... I'll take a closer look at it later.

I convert everything to everything (and its 4 types) as well as points to points, point/vector to vector and point&vector to point&vector. We can omit the last case if that is not needed, especially if I change the point/vector->vector interface now anyways.

[mateuszbaran]

Looks OK now 👍 .

[mateuszbaran]

Oh well... it fails on MacOS but not Ubuntu or Windows...

[mateuszbaran]

I think it works now 🎉.