Wrong parameter for GroupExponentialRetraction

[olivierverdier]

Currently, GroupExponentialRetraction depends on a type ActionDirectionAndSide, but, unless I am mistaken, it should only depend on GroupActionSide.

Put differently, the GroupExponentialRetraction{LeftBackwardAction} and GroupExponentialRetraction{RightForwardAction} make no sense.

Indeed: in the code to retract, there is a call to inverse_translate_diff(G, p, p, X, conv), which is supposed to send the tangent vector X to the Lie algebra. But this only holds if conv is either LeftForwardAction or RightBackwardAction.

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Suggested solution:
Implement the generally useful translate_to_id which translates a tangent vector to the identity (to the Lie algebra) like so

_conv_from_side(::LeftSide) = LeftForwardAction()
_conv_from_side(::RightSide) = RightBackwardAction()

translate_to_id(G, p, X, side::GroupActionSide) = translate_diff(G, p, p, X, switch_side(_inv_conv_from_side(side)))

then change the implementation of GroupExponentialRetraction as

struct GroupExponentialRetraction{D<:GroupActionSide} <: AbstractRetractionMethod end

function GroupExponentialRetraction(side::GroupActionSide=LeftSide())
    return GroupExponentialRetraction{typeof(side)}()
end

group_action_side(::GroupExponentialRetraction{D}) where {D} = D()

then essentially keep the current implementation of retract as

function retract(
    ::TraitList{<:IsGroupManifold},
    G::AbstractDecoratorManifold,
    p,
    X,
    method::GroupExponentialRetraction,
)
    side = group_side(method)
    Xₑ = translate_to_id(G, p, X, side)
    pinvq = exp_lie(G, Xₑ)
    q = translate(G, p, pinvq, _conv_from_side(side))
    return q
end

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Note 1 there is probably exactly the same problem (solved in the same manner) for GroupLogarithmicInverseRetraction.

Note 2 inverse_translate_diff may be superfluous in general, or at least it should be implemented as inverse_translate_diff(G, p, q, X, conv) = inverse_translate_diff(G, p, q, X, switch_direction(conv)).

Let me know if the problem and/or the solution make any sense to you.

[mateuszbaran]

Currently, GroupExponentialRetraction depends on a type ActionDirectionAndSide, but, unless I am mistaken, it should only depend on GroupActionSide.

Put differently, the GroupExponentialRetraction{LeftBackwardAction} and GroupExponentialRetraction{RightForwardAction} make no sense.

You're right, GroupExponentialRetraction and GroupLogarithmicInverseRetraction should be restricted to left and right side.

Suggested solution:
Implement the generally useful translate_to_id which translates a tangent vector to the identity (to the Lie algebra) like so

This looks like the right solution 👍 . It's technically breaking but I'd consider it a bugfix so we could do it without a breaking release.

Note 2 inverse_translate_diff may be superfluous in general, or at least it should be implemented as inverse_translate_diff(G, p, q, X, conv) = inverse_translate_diff(G, p, q, X, switch_direction(conv)).

Yes, it is currently superfluous. Originally when it was introduced we didn't have right-forward and left-backward actions so it was useful but right now is should be deprecated in favor of translate_diff with these conventions.