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metric.jl
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metric.jl
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@doc raw"""
has_approx_invariant_metric(
G::AbstractDecoratorManifold,
p,
X,
Y,
qs::AbstractVector,
conv::ActionDirectionAndSide = LeftForwardAction();
kwargs...,
) -> Bool
Check whether the metric on the group $\mathcal{G}$ is (approximately) invariant using a set of predefined
points. Namely, for $p ∈ \mathcal{G}$, $X,Y ∈ T_p \mathcal{G}$, a metric $g$, and a
translation map $τ_q$ in the specified direction, check for each $q ∈ \mathcal{G}$ that the
following condition holds:
````math
g_p(X, Y) ≈ g_{τ_q p}((\mathrm{d}τ_q)_p X, (\mathrm{d}τ_q)_p Y).
````
This is necessary but not sufficient for invariance.
Optionally, `kwargs` passed to `isapprox` may be provided.
"""
has_approx_invariant_metric(
::AbstractDecoratorManifold,
p,
X,
Y,
qs,
::ActionDirectionAndSide,
)
@trait_function has_approx_invariant_metric(
M::AbstractDecoratorManifold,
p,
X,
Y,
qs,
conv::ActionDirectionAndSide=LeftForwardAction();
kwargs...,
)
function has_approx_invariant_metric(
::TraitList{<:IsGroupManifold},
M::AbstractDecoratorManifold,
p,
X,
Y,
qs,
conv::ActionDirectionAndSide;
kwargs...,
)
gpXY = inner(M, p, X, Y)
for q in qs
τq = translate(M, q, p, conv)
dτqX = translate_diff(M, q, p, X, conv)
dτqY = translate_diff(M, q, p, Y, conv)
isapprox(gpXY, inner(M, τq, dτqX, dτqY); kwargs...) || return false
end
return true
end
"""
direction(::AbstractDecoratorManifold) -> AD
Get the direction of the action a certain Lie group with its implicit metric has.
"""
direction(::AbstractDecoratorManifold)
@trait_function direction(M::AbstractDecoratorManifold)
function direction(::TraitList{HasLeftInvariantMetric}, ::AbstractDecoratorManifold)
return LeftAction()
end
function direction(::TraitList{HasRightInvariantMetric}, ::AbstractDecoratorManifold)
return RightAction()
end
@trait_function direction_and_side(M::AbstractDecoratorManifold)
function direction_and_side(
::TraitList{HasLeftInvariantMetric},
::AbstractDecoratorManifold,
)
return LeftForwardAction()
end
function direction_and_side(
::TraitList{HasRightInvariantMetric},
::AbstractDecoratorManifold,
)
return RightBackwardAction()
end
function exp(::TraitList{HasLeftInvariantMetric}, M::AbstractDecoratorManifold, p, X)
return retract(M, p, X, GroupExponentialRetraction(LeftForwardAction()))
end
function exp(
::TraitList{HasLeftInvariantMetric},
M::AbstractDecoratorManifold,
p,
X,
t::Number,
)
return retract(M, p, X, t, GroupExponentialRetraction(LeftForwardAction()))
end
function exp!(::TraitList{HasLeftInvariantMetric}, M::AbstractDecoratorManifold, q, p, X)
return retract!(M, q, p, X, GroupExponentialRetraction(LeftForwardAction()))
end
function exp!(
::TraitList{HasLeftInvariantMetric},
M::AbstractDecoratorManifold,
q,
p,
X,
t::Number,
)
return retract!(M, q, p, X, t, GroupExponentialRetraction(LeftForwardAction()))
end
function exp(::TraitList{HasRightInvariantMetric}, M::AbstractDecoratorManifold, p, X)
return retract(M, p, X, GroupExponentialRetraction(RightBackwardAction()))
end
function exp(
::TraitList{HasRightInvariantMetric},
M::AbstractDecoratorManifold,
p,
X,
t::Number,
)
return retract(M, p, X, t, GroupExponentialRetraction(RightBackwardAction()))
end
function exp!(::TraitList{HasRightInvariantMetric}, M::AbstractDecoratorManifold, q, p, X)
return retract!(M, q, p, X, GroupExponentialRetraction(RightBackwardAction()))
end
function exp!(
::TraitList{HasRightInvariantMetric},
M::AbstractDecoratorManifold,
q,
p,
X,
t::Number,
)
return retract!(M, q, p, X, t, GroupExponentialRetraction(RightBackwardAction()))
end
function exp(::TraitList{HasBiinvariantMetric}, M::MetricManifold, p, X)
return exp(M.manifold, p, X)
end
function exp(::TraitList{HasBiinvariantMetric}, M::MetricManifold, p, X, t::Number)
return exp(M.manifold, p, X, t)
end
function exp!(::TraitList{HasBiinvariantMetric}, M::MetricManifold, q, p, X)
return exp!(M.manifold, q, p, X)
end
function exp!(::TraitList{HasBiinvariantMetric}, M::MetricManifold, q, p, X, t::Number)
return exp!(M.manifold, q, p, X, t)
end
function get_coordinates(
t::TraitList{IT},
M::MetricManifold,
p,
X,
B::AbstractBasis,
) where {IT<:AbstractInvarianceTrait}
conv = direction_and_side(t, M)
Xₑ = inverse_translate_diff(M, p, p, X, conv)
return get_coordinates_lie(next_trait(t), M, Xₑ, B)
end
function get_coordinates!(
t::TraitList{IT},
M::MetricManifold,
c,
p,
X,
B::AbstractBasis,
) where {IT<:AbstractInvarianceTrait}
conv = direction_and_side(t, M)
Xₑ = inverse_translate_diff(M, p, p, X, conv)
return get_coordinates_lie!(next_trait(t), M, c, Xₑ, B)
end
function get_vector(
t::TraitList{IT},
M::MetricManifold,
p,
c,
B::AbstractBasis,
) where {IT<:AbstractInvarianceTrait}
conv = direction_and_side(t, M)
Xₑ = get_vector_lie(next_trait(t), M, c, B)
return translate_diff(M, p, Identity(M), Xₑ, conv)
end
function get_vector!(
t::TraitList{IT},
M::MetricManifold,
X,
p,
c,
B::AbstractBasis,
) where {IT<:AbstractInvarianceTrait}
conv = direction_and_side(t, M)
Xₑ = get_vector_lie(next_trait(t), M, c, B)
return translate_diff!(M, X, p, Identity(M), Xₑ, conv)
end
@trait_function has_invariant_metric(
M::AbstractDecoratorManifold,
op::ActionDirectionAndSide,
)
# Fallbacks / false
has_invariant_metric(::AbstractManifold, op::ActionDirectionAndSide) = false
function has_invariant_metric(
::TraitList{<:HasLeftInvariantMetric},
::AbstractDecoratorManifold,
::LeftForwardAction,
)
return true
end
function has_invariant_metric(
::TraitList{<:HasRightInvariantMetric},
::AbstractDecoratorManifold,
::RightBackwardAction,
)
return true
end
@trait_function has_biinvariant_metric(M::AbstractDecoratorManifold)
# fallback / default: false
has_biinvariant_metric(::AbstractManifold) = false
function has_biinvariant_metric(
::TraitList{<:HasBiinvariantMetric},
::AbstractDecoratorManifold,
)
return true
end
function inner(
t::TraitList{IT},
M::AbstractDecoratorManifold,
p,
X,
Y,
) where {IT<:AbstractInvarianceTrait}
conv = direction_and_side(t, M)
Xₑ = inverse_translate_diff(M, p, p, X, conv)
Yₑ = inverse_translate_diff(M, p, p, Y, conv)
return inner(next_trait(t), M, Identity(M), Xₑ, Yₑ)
end
function inner(
t::TraitList{<:IsGroupManifold},
M::AbstractDecoratorManifold,
::Identity,
X,
Y,
)
return inner(next_trait(t), M, identity_element(M), X, Y)
end
function inverse_translate_diff(
::TraitList{IsMetricManifold},
M::MetricManifold,
p,
q,
X,
conv::ActionDirectionAndSide,
)
return inverse_translate_diff(M.manifold, p, q, X, conv)
end
function inverse_translate_diff!(
::TraitList{IsMetricManifold},
M::MetricManifold,
Y,
p,
q,
X,
conv::ActionDirectionAndSide,
)
return inverse_translate_diff!(M.manifold, Y, p, q, X, conv)
end
function log(::TraitList{HasLeftInvariantMetric}, M::AbstractDecoratorManifold, p, q)
return inverse_retract(M, p, q, GroupLogarithmicInverseRetraction(LeftForwardAction()))
end
function log!(::TraitList{HasLeftInvariantMetric}, M::AbstractDecoratorManifold, X, p, q)
return inverse_retract!(
M,
X,
p,
q,
GroupLogarithmicInverseRetraction(LeftForwardAction()),
)
end
function log(::TraitList{HasRightInvariantMetric}, M::AbstractDecoratorManifold, p, q)
return inverse_retract(
M,
p,
q,
GroupLogarithmicInverseRetraction(RightBackwardAction()),
)
end
function log!(::TraitList{HasRightInvariantMetric}, M::AbstractDecoratorManifold, X, p, q)
return inverse_retract!(
M,
X,
p,
q,
GroupLogarithmicInverseRetraction(RightBackwardAction()),
)
end
function log(::TraitList{HasBiinvariantMetric}, M::MetricManifold, p, q)
return log(M.manifold, p, q)
end
function log!(::TraitList{HasBiinvariantMetric}, M::MetricManifold, X, p, q)
return log!(M.manifold, X, p, q)
end
function LinearAlgebra.norm(
t::TraitList{IT},
M::AbstractDecoratorManifold,
p,
X,
) where {IT<:AbstractInvarianceTrait}
conv = direction_and_side(t, M)
Xₑ = inverse_translate_diff(M, p, p, X, conv)
return norm(next_trait(t), M, Identity(M), Xₑ)
end
function LinearAlgebra.norm(
t::TraitList{<:IsGroupManifold},
M::AbstractDecoratorManifold,
::Identity,
X,
)
return norm(next_trait(t), M, identity_element(M), X)
end
function translate_diff(
::TraitList{IsMetricManifold},
M::MetricManifold,
p,
q,
X,
conv::ActionDirectionAndSide,
)
return translate_diff(M.manifold, p, q, X, conv)
end
function translate_diff!(
::TraitList{IsMetricManifold},
M::MetricManifold,
Y,
p,
q,
X,
conv::ActionDirectionAndSide,
)
return translate_diff!(M.manifold, Y, p, q, X, conv)
end
"""
LeftInvariantMetric <: AbstractMetric
An [`AbstractMetric`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/manifolds.html#ManifoldsBase.AbstractMetric)
that changes the metric of a Lie group to the left-invariant
metric obtained by left-translations to the identity. Adds the
[`HasLeftInvariantMetric`](@ref) trait.
"""
struct LeftInvariantMetric <: AbstractMetric end
"""
RightInvariantMetric <: AbstractMetric
An [`AbstractMetric`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/manifolds.html#ManifoldsBase.AbstractMetric)
that changes the metric of a Lie group to the right-invariant
metric obtained by right-translations to the identity. Adds the
[`HasRightInvariantMetric`](@ref) trait.
"""
struct RightInvariantMetric <: AbstractMetric end
@inline function active_traits(
f,
::MetricManifold{<:Any,<:AbstractManifold,LeftInvariantMetric},
args...,
)
return merge_traits(HasLeftInvariantMetric(), IsExplicitDecorator())
end
@inline function active_traits(
f,
::MetricManifold{<:Any,<:AbstractManifold,RightInvariantMetric},
args...,
)
return merge_traits(HasRightInvariantMetric(), IsExplicitDecorator())
end
direction(::LeftInvariantMetric) = LeftAction()
direction(::RightInvariantMetric) = RightAction()