Group manifolds and actions

AbstractGroupManifold{𝔽,O<:AbstractGroupOperation} <: AbstractDecoratorManifold{𝔽}

Abstract type for a Lie group, a group that is also a smooth manifold with an AbstractGroupOperation, a smooth binary operation. AbstractGroupManifolds must implement at least inv, identity, compose, and translate_diff.

AbstractGroupOperation

Abstract type for smooth binary operations $∘$ on elements of a Lie group $\mathcal{G}$:

An operation can be either defined for a specific AbstractGroupManifold over number system 𝔽 or in general, by defining for an operation Op the following methods:

identity!(::AbstractGroupManifold{𝔽,Op}, q, q)
identity(::AbstractGroupManifold{𝔽,Op}, p)
inv!(::AbstractGroupManifold{𝔽,Op}, q, p)
inv(::AbstractGroupManifold{𝔽,Op}, p)
compose(::AbstractGroupManifold{𝔽,Op}, p, q)
compose!(::AbstractGroupManifold{𝔽,Op}, x, p, q)

Note that a manifold is connected with an operation by wrapping it with a decorator, AbstractGroupManifold. In typical cases the concrete wrapper GroupManifold can be used.

ActionDirection

Direction of action on a manifold, either LeftAction or RightAction.

AdditionOperation <: AbstractGroupOperation

Group operation that consists of simple addition.

GroupExponentialRetraction{D<:ActionDirection} <: AbstractRetractionMethod

Retraction using the group exponential group_exp "translated" to any point on the manifold.

For more details, see retract.

Constructor

GroupExponentialRetraction(conv::ActionDirection = LeftAction())

GroupLogarithmicInverseRetraction{D<:ActionDirection} <: AbstractInverseRetractionMethod

Retraction using the group logarithm group_log "translated" to any point on the manifold.

For more details, see inverse_retract.

Constructor

GroupLogarithmicInverseRetraction(conv::ActionDirection = LeftAction())

GroupManifold{𝔽,M<:Manifold{𝔽},O<:AbstractGroupOperation} <: AbstractGroupManifold{𝔽,O}

Decorator for a smooth manifold that equips the manifold with a group operation, thus making it a Lie group. See AbstractGroupManifold for more details.

Group manifolds by default forward metric-related operations to the wrapped manifold.

Constructor

GroupManifold(manifold, op)

Identity(G::AbstractGroupManifold, p)

The group identity element $e ∈ \mathcal{G}$ represented by point p.

LeftAction()

Left action of a group on a manifold.

MultiplicationOperation <: AbstractGroupOperation

Group operation that consists of multiplication.

RightAction()

Right action of a group on a manifold.

identity(G::AbstractGroupManifold, p)

Identity element $e ∈ \mathcal{G}$, such that for any element $p ∈ \mathcal{G}$, $p \circ e = e \circ p = p$. The returned element is of a similar type to p.

inv(G::AbstractGroupManifold, p)

Inverse $p^{-1} ∈ \mathcal{G}$ of an element $p ∈ \mathcal{G}$, such that $p \circ p^{-1} = p^{-1} \circ p = e ∈ \mathcal{G}$, where $e$ is the identity element of $\mathcal{G}$.

base_group(M::Manifold) -> AbstractGroupManifold

Un-decorate M until an AbstractGroupManifold is encountered. Return an error if the base_manifold is reached without encountering a group.

compose(G::AbstractGroupManifold, p, q)

Compose elements $p,q ∈ \mathcal{G}$ using the group operation $p \circ q$.

group_exp(G::AbstractGroupManifold, X)

Compute the group exponential of the Lie algebra element X.

Given an element $X ∈ 𝔤 = T_e \mathcal{G}$, where $e$ is the identity element of the group $\mathcal{G}$, and $𝔤$ is its Lie algebra, the group exponential is the map

such that for $t,s ∈ ℝ$, $γ(t) = \exp (t X)$ defines a one-parameter subgroup with the following properties:

group_exp(G::AbstractGroupManifold{𝔽,AdditionOperation}, X) where {𝔽}

Compute $q = X$.

group_exp(G::AbstractGroupManifold{𝔽,MultiplicationOperation}, X) where {𝔽}

For Number and AbstractMatrix types of X, compute the usual numeric/matrix exponential,

group_log(G::AbstractGroupManifold, q)

Compute the group logarithm of the group element q.

Given an element $q ∈ \mathcal{G}$, compute the right inverse of the group exponential map group_exp, that is, the element $\log q = X ∈ 𝔤 = T_e \mathcal{G}$, such that $q = \exp X$

group_log(G::AbstractGroupManifold{𝔽,AdditionOperation}, q) where {𝔽}

Compute $X = q$.

group_log(G::AbstractGroupManifold{𝔽,MultiplicationOperation}, q) where {𝔽}

For Number and AbstractMatrix types of q, compute the usual numeric/matrix logarithm:

where $e$ here is the identity element, that is, $1$ for numeric $q$ or the identity matrix $I_m$ for matrix $q ∈ ℝ^{m × m}$.

inverse_translate(G::AbstractGroupManifold, p, q)
inverse_translate(G::AbstractGroupManifold, p, q, conv::ActionDirection=LeftAction())

Inverse translate group element $q$ by $p$ with the inverse translation $τ_p^{-1}$ with the specified convention, either left ($L_p^{-1}$) or right ($R_p^{-1}$), defined as

inverse_translate_diff(G::AbstractGroupManifold, p, q, X)
inverse_translate_diff(G::AbstractGroupManifold, p, q, X, conv::ActionDirection=LeftAction())

For group elements $p, q ∈ \mathcal{G}$ and tangent vector $X ∈ T_q \mathcal{G}$, compute the action on $X$ of the differential of the inverse translation $τ_p$ by $p$, with the specified left or right convention. The differential transports vectors:

switch_direction(::ActionDirection)

Returns a RightAction when given a LeftAction and vice versa.

translate(G::AbstractGroupManifold, p, q)
translate(G::AbstractGroupManifold, p, q, conv::ActionDirection=LeftAction()])

Translate group element $q$ by $p$ with the translation $τ_p$ with the specified convention, either left ($L_p$) or right ($R_p$), defined as

translate_diff(G::AbstractGroupManifold, p, q, X)
translate_diff(G::AbstractGroupManifold, p, q, X, conv::ActionDirection=LeftAction())

For group elements $p, q ∈ \mathcal{G}$ and tangent vector $X ∈ T_q \mathcal{G}$, compute the action of the differential of the translation $τ_p$ by $p$ on $X$, with the specified left or right convention. The differential transports vectors:

inverse_retract(
    G::AbstractGroupManifold,
    p,
    X,
    method::GroupLogarithmicInverseRetraction{<:ActionDirection},
)

Compute the inverse retraction using the group logarithm group_log "translated" to any point on the manifold. With a group translation (translate) $τ_p$ in a specified direction, the retraction is

where $\log$ is the group logarithm (group_log), and $(\mathrm{d}τ_p)_e$ is the action of the differential of translation $τ_p$ evaluated at the identity element $e$ (see translate_diff).

retract(
    G::AbstractGroupManifold,
    p,
    X,
    method::GroupExponentialRetraction{<:ActionDirection},
)

Compute the retraction using the group exponential group_exp "translated" to any point on the manifold. With a group translation (translate) $τ_p$ in a specified direction, the retraction is

where $\exp$ is the group exponential (group_exp), and $(\mathrm{d}τ_p^{-1})_p$ is the action of the differential of inverse translation $τ_p^{-1}$ evaluated at $p$ (see inverse_translate_diff).

ProductGroup{𝔽,T} <: GroupManifold{𝔽,ProductManifold{T},ProductOperation}

Decorate a product manifold with a ProductOperation.

Each submanifold must also be an AbstractGroupManifold or a decorated instance of one. This type is mostly useful for equipping the direct product of group manifolds with an Identity element.

Constructor

ProductGroup(manifold::ProductManifold)

ProductOperation <: AbstractGroupOperation

Direct product group operation.

SemidirectProductGroup(N::GroupManifold, H::GroupManifold, A::AbstractGroupAction)

A group that is the semidirect product of a normal group $\mathcal{N}$ and a subgroup $\mathcal{H}$, written $\mathcal{G} = \mathcal{N} ⋊_θ \mathcal{H}$, where $θ: \mathcal{H} × \mathcal{N} → \mathcal{N}$ is an automorphism action of $\mathcal{H}$ on $\mathcal{N}$. The group $\mathcal{G}$ has the composition rule

and the inverse

SemidirectProductOperation(action::AbstractGroupAction)

Group operation of a semidirect product group. The operation consists of the operation opN on a normal subgroup N, the operation opH on a subgroup H, and an automorphism action of elements of H on N. Only the action is stored.

CircleGroup <: GroupManifold{Circle{ℂ},MultiplicationOperation}

The circle group is the complex circle (Circle(ℂ)) equipped with the group operation of complex multiplication (MultiplicationOperation).

SpecialOrthogonal{n} <: GroupManifold{ℝ,Rotations{n},MultiplicationOperation}

Special orthogonal group $\mathrm{SO}(n)$ represented by rotation matrices.

Constructor

SpecialOrthogonal(n)

TranslationGroup{T<:Tuple,𝔽} <: GroupManifold{Euclidean{T,𝔽},AdditionOperation}

Translation group $\mathrm{T}(n)$ represented by translation arrays.

Constructor

TranslationGroup(n₁,...,nᵢ; field = 𝔽)

Generate the translation group on $𝔽^{n₁,…,nᵢ}$ = Euclidean(n₁,...,nᵢ; field = 𝔽), which is isomorphic to the group itself.

SpecialEuclidean(n)

Special Euclidean group $\mathrm{SE}(n)$, the group of rigid motions.

$\mathrm{SE}(n)$ is the semidirect product of the TranslationGroup on $ℝ^n$ and SpecialOrthogonal(n)

where $θ$ is the canonical action of $\mathrm{SO}(n)$ on $\mathrm{T}(n)$ by vector rotation.

This constructor is equivalent to calling

Tn = TranslationGroup(n)
SOn = SpecialOrthogonal(n)
SemidirectProductGroup(Tn, SOn, RotationAction(Tn, SOn))

Points on $\mathrm{SE}(n)$ may be represented as points on the underlying product manifold $\mathrm{T}(n) × \mathrm{SO}(n)$. For group-specific functions, they may also be represented as affine matrices with size (n + 1, n + 1) (see affine_matrix), for which the group operation is MultiplicationOperation.

affine_matrix(G::SpecialEuclidean, p) -> AbstractMatrix

Represent the point $p ∈ \mathrm{SE}(n)$ as an affine matrix. For $p = (t, R) ∈ \mathrm{SE}(n)$, where $t ∈ \mathrm{T}(n), R ∈ \mathrm{SO}(n)$, the affine representation is the $n + 1 × n + 1$ matrix

See also screw_matrix for matrix representations of the Lie algebra.

group_exp(G::SpecialEuclidean{2}, X)

Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(2)$, where $b ∈ 𝔱(2)$ and $Ω ∈ 𝔰𝔬(2)$:

where $t ∈ \mathrm{T}(2)$, $R = \exp Ω$ is the group exponential on $\mathrm{SO}(2)$,

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

group_exp(G::SpecialEuclidean{3}, X)

Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(3)$, where $b ∈ 𝔱(3)$ and $Ω ∈ 𝔰𝔬(3)$:

where $t ∈ \mathrm{T}(3)$, $R = \exp Ω$ is the group exponential on $\mathrm{SO}(3)$,

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

group_exp(G::SpecialEuclidean{n}, X)

Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(n)$, where $b ∈ 𝔱(n)$ and $Ω ∈ 𝔰𝔬(n)$:

where $t ∈ \mathrm{T}(n)$ and $R = \exp Ω$ is the group exponential on $\mathrm{SO}(n)$.

In the screw_matrix representation, the group exponential is the matrix exponential (see group_exp).

group_log(G::SpecialEuclidean{2}, p)

Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(2)$, where $t ∈ \mathrm{T}(2)$ and $R ∈ \mathrm{SO}(2)$:

where $b ∈ 𝔱(2)$, $Ω = \log R ∈ 𝔰𝔬(2)$ is the group logarithm on $\mathrm{SO}(2)$,

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

group_log(G::SpecialEuclidean{3}, p)

Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(3)$, where $t ∈ \mathrm{T}(3)$ and $R ∈ \mathrm{SO}(3)$:

where $b ∈ 𝔱(3)$, $Ω = \log R ∈ 𝔰𝔬(3)$ is the group logarithm on $\mathrm{SO}(3)$,

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

group_log(G::SpecialEuclidean{n}, p) where {n}

Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(n)$, where $t ∈ \mathrm{T}(n)$ and $R ∈ \mathrm{SO}(n)$:

where $b ∈ 𝔱(n)$ and $Ω = \log R ∈ 𝔰𝔬(n)$ is the group logarithm on $\mathrm{SO}(n)$.

In the affine_matrix representation, the group logarithm is the matrix logarithm (see group_log):

screw_matrix(G::SpecialEuclidean, X) -> AbstractMatrix

Represent the Lie algebra element $X ∈ 𝔰𝔢(n) = T_e \mathrm{SE}(n)$ as a screw matrix. For $X = (b, Ω) ∈ 𝔰𝔢(n)$, where $Ω ∈ 𝔰𝔬(n) = T_e \mathrm{SO}(n)$, the screw representation is the $n + 1 × n + 1$ matrix

See also affine_matrix for matrix representations of the Lie group.

AbstractGroupAction

An abstract group action on a manifold.

apply!(A::AbstractGroupAction, q, a, p)

Apply action a to the point p with the rule specified by A. The result is saved in q.

apply(A::AbstractGroupAction, a, p)

Apply action a to the point p using map $τ_a$, specified by A. Unless otherwise specified, the right action is defined in terms of the left action:

apply_diff(A::AbstractGroupAction, a, p, X)

For group point $p ∈ \mathcal M$ and tangent vector $X ∈ T_p \mathcal M$, compute the action on $X$ of the differential of the action of $a ∈ \mathcal{G}$, specified by rule A. Written as $(\mathrm{d}τ_a)_p$, with the specified left or right convention, the differential transports vectors

base_group(A::AbstractGroupAction)

The group that acts in action A.

center_of_orbit(
    A::AbstractGroupAction,
    pts,
    p,
    mean_method::AbstractEstimationMethod = GradientDescentEstimation(),
)

Calculate an action element $a$ of action A that is the mean element of the orbit of p with respect to given set of points pts. The mean is calculated using the method mean_method.

The orbit of $p$ with respect to the action of a group $\mathcal{G}$ is the set

This function is useful for computing means on quotients of manifolds by a Lie group action.

direction(::AbstractGroupAction{AD}) -> AD

Get the direction of the action

g_manifold(A::AbstractGroupAction)

The manifold the action A acts upon.

inverse_apply!(A::AbstractGroupAction, q, a, p)

Apply inverse of action a to the point p with the rule specified by A. The result is saved in q.

inverse_apply(A::AbstractGroupAction, a, p)

Apply inverse of action a to the point p. The action is specified by A.

inverse_apply_diff(A::AbstractGroupAction, a, p, X)

For group point $p ∈ \mathcal M$ and tangent vector $X ∈ T_p \mathcal M$, compute the action on $X$ of the differential of the inverse action of $a ∈ \mathcal{G}$, specified by rule A. Written as $(\mathrm{d}τ_a^{-1})_p$, with the specified left or right convention, the differential transports vectors

optimal_alignment!(A::AbstractGroupAction, x, p, q)

Calculate an action element of action A that acts upon p to produce the element closest to q. The result is written to x.

optimal_alignment(A::AbstractGroupAction, p, q)

Calculate an action element $a$ of action A that acts upon p to produce the element closest to q in the metric of the G-manifold:

where $\mathcal{G}$ is the group that acts on the G-manifold $\mathcal M$.

GroupOperationAction(group::AbstractGroupManifold, AD::ActionDirection = LeftAction())

Action of a group upon itself via left or right translation.

RotationAction(
    M::Manifold,
    SOn::SpecialOrthogonal,
    AD::ActionDirection = LeftAction(),
)

Space of actions of the SpecialOrthogonal group $\mathrm{SO}(n)$ on a Euclidean-like manifold M of dimension n.

TranslationAction(
    M::Manifold,
    Rn::TranslationGroup,
    AD::ActionDirection = LeftAction(),
)

Space of actions of the TranslationGroup $\mathrm{T}(n)$ on a Euclidean-like manifold M.

The left and right actions are equivalent.

InvariantMetric{G<:Metric,D<:ActionDirection} <: Metric

Extend a metric on the Lie algebra of an AbstractGroupManifold to the whole group via translation in the specified direction.

Given a group $\mathcal{G}$ and a left- or right group translation map $τ$ on the group, a metric $g$ is $τ$-invariant if it has the inner product

for all $p,q ∈ \mathcal{G}$ and $X,Y ∈ T_p \mathcal{G}$, where $(\mathrm{d}τ_q)_p$ is the differential of translation by $q$ evaluated at $p$ (see translate_diff).

InvariantMetric constructs an (assumed) $τ$-invariant metric by extending the inner product of a metric $h_e$ on the Lie algebra to the whole group:

The convenient aliases LeftInvariantMetric and RightInvariantMetric are provided.

Constructor

InvariantMetric(metric::Metric, conv::ActionDirection = LeftAction())

LeftInvariantMetric(metric::Metric)

Alias for a left-InvariantMetric.

RightInvariantMetric(metric::Metric)

Alias for a right-InvariantMetric.

biinvariant_metric_dispatch(G::AbstractGroupManifold) -> Val

Return Val(true) if the metric on the manifold is bi-invariant, that is, if the metric is both left- and right-invariant (see invariant_metric_dispatch).

has_approx_invariant_metric(
    G::AbstractGroupManifold,
    p,
    X,
    Y,
    qs::AbstractVector,
    conv::ActionDirection = LeftAction();
    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:

This is necessary but not sufficient for invariance.

Optionally, kwargs passed to isapprox may be provided.

invariant_metric_dispatch(G::AbstractGroupManifold, conv::ActionDirection) -> Val

Return Val(true) if the metric on the group $\mathcal{G}$ is invariant under translations by the specified direction, that is, given a group $\mathcal{G}$, a left- or right group translation map $τ$, and a metric $g_e$ on the Lie algebra, a $τ$-invariant metric at any point $p ∈ \mathcal{G}$ is defined as a metric with the inner product

for $X,Y ∈ T_q \mathcal{G}$ and all $q ∈ \mathcal{G}$, where $(\mathrm{d}τ_q)_p$ is the differential of translation by $q$ evaluated at $p$ (see translate_diff).