Meta Manifolds

AbstractPowerManifold{𝔽,M,TPR} <: AbstractManifold{𝔽}

An abstract AbstractManifold to represent manifolds that are build as powers of another AbstractManifoldM with representation type TPR, a subtype of AbstractPowerRepresentation.

AbstractPowerRepresentation

An abstract representation type of points and tangent vectors on a power manifold.

NestedPowerRepresentation

Representation of points and tangent vectors on a power manifold using arrays of size equal to TSize of a PowerManifold. Each element of such array stores a single point or tangent vector.

For modifying operations, each element of the outer array is modified in-place, differently than in NestedReplacingPowerRepresentation.

NestedReplacingPowerRepresentation

Representation of points and tangent vectors on a power manifold using arrays of size equal to TSize of a PowerManifold. Each element of such array stores a single point or tangent vector.

For modifying operations, each element of the outer array is replaced using non-modifying operations, differently than for NestedReplacingPowerRepresentation.

PowerBasisData{TB<:AbstractArray}

Data storage for an array of basis data.

PowerManifold{𝔽,TM<:AbstractManifold,TSize,TPR<:AbstractPowerRepresentation} <: AbstractPowerManifold{𝔽,TM}

The power manifold $\mathcal M^{n_1× n_2 × … × n_d}$ with power geometry. TSize defines the number of elements along each axis, either statically using TypeParameter or storing it in a field.

For example, a manifold-valued time series would be represented by a power manifold with $d$ equal to 1 and $n_1$ equal to the number of samples. A manifold-valued image (for example in diffusion tensor imaging) would be represented by a two-axis power manifold ($d=2$) with $n_1$ and $n_2$ equal to width and height of the image.

While the size of the manifold is static, points on the power manifold would not be represented by statically-sized arrays.

Constructor

PowerManifold(M::PowerManifold, N_1, N_2, ..., N_d; parameter::Symbol=:field)
PowerManifold(M::AbstractManifold, NestedPowerRepresentation(), N_1, N_2, ..., N_d; parameter::Symbol=:field)
M^(N_1, N_2, ..., N_d)

Generate the power manifold $M^{N_1 × N_2 × … × N_d}$. By default, a PowerManifold is expanded further, i.e. for M=PowerManifold(N, 3)PowerManifold(M, 2) is equivalent to PowerManifold(N, 3, 2). Points are then 3×2 matrices of points on N. Providing a NestedPowerRepresentation as the second argument to the constructor can be used to nest manifold, i.e. PowerManifold(M, NestedPowerRepresentation(), 2) represents vectors of length 2 whose elements are vectors of length 3 of points on N in a nested array representation.

Since there is no default AbstractPowerRepresentation within this interface, the ^ operator is only available for PowerManifolds and concatenates dimensions.

parameter: whether a type parameter should be used to store n. By default size is stored in a field. Value can either be :field or :type.

copyto!(M::PowerManifoldNested, Y, p, X)

Copy the values elementwise, i.e. call copyto!(M.manifold, B, a, A) for all elements A, a and B of X, p, and Y, respectively.

copyto!(M::PowerManifoldNested, q, p)

Copy the values elementwise, i.e. call copyto!(M.manifold, b, a) for all elements a and b of p and q, respectively.

exp(M::AbstractPowerManifold, p, X)

Compute the exponential map from p in direction X on the AbstractPowerManifoldM, which can be computed using the base manifolds exponential map elementwise.

getindex(p, M::AbstractPowerManifold, i::Union{Integer,Colon,AbstractVector}...)
p[M::AbstractPowerManifold, i...]

Access the element(s) at index [i...] of a point p on an AbstractPowerManifoldM by linear or multidimensional indexing. See also Array Indexing in Julia.

log(M::AbstractPowerManifold, p, q)

Compute the logarithmic map from p to q on the AbstractPowerManifoldM, which can be computed using the base manifolds logarithmic map elementwise.

setindex!(q, p, M::AbstractPowerManifold, i::Union{Integer,Colon,AbstractVector}...)
q[M::AbstractPowerManifold, i...] = p

Set the element(s) at index [i...] of a point q on an AbstractPowerManifoldM by linear or multidimensional indexing to q. See also Array Indexing in Julia.

view(p, M::PowerManifoldNested, i::Union{Integer,Colon,AbstractVector}...)

Get the view of the element(s) at index [i...] of a point p on an AbstractPowerManifoldM by linear or multidimensional indexing.

norm(M::AbstractPowerManifold, p, X)

Compute the norm of X from the tangent space of p on an AbstractPowerManifoldM, i.e. from the element wise norms the Frobenius norm is computed.

Y = Weingarten(M::AbstractPowerManifold, p, X, V)
Weingarten!(M::AbstractPowerManifold, Y, p, X, V)

Since the metric decouples, also the computation of the Weingarten map $\mathcal W_p$ can be computed elementwise on the single elements of the PowerManifoldM.

_allocate_access_nested(M::PowerManifoldNested, y, i)

Helper function for allocate_result on PowerManifoldNested. In allocation y can be a number in which case _access_nested wouldn't work.

_parameter_symbol(M::PowerManifold)

Return :field if size of PowerManifoldM is stored in a field and :type if in a TypeParameter.

change_metric(M::AbstractPowerManifold, ::AbstractMetric, p, X)

Since the metric on a power manifold decouples, the change of metric can be done elementwise.

change_representer(M::AbstractPowerManifold, ::AbstractMetric, p, X)

Since the metric on a power manifold decouples, the change of a representer can be done elementwise

check_point(M::AbstractPowerManifold, p; kwargs...)

Check whether p is a valid point on an AbstractPowerManifoldM, i.e. each element of p has to be a valid point on the base manifold. If p is not a point on M a CompositeManifoldError consisting of all error messages of the components, for which the tests fail is returned.

The tolerance for the last test can be set using the kwargs....

check_power_size(M, p)
check_power_size(M, p, X)

Check whether phas the right size to represent points onM`` generically, i.e. just checking the overall sizes, not the individual ones per manifold.

check_vector(M::AbstractPowerManifold, p, X; kwargs... )

Check whether X is a tangent vector to p an the AbstractPowerManifoldM, i.e. atfer check_point(M, p), and all projections to base manifolds must be respective tangent vectors. If X is not a tangent vector to p on M a CompositeManifoldError consisting of all error messages of the components, for which the tests fail is returned.

The tolerance for the last test can be set using the kwargs....

default_inverse_retraction_method(M::PowerManifold)

Use the default inverse retraction method of the internal M.manifold also in defaults of functions defined for the power manifold, meaning that this is used elementwise.

default_retraction_method(M::PowerManifold)

Use the default retraction method of the internal M.manifold also in defaults of functions defined for the power manifold, meaning that this is used elementwise.

default_vector_transport_method(M::PowerManifold)

Use the default vector transport method of the internal M.manifold also in defaults of functions defined for the power manifold, meaning that this is used elementwise.

distance(M::AbstractPowerManifold, p, q)

Compute the distance between q and p on an AbstractPowerManifold, i.e. from the element wise distances the Forbenius norm is computed.

get_component(M::AbstractPowerManifold, p, idx...)

Get the component of a point p on an AbstractPowerManifoldM at index idx.

injectivity_radius(M::AbstractPowerManifold[, p])

the injectivity radius on an AbstractPowerManifold is for the global case equal to the one of its base manifold. For a given point p it's equal to the minimum of all radii in the array entries.

inner(M::AbstractPowerManifold, p, X, Y)

Compute the inner product of X and Y from the tangent space at p on an AbstractPowerManifoldM, i.e. for each arrays entry the tangent vector entries from X and Y are in the tangent space of the corresponding element from p. The inner product is then the sum of the elementwise inner products.

inverse_retract(M::AbstractPowerManifold, p, q, m::AbstractInverseRetractionMethod)

Compute the inverse retraction from p with respect to q on an AbstractPowerManifoldM using an AbstractInverseRetractionMethod. Then this method is performed elementwise, so the inverse retraction method has to be one that is available on the base AbstractManifold.

is_flat(M::AbstractPowerManifold)

Return true if AbstractPowerManifold is flat. It is flat if and only if the wrapped manifold is flat.

manifold_dimension(M::PowerManifold)

Returns the manifold-dimension of an PowerManifoldM$=\mathcal N = (\mathcal M)^{n_1,…,n_d}$, i.e. with $n=(n_1,…,n_d)$ the array size of the power manifold and $d_{\mathcal M}$ the dimension of the base manifold $\mathcal M$, the manifold is of dimension

\[\dim(\mathcal N) = \dim(\mathcal M)\prod_{i=1}^d n_i = n_1n_2⋅…⋅ n_d \dim(\mathcal M).\]

power_dimensions(M::PowerManifold)

return the power of M,

project(M::AbstractPowerManifold, p, X)

Project the point X onto the tangent space at p on the AbstractPowerManifoldM by projecting all components.

project(M::AbstractPowerManifold, p)

Project the point p from the embedding onto the AbstractPowerManifoldM by projecting all components.

retract(M::AbstractPowerManifold, p, X, method::AbstractRetractionMethod)

Compute the retraction from p with tangent vector X on an AbstractPowerManifoldM using a AbstractRetractionMethod. Then this method is performed elementwise, so the retraction method has to be one that is available on the base AbstractManifold.

riemann_tensor(M::AbstractPowerManifold, p, X, Y, Z)

Compute the Riemann tensor at point from p with tangent vectors X, Y and Z on the AbstractPowerManifoldM.

sectional_curvature(M::AbstractPowerManifold, p, X, Y)

Compute the sectional curvature of a power manifold manifold $\mathcal M$ at a point $p \in \mathcal M$ on two linearly independent tangent vectors at $p$. It may be 0 for if projections of X and Y on subspaces corresponding to component manifolds are not linearly independent.

sectional_curvature_max(M::AbstractPowerManifold)

Upper bound on sectional curvature of AbstractPowerManifoldM. It is the maximum of sectional curvature of the wrapped manifold and 0 in case there are two or more component manifolds, as the sectional curvature corresponding to the plane spanned by vectors (X_1, 0, ... 0) and (0, X_2, 0, ..., 0) is 0.

sectional_curvature_min(M::AbstractPowerManifold)

Lower bound on sectional curvature of AbstractPowerManifoldM. It is the minimum of sectional curvature of the wrapped manifold and 0 in case there are two or more component manifolds, as the sectional curvature corresponding to the plane spanned by vectors (X_1, 0, ... 0) and (0, X_2, 0, ..., 0) is 0.

set_component!(M::AbstractPowerManifold, q, p, idx...)

Set the component of a point q on an AbstractPowerManifoldM at index idx to p, which itself is a point on the AbstractManifold the power manifold is build on.

vector_transport_to(M::AbstractPowerManifold, p, X, q, method::AbstractVectorTransportMethod)

Compute the vector transport the tangent vector Xat p to q on the PowerManifoldM using an AbstractVectorTransportMethodm. This method is performed elementwise, i.e. the method m has to be implemented on the base manifold.

InverseProductRetraction(retractions::AbstractInverseRetractionMethod...)

Product inverse retraction of inverse retractions. Works on ProductManifold.

ProductBasisData

A typed tuple to store tuples of data of stored/precomputed bases for a ProductManifold.

ProductManifold{𝔽,TM<:Tuple} <: AbstractManifold{𝔽}

Product manifold $M_1 × M_2 × … × M_n$ with product geometry.

Constructor

ProductManifold(M_1, M_2, ..., M_n)

generates the product manifold $M_1 × M_2 × … × M_n$. Alternatively, the same manifold can be contructed using the × operator: M_1 × M_2 × M_3.

ProductMetric <: AbstractMetric

A type to represent the product of metrics for a ProductManifold.

ProductRetraction(retractions::AbstractRetractionMethod...)

Product retraction of retractions. Works on ProductManifold.

ProductVectorTransport(methods::AbstractVectorTransportMethod...)

Product vector transport type of methods. Works on ProductManifold.

exp(M::ProductManifold, p, X)

compute the exponential map from p in the direction of X on the ProductManifoldM, which is the elementwise exponential map on the internal manifolds that build M.

getindex(M::ProductManifold, i)
M[i]

access the ith manifold component from the ProductManifoldM.

log(M::ProductManifold, p, q)

Compute the logarithmic map from p to q on the ProductManifoldM, which can be computed using the logarithmic maps of the manifolds elementwise.

×(m, n)
cross(m, n)
cross(m1, m2, m3,...)

Return the InverseProductRetraction For two or more AbstractInverseRetractionMethods, where for the case that one of them is a InverseProductRetraction itself, the other is either prepended (if r is a product) or appenden (if s) is. If both InverseProductRetractions, they are combined into one keeping the order.

×(M, N)
cross(M, N)
cross(M1, M2, M3,...)

Return the ProductManifold For two AbstractManifolds M and N, where for the case that one of them is a ProductManifold itself, the other is either prepended (if N is a product) or appenden (if M) is. If both are product manifold, they are combined into one product manifold, keeping the order.

For the case that more than one is a product manifold of these is build with the same approach as above

×(m, n)
cross(m, n)
cross(m1, m2, m3,...)

Return the ProductRetraction For two or more AbstractRetractionMethods, where for the case that one of them is a ProductRetraction itself, the other is either prepended (if m is a product) or appenden (if n) is. If both ProductRetractions, they are combined into one keeping the order.

×(m, n)
cross(m, n)
cross(m1, m2, m3,...)

Return the ProductVectorTransport For two or more AbstractVectorTransportMethods, where for the case that one of them is a ProductVectorTransport itself, the other is either prepended (if r is a product) or appenden (if s) is. If both ProductVectorTransports, they are combined into one keeping the order.

norm(M::ProductManifold, p, X)

Compute the norm of X from the tangent space of p on the ProductManifold, i.e. from the element wise norms the 2-norm is computed.

Y = Weingarten(M::ProductManifold, p, X, V)
Weingarten!(M::ProductManifold, Y, p, X, V)

Since the metric decouples, also the computation of the Weingarten map $\mathcal W_p$ can be computed elementwise on the single elements of the ProductManifoldM.

change_metric(M::ProductManifold, ::AbstractMetric, p, X)

Since the metric on a product manifold decouples, the change of metric can be done elementwise.

change_representer(M::ProductManifold, ::AbstractMetric, p, X)

Since the metric on a product manifold decouples, the change of a representer can be done elementwise

check_point(M::ProductManifold, p; kwargs...)

Check whether p is a valid point on the ProductManifoldM. If p is not a point on M a CompositeManifoldError.consisting of all error messages of the components, for which the tests fail is returned.

The tolerance for the last test can be set using the kwargs....

check_size(M::ProductManifold, p; kwargs...)

Check whether p is of valid size on the ProductManifoldM. If p has components of wrong size a CompositeManifoldError.consisting of all error messages of the components, for which the tests fail is returned.

The tolerance for the last test can be set using the kwargs....

check_vector(M::ProductManifold, p, X; kwargs... )

Check whether X is a tangent vector to p on the ProductManifoldM, i.e. all projections to base manifolds must be respective tangent vectors. If X is not a tangent vector to p on M a CompositeManifoldError.consisting of all error messages of the components, for which the tests fail is returned.

The tolerance for the last test can be set using the kwargs....

distance(M::ProductManifold, p, q)

Compute the distance between two points p and q on the ProductManifoldM, which is the 2-norm of the elementwise distances on the internal manifolds that build M.

get_component(M::ProductManifold, p, i)

Get the ith component of a point p on a ProductManifoldM.

injectivity_radius(M::ProductManifold)
injectivity_radius(M::ProductManifold, x)

Compute the injectivity radius on the ProductManifold, which is the minimum of the factor manifolds.

inner(M::ProductManifold, p, X, Y)

compute the inner product of two tangent vectors X, Y from the tangent space at p on the ProductManifoldM, which is just the sum of the internal manifolds that build M.

inverse_retract(M::ProductManifold, p, q, m::AbstractInverseRetractionMethod)

Compute the inverse retraction from p with respect to q on the ProductManifoldM using an AbstractInverseRetractionMethod, which is used on each manifold of the product.

inverse_retract(M::ProductManifold, p, q, m::InverseProductRetraction)

Compute the inverse retraction from p with respect to q on the ProductManifoldM using an InverseProductRetraction, which by default encapsulates a inverse retraction for each manifold of the product. Then this method is performed elementwise, so the encapsulated inverse retraction methods have to be available per factor.

is_flat(::ProductManifold)

Return true if and only if all component manifolds of ProductManifoldM are flat.

manifold_dimension(M::ProductManifold)

Return the manifold dimension of the ProductManifold, which is the sum of the manifold dimensions the product is made of.

number_of_components(M::ProductManifold{<:NTuple{N,Any}}) where {N}

Calculate the number of manifolds multiplied in the given ProductManifoldM.

retract(M::ProductManifold, p, X, m::AbstractRetractionMethod)

Compute the retraction from p with tangent vector X on the ProductManifoldM using the AbstractRetractionMethodm on every manifold.

retract(M::ProductManifold, p, X, m::ProductRetraction)

Compute the retraction from p with tangent vector X on the ProductManifoldM using an ProductRetraction, which by default encapsulates retractions of the base manifolds. Then this method is performed elementwise, so the encapsulated retractions method has to be one that is available on the manifolds.

riemann_tensor(M::ProductManifold, p, X, Y, Z)

Compute the Riemann tensor at point from p with tangent vectors X, Y and Z on the ProductManifoldM.

sectional_curvature(M::ProductManifold, p, X, Y)

Compute the sectional curvature of a manifold $\mathcal M$ at a point $p \in \mathcal M$ on two linearly independent tangent vectors at $p$. It may be 0 for a product of non-flat manifolds if projections of X and Y on subspaces corresponding to component manifolds are not linearly independent.

sectional_curvature_max(M::ProductManifold)

Upper bound on sectional curvature of ProductManifoldM. It is the maximum of sectional curvatures of component manifolds and 0 in case there are two or more component manifolds, as the sectional curvature corresponding to the plane spanned by vectors (X_1, 0) and (0, X_2) is 0.

sectional_curvature_min(M::ProductManifold)

Lower bound on sectional curvature of ProductManifoldM. It is the minimum of sectional curvatures of component manifolds and 0 in case there are two or more component manifolds, as the sectional curvature corresponding to the plane spanned by vectors (X_1, 0) and (0, X_2) is 0.

select_from_tuple(t::NTuple{N, Any}, positions::Val{P})

Selects elements of tuple t at positions specified by the second argument. For example select_from_tuple(("a", "b", "c"), Val((3, 1, 1))) returns ("c", "a", "a").

set_component!(M::ProductManifold, q, p, i)

Set the ith component of a point q on a ProductManifoldM to p, where p is a point on the AbstractManifold this factor of the product manifold consists of.

submanifold(M::ProductManifold, i::Integer)

Extract the ith factor of the product manifold M.

submanifold(M::ProductManifold, i::Val)
submanifold(M::ProductManifold, i::AbstractVector)

Extract the factor of the product manifold M indicated by indices in i. For example, for i equal to Val((1, 3)) the product manifold constructed from the first and the third factor is returned.

The version with AbstractVector is not type-stable, for better preformance use Val.

submanifold_component(M::AbstractManifold, p, i::Integer)
submanifold_component(M::AbstractManifold, p, ::Val{i}) where {i}
submanifold_component(p, i::Integer)
submanifold_component(p, ::Val{i}) where {i}

Project the product array p on M to its ith component. A new array is returned.

submanifold_components(M::AbstractManifold, p)
submanifold_components(p)

Get the projected components of p on the submanifolds of M. The components are returned in a Tuple.

vector_transport_to(M::ProductManifold, p, X, q, m::AbstractVectorTransportMethod)

Compute the vector transport the tangent vector X at p to q on the ProductManifoldM using an AbstractVectorTransportMethodm on each manifold.

vector_transport_to(M::ProductManifold, p, X, q, m::ProductVectorTransport)

Compute the vector transport the tangent vector X at p to q on the ProductManifoldM using a ProductVectorTransportm.

ziptuples(a, b[, c[, d[, e]]])

Zips tuples a, b, and remaining in a fast, type-stable way. If they have different lengths, the result is trimmed to the length of the shorter tuple.

Fiber{𝔽,TFiber<:FiberType,TM<:AbstractManifold{𝔽},TX} <: AbstractManifold{𝔽}

A fiber of a fiber bundle at a point p on the manifold.

This fiber itself is also a manifold. For vector fibers it's by default flat and hence isometric to the Euclidean manifold.

Fields

• manifold – base space of the fiber bundle
• point – a point $p$ from the base space; the fiber corresponds to the preimage by bundle projection $\pi^{-1}(\{p\})$.

Constructor

Fiber(M::AbstractManifold, p, fiber_type::FiberType)

A fiber of type fiber_type at point p from the manifold manifold.

abstract type FiberType end

An abstract type for fiber types that can be used within Fiber.

VectorSpaceFiber{𝔽,M,TSpaceType} = Fiber{𝔽,TSpaceType,M}
    where {𝔽,M<:AbstractManifold{𝔽},TSpaceType<:VectorSpaceType}

Alias for a Fiber when the fiber is a vector space.

CotangentSpace{𝔽,M} = Fiber{𝔽,CotangentSpaceType,M} where {𝔽,M<:AbstractManifold{𝔽}}

A manifold for the Cotangent space $T^*_p\mathcal M$ at a point $p\in\mathcal M$. This is modelled as an alias for VectorSpaceFiber corresponding to CotangentSpaceType.

Constructor

CotangentSpace(M::AbstractManifold, p)

Return the manifold (vector space) representing the cotangent space $T^*_p\mathcal M$ at point p, $p\in\mathcal M$.

TangentSpace{𝔽,M} = Fiber{𝔽,TangentSpaceType,M} where {𝔽,M<:AbstractManifold{𝔽}}

A manifold for the tangent space $T_p\mathcal M$ at a point $p\in\mathcal M$. This is modelled as an alias for VectorSpaceFiber corresponding to TangentSpaceType.

Constructor

TangentSpace(M::AbstractManifold, p)

Return the manifold (vector space) representing the tangent space $T_p\mathcal M$ at point p, $p\in\mathcal M$.

exp(TpM::TangentSpace, X, V)

Exponential map of tangent vectors X from TpM and a direction V, which is also from the TangentSpaceTpM since we identify the tangent space of TpM with TpM. The exponential map then simplifies to the sum X+V.

log(TpM::TangentSpace, X, Y)

Logarithmic map on the TangentSpaceTpM, calculated as the difference of tangent vectors q and p from TpM.

Y = Weingarten(TpM::TangentSpace, X, V, A)
Weingarten!(TpM::TangentSpace, Y, p, X, V)

Compute the Weingarten map $\mathcal W_X$ at X on the TangentSpaceTpM with respect to the tangent vector $V \in T_p\mathcal M$ and the normal vector $A \in N_p\mathcal M$.

Since this a flat space by itself, the result is always the zero tangent vector.

distance(M::TangentSpace, X, Y)

Distance between vectors X and Y from the TangentSpaceTpM. It is calculated as the norm (induced by the metric on TpM) of their difference.

injectivity_radius(TpM::TangentSpace)

Return the injectivity radius on the TangentSpaceTpM, which is $∞$.

inner(M::TangentSpace, X, V, W)

For any $X ∈ T_p\mathcal M$ we identify the tangent space $T_X(T_p\mathcal M)$ with $T_p\mathcal M$ again. Hence an inner product of $V,W$ is just the inner product of the tangent space itself. $⟨V,W⟩_X = ⟨V,W⟩_p$.

is_flat(::TangentSpace)

The TangentSpace is a flat manifold, so this returns true.

manifold_dimension(TpM::TangentSpace)

Return the dimension of the TangentSpace$T_p\mathcal M$ at $p∈\mathcal M$, which is the same as the dimension of the manifold $\mathcal M$.

parallel_transport_to(::TangentSpace, X, V, Y)

Transport the tangent vector $Z ∈ T_X(T_p\mathcal M)$ from X to Y. Since we identify $T_X(T_p\mathcal M) = T_p\mathcal M$ and the tangent space is a vector space, parallel transport simplifies to the identity, so this function yields $V$ as a result.

project(TpM::TangentSpace, X, V)

Project the vector V from the embedding of the tangent space TpM (identified with $T_X(T_p\mathcal M)$), that is project the vector V onto the tangent space at TpM.point.

project(TpM::TangentSpace, X)

Project the point X from embedding of the TangentSpaceTpM onto TpM.

zero_vector(TpM::TangentSpace, X)

Zero tangent vector at point X from the TangentSpaceTpM, that is the zero tangent vector at point TpM.point, since we identify the tangent space $T_X(T_p\mathcal M)$ with $T_p\mathcal M$.

zero_vector(TpM::TangentSpace)

Zero tangent vector in the TangentSpaceTpM, that is the zero tangent vector at point TpM.point.