Product 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.

ProductFVectorDistribution([type::VectorBundleFibers], [x], distrs...)

Generates a random vector at point x from vector space (a fiber of a tangent bundle) of type type using the product distribution of given distributions.

Vector space type and x can be automatically inferred from distributions distrs.

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.

ProductPointDistribution(M::ProductManifold, distributions)

Product distribution on manifold M, combined from distributions.

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 ProductManifold M, 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 ProductManifold M.

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

Access the element(s) at index i of a point p on a ProductManifold M by linear indexing. See also Array Indexing in Julia.

log(M::ProductManifold, p, q)

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

rand(M::ProductManifold; parts_kwargs = map(_ -> (;), M.manifolds))

Return a random point on ProductManifold M. parts_kwargs is a tuple of keyword arguments for rand on each manifold in M.manifolds.

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

set the element [i...] of a point q on a ProductManifold by linear indexing to q. See also Array Indexing in Julia.

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

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 = riemannian_Hessian(M::ProductManifold, p, G, H, X)
riemannian_Hessian!(M::ProductManifold, Y, p, G, H, X)

Compute the Riemannian Hessian $\operatorname{Hess} f(p)[X]$ given the Euclidean gradient $∇ f(\tilde p)$ in G and the Euclidean Hessian $∇^2 f(\tilde p)[\tilde X]$ in H, where $\tilde p, \tilde X$ are the representations of $p,X$ in the embedding,.

On a product manifold, this decouples and can be computed elementwise.

flat(M::ProductManifold, p, X::FVector{TangentSpaceType})

use the musical isomorphism to transform the tangent vector X from the tangent space at p on the ProductManifold M to a cotangent vector. This can be done elementwise for every entry of X (with respect to the corresponding entry in p) separately.

manifold_dimension(M::ProductManifold)

Return the volume of ProductManifold M, i.e. product of volumes of the manifolds M is constructed from.

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

Calculate the number of manifolds multiplied in the given ProductManifold M.

sharp(M::ProductManifold, p, ξ::FVector{CotangentSpaceType})

Use the musical isomorphism to transform the cotangent vector ξ from the tangent space at p on the ProductManifold M to a tangent vector. This can be done elementwise for every entry of ξ (and p) separately

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.

volume_density(M::ProductManifold, p, X)

Return volume density on the ProductManifold M, i.e. product of constituent volume densities.

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 ProductManifold M.

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 ProductManifold M. 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 ProductManifold M. 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 ProductManifold M, 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 ProductManifold M, 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 ProductManifold M.

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 ProductManifold M, which is just the sum of the internal manifolds that build M.

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

Compute the inverse retraction from p with respect to q on the ProductManifold M 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 ProductManifold M 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.

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

Compute the retraction from p with tangent vector X on the ProductManifold M 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 ProductManifold M.

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

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

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

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