Hyperbolic space

Hyperbolic{N} <: AbstractEmbeddedManifold{ℝ,DefaultIsometricEmbeddingType}

The hyperbolic space $\mathcal H^n$ represented by $n+1$-Tuples, i.e. embedded in the Lorentzian manifold equipped with the MinkowskiMetric$⟨\cdot,\cdot⟩_{\mathrm{M}}$. The space is defined as

\[\mathcal H^n = \Bigl\{p ∈ ℝ^{n+1}\ \Big|\ ⟨p,p⟩_{\mathrm{M}}= -p_{n+1}^2 + \displaystyle\sum_{k=1}^n p_k^2 = -1, p_{n+1} > 0\Bigr\},.\]

The tangent space $T_p \mathcal H^n$ is given by

\[T_p \mathcal H^n := \bigl\{ X ∈ ℝ^{n+1} : ⟨p,X⟩_{\mathrm{M}} = 0 \bigr\}.\]

Note that while the MinkowskiMetric renders the Lorentz manifold (only) pseudo-Riemannian, on the tangent bundle of the Hyperbolic space it induces a Riemannian metric. The corresponding sectional curvature is $-1$.

If p and X are Vectors of length n+1 they are assumed to be a HyperboloidPoint and a HyperboloidTVector, respectively

Other models are the Poincaré ball model, see PoincareBallPoint and PoincareBallTVector, respectiely and the Poincaré half space model, see PoincareHalfSpacePoint and PoincareHalfSpaceTVector, respectively.

Constructor

Hyperbolic(n)

Generate the Hyperbolic manifold of dimension n.

HyperboloidPoint <: MPoint

In the Hyperboloid model of the Hyperbolic$\mathcal H^n$ points are represented as vectors in $ℝ^{n+1}$ with MinkowskiMetric equal to $-1$.

This representation is the default, i.e. AbstractVectors are assumed to have this repesentation.

HyperboloidTVector <: TVector

In the Hyperboloid model of the Hyperbolic$\mathcal H^n$ tangent vctors are represented as vectors in $ℝ^{n+1}$ with MinkowskiMetric$⟨p,X⟩_{\mathrm{M}}=0$ to their base point $p$.

This representation is the default, i.e. vectors are assumed to have this repesentation.

PoincareBallPoint <: MPoint

A point on the Hyperbolic manifold $\mathcal H^n$ can be represented as a vector of norm less than one in $\mathbb R^n$.

PoincareBallTVector <: TVector

In the Poincaré ball model of the Hyperbolic$\mathcal H^n$ tangent vectors are represented as vectors in $ℝ^{n}$.

PoincareHalfSpacePoint <: MPoint

A point on the Hyperbolic manifold $\mathcal H^n$ can be represented as a vector in the half plane, i.e. $x ∈ ℝ^n$ with $x_d > 0$.

PoincareHalfPlaneTVector <: TVector

In the Poincaré half plane model of the Hyperbolic$\mathcal H^n$ tangent vectors are represented as vectors in $ℝ^{n}$.

exp(M::Hyperbolic, p, X)

Compute the exponential map on the Hyperbolic space $\mathcal H^n$ emanating from p towards X. The formula reads

\[\exp_p X = \cosh(\sqrt{⟨X,X⟩_{\mathrm{M}}})p + \sinh(\sqrt{⟨X,X⟩_{\mathrm{M}}})\frac{X}{\sqrt{⟨X,X⟩_{\mathrm{M}}}},\]

where $⟨\cdot,\cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

log(M::Hyperbolic, p, q)

Compute the logarithmic map on the Hyperbolic space $\mathcal H^n$, the tangent vector representing the geodesic starting from p reaches q after time 1. The formula reads for $p ≠ q$

\[\log_p q = d_{\mathcal H^n}(p,q) \frac{q-⟨p,q⟩_{\mathrm{M}} p}{\lVert q-⟨p,q⟩_{\mathrm{M}} p \rVert_2},\]

where $⟨\cdot,\cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold. For $p=q$ the logarihmic map is equal to the zero vector.

check_manifold_point(M::Hyperbolic, p; kwargs...)

Check whether p is a valid point on the HyperbolicM.

For the HyperboloidPoint or plain vectors this means that, p is a vector of length $n+1$ with inner product in the embedding of -1, see MinkowskiMetric. The tolerance for the last test can be set using the kwargs....

For the PoincareBallPoint a valid point is a vector $p ∈ ℝ^n$ with a norm stricly less than 1.

For the PoincareHalfSpacePoint a valid point is a vector from $p ∈ ℝ^n$ with a positive last entry, i.e. $p_n>0$

check_tangent_vector(M::Hyperbolic{n}, p, X; check_base_point = true, kwargs... )

Check whether X is a tangent vector to p on the HyperbolicM, i.e. after check_manifold_point(M,p), X has to be of the same dimension as p. The optional parameter check_base_point indicates whether to call check_manifold_point for p. The tolerance for the last test can be set using the kwargs....

For a the hyperboloid model or vectors, X has to be orthogonal to p with respect to the inner product from the embedding, see MinkowskiMetric.

For a the Poincaré ball as well as the Poincaré half plane model, X has to be a vector from $ℝ^{n}$.

injectivity_radius(M::Hyperbolic)
injectivity_radius(M::Hyperbolic, p)

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

manifold_dimension(M::Hyperbolic)

Return the dimension of the hyperbolic space manifold $\mathcal H^n$, i.e. $\dim(\mathcal H^n) = n$.

project(M::Hyperbolic, p, X)

Perform an orthogonal projection with respect to the Minkowski inner product of X onto the tangent space at p of the Hyperbolic space M.

The formula reads

\[Y = X + ⟨p,X⟩_{\mathrm{M}} p,\]

where $⟨\cdot, \cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

vector_transport_to(M::Hyperbolic, p, X, q, ::ParallelTransport)

Compute the paralllel transport of the X from the tangent space at p on the Hyperbolic space $\mathcal H^n$ to the tangent at q along the geodesic connecting p and q. The formula reads

\[\mathcal P_{q←p}X = X - \frac{⟨\log_p q,X⟩_p}{d^2_{\mathcal H^n}(p,q)} \bigl(\log_p q + \log_qp \bigr),\]

where $⟨\cdot,\cdot⟩_p$ denotes the inner product in the tangent space at p.

mean(
    M::Hyperbolic,
    x::AbstractVector,
    [w::AbstractWeights,]
    method = CyclicProximalPointEstimation();
    kwargs...,
)

Compute the Riemannian mean of x on the Hyperbolic space using CyclicProximalPointEstimation.

convert(::Type{HyperboloidPoint}, p::PoincareBallPoint)
convert(::Type{AbstractVector}, p::PoincareBallPoint)

convert a point PoincareBallPointx (from $ℝ^n$) from the Poincaré ball model of the Hyperbolic manifold $\mathcal H^n$ to a HyperboloidPoint$π(p) ∈ ℝ^{n+1}$. The isometry is defined by

\[π(p) = \frac{1}{1-\lVert p \rVert^2} \begin{pmatrix}2p_1\\⋮\\2p_n\\1+\lVert p \rVert^2\end{pmatrix}\]

Note that this is also used, when the type to convert to is a vector.

convert(::Type{HyperboloidPoint}, p::PoincareHalfSpacePoint)
convert(::Type{AbstractVector}, p::PoincareHalfSpacePoint)

convert a point PoincareHalfSpacePointp (from $ℝ^n$) from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a HyperboloidPoint$π(p) ∈ ℝ^{n+1}$.

This is done in two steps, namely transforming it to a Poincare ball point and from there further on to a Hyperboloid point.

convert(::Type{HyperboloidTVector}, p::PoincareBallPoint, X::PoincareBallTVector)
convert(::Type{AbstractVector}, p::PoincareBallPoint, X::PoincareBallTVector)

Convert the PoincareBallTVectorX from the tangent space at p to a HyperboloidTVector by computing the push forward of the isometric map, cf. convert(::Type{HyperboloidPoint}, p::PoincareBallPoint).

The push forward $π_*(p)$ maps from $ℝ^n$ to a subspace of $ℝ^{n+1}$, the formula reads

\[π_*(p)[X] = \begin{pmatrix} \frac{2X_1}{1-\lVert p \rVert^2} + \frac{4}{(1-\lVert p \rVert^2)^2}⟨X,p⟩p_1\\ ⋮\\ \frac{2X_n}{1-\lVert p \rVert^2} + \frac{4}{(1-\lVert p \rVert^2)^2}⟨X,p⟩p_n\\ \frac{4}{(1-\lVert p \rVert^2)^2}⟨X,p⟩ \end{pmatrix}.\]

convert(::Type{HyperboloidTVector}, p::PoincareHalfSpacePoint, X::PoincareHalfSpaceTVector)
convert(::Type{AbstractVector}, p::PoincareHalfSpacePoint, X::PoincareHalfSpaceTVector)

convert a point PoincareHalfSpaceTVectorX (from $ℝ^n$) at p from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a HyperboloidTVector$π(p) ∈ ℝ^{n+1}$.

This is done in two steps, namely transforming it to a Poincare ball point and from there further on to a Hyperboloid point.

convert(
    ::Type{Tuple{HyperboloidPoint,HyperboloidTVector}}.
    (p,X)::Tuple{PoincareBallPoint,PoincareBallTVector}
)
convert(
    ::Type{Tuple{P,T}},
    (p, X)::Tuple{PoincareBallPoint,PoincareBallTVector},
) where {P<:AbstractVector, T <: AbstractVector}

Convert a PoincareBallPointp and a PoincareBallTVectorX to a HyperboloidPoint and a HyperboloidTVector simultaneously, see convert(::Type{HyperboloidPoint}, ::PoincareBallPoint) and convert(::Type{HyperboloidTVector}, ::PoincareBallPoint, ::PoincareBallTVector) for the formulae.

convert(
    ::Type{Tuple{HyperboloidPoint,HyperboloidTVector},
    (p,X)::Tuple{PoincareHalfSpacePoint, PoincareHalfSpaceTVector}
)
convert(
    ::Type{Tuple{T,T},
    (p,X)::Tuple{PoincareHalfSpacePoint, PoincareHalfSpaceTVector}
) where {T<:AbstractVector}

convert a point PoincareHalfSpaceTVectorX (from $ℝ^n$) at p from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a tuple of a HyperboloidPoint and a HyperboloidTVector$π(p) ∈ ℝ^{n+1}$ simultaneously.

This is done in two steps, namely transforming it to the Poincare ball model and from there further on to a Hyperboloid.

_hyperbolize(M, p, Y)

Given the Hyperbolic(n) manifold using the hyperboloid model and a point p thereon, we can put a vector $Y\in ℝ^n$ into the tangent space by computing its last component such that for the resulting p we have that its minkowski_metric is $⟨p,X⟩_{\mathrm{M}} = 0$, i.e. $X_{n+1} = \frac{⟨\tilde p, Y⟩}{p_{n+1}}$, where $\tilde p = (p_1,\ldots,p_n)$.

_hyperbolize(M,q)

Given the Hyperbolic(n) manifold using the hyperboloid model, a point from the $q\in ℝ^n$ can be set onto the manifold by computing its last component such that for the resulting p we have that its minkowski_metric is $⟨p,p⟩_{\mathrm{M}} = - 1$, i.e. $p_{n+1} = \sqrt{\lVert q \rVert^2-^}$

distance(M::Hyperbolic, p, q)
distance(M::Hyperbolic, p::HyperboloidPoint, q::HyperboloidPoint)

Compute the distance on the HyperbolicM, which reads

\[d_{\mathcal H^n}(p,q) = \operatorname{acosh}( - ⟨p, q⟩_{\mathrm{M}}),\]

where $⟨\cdot,\cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

get_coordinates(M::Hyperbolic, p, X, ::DefaultOrthonormalBasis)

Compute the coordinates of the vector X with respect to the orthogonalized version of the unit vectors from $ℝ^n$, where $n$ is the manifold dimension of the HyperbolicM, utting them intop the tangent space at p and orthonormalizing them.

get_vector(M::Hyperbolic, p, c, ::DefaultOrthonormalBasis)

Compute the vector from the coordinates with respect to the orthogonalized version of the unit vectors from $ℝ^n$, where $n$ is the manifold dimension of the HyperbolicM, utting them intop the tangent space at p and orthonormalizing them.

inner(M::Hyperbolic{n}, p, X, Y)
inner(M::Hyperbolic{n}, p::HyperboloidPoint, X::HyperboloidTVector, Y::HyperboloidTVector)

Cmpute the inner product in the Hyperboloid model, i.e. the minkowski_metric in the embedding. The formula reads

\[g_p(X,Y) = ⟨X,Y⟩_{\mathrm{M}} = -X_{n}Y_{n} + \displaystyle\sum_{k=1}^{n-1} X_kY_k.\]

This employs the metric of the embedding, see Lorentz space.

convert(::Type{PoincareBallPoint}, p::HyperboloidPoint)
convert(::Type{PoincareBallPoint}, p::T) where {T<:AbstractVector}

convert a HyperboloidPoint$p∈ℝ^{n+1}$ from the hyperboloid model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareBallPoint$π(p)∈ℝ^{n}$ in the Poincaré ball model. The isometry is defined by

\[π(p) = \frac{1}{1+p_{n+1}} \begin{pmatrix}p_1\\⋮\\p_n\end{pmatrix}\]

Note that this is also used, when x is a vector.

convert(::Type{PoincareBallPoint}, p::PoincareHalfSpacePoint)

convert a point PoincareHalfSpacePointp (from $ℝ^n$) from the Poincaré half plane model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareBallPoint$π(p) ∈ ℝ^n$. Denote by $\tilde p = (p_1,\ldots,p_{d-1})^{\mathrm{T}}$. Then the isometry is defined by

\[π(p) = \frac{1}{\lVert \tilde p \rVert^2 + (p_n+1)^2} \begin{pmatrix}2p_1\\⋮\\2p_{n-1}\\\lVert p\rVert^2 - 1\end{pmatrix}.\]

convert(::Type{PoincareBallTVector}, p::HyperboloidPoint, X::HyperboloidTVector)
convert(::Type{PoincareBallTVector}, p::P, X::T) where {P<:AbstractVector, T<:AbstractVector}

convert a HyperboloidTVectorX at p to a PoincareBallTVector on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Hyperboloid to the Poincaré ball, cf. convert(::Type{PoincareBallPoint}, ::HyperboloidPoint).

The formula reads

\[π_*(p)[X] = \frac{1}{p_{n+1}+1}\Bigl(\tilde X - \frac{X_{n+1}}{p_{n+1}+1}\tilde p \Bigl),\]

where $\tilde X = \begin{pmatrix}X_1\\⋮\\X_n\end{pmatrix}$ and $\tilde p = \begin{pmatrix}p_1\\⋮\\p_n\end{pmatrix}$.

convert(
    ::Type{PoincareBallTVector},
    p::PoincareHalfSpacePoint,
    X::PoincareHalfSpaceTVector
)

convert a PoincareHalfSpaceTVectorX at p to a PoincareBallTVector on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Poincaré half space to the Poincaré ball, cf. convert(::Type{PoincareBallPoint}, ::PoincareHalfSpacePoint).

The formula reads

\[π_*(p)[X] = \frac{1}{\lVert \tilde p\rVert^2 + (1+p_n)^2} \begin{pmatrix} 2X_1\\ ⋮\\ 2X_{n-1}\\ 2⟨X,p⟩ \end{pmatrix} - \frac{2}{(\lVert \tilde p\rVert^2 + (1+p_n)^2)^2} \begin{pmatrix} 2p_1(⟨X,p⟩+X_n)\\ ⋮\\ 2p_{n-1}(⟨X,p⟩+X_n)\\ (\lVert p \rVert^2-1)(⟨X,p⟩+X_n) \end{pmatrix}\]

where $\tilde p = \begin{pmatrix}p_1\\⋮\\p_{n-1}\end{pmatrix}$.

convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p,X)::Tuple{HyperboloidPoint,HyperboloidTVector}
)
convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p, X)::Tuple{P,T},
) where {P<:AbstractVector, T <: AbstractVector}

Convert a HyperboloidPointp and a HyperboloidTVectorX to a PoincareBallPoint and a PoincareBallTVector simultaneously, see convert(::Type{PoincareBallPoint}, ::HyperboloidPoint) and convert(::Type{PoincareBallTVector}, ::HyperboloidPoint, ::HyperboloidTVector) for the formulae.

convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p,X)::Tuple{HyperboloidPoint,HyperboloidTVector}
)
convert(
    ::Type{Tuple{PoincareBallPoint,PoincareBallTVector}},
    (p, X)::Tuple{T,T},
) where {T <: AbstractVector}

Convert a PoincareHalfSpacePointp and a PoincareHalfSpaceTVectorX to a PoincareBallPoint and a PoincareBallTVector simultaneously, see convert(::Type{PoincareBallPoint}, ::PoincareHalfSpacePoint) and convert(::Type{PoincareBallTVector}, ::PoincareHalfSpacePoint, ::PoincareHalfSpaceTVector) for the formulae.

distance(::Hyperbolic, p::PoincareBallPoint, q::PoincareBallPoint)

Compute the distance on the Hyperbolic manifold $\mathcal H^n$ represented in the Poincaré ball model. The formula reads

\[d_{\mathcal H^n}(p,q) = \operatorname{acosh}\Bigl( 1 + \frac{2\lVert p - q \rVert^2}{(1-\lVert p\rVert^2)(1-\lVert q\rVert^2)} \Bigr)\]

inner(::Hyperbolic, p::PoincareBallPoint, X::PoincareBallTVector, Y::PoincareBallTVector)

Compute the inner producz in the Poincaré ball model. The formula reads

\[g_p(X,Y) = \frac{4}{(1-\lVert p \rVert^2)^2} ⟨X, Y⟩ .\]

project(::Hyperbolic, ::PoincareBallPoint, ::PoincareBallTVector)

projction of tangent vectors in the Poincaré ball model is just the identity, since the tangent space consists of all $ℝ^n$.

convert(::Type{PoincareHalfSpacePoint}, p::Hyperboloid)
convert(::Type{PoincareHalfSpacePoint}, p)

convert a HyperboloidPoint or Vectorp (from $ℝ^{n+1}$) from the Hyperboloid model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareHalfSpacePoint$π(x) ∈ ℝ^{n}$.

This is done in two steps, namely transforming it to a Poincare ball point and from there further on to a PoincareHalfSpacePoint point.

convert(::Type{PoincareHalfSpacePoint}, p::PoincareBallPoint)

convert a point PoincareBallPointp (from $ℝ^n$) from the Poincaré ball model of the Hyperbolic manifold $\mathcal H^n$ to a PoincareHalfSpacePoint$π(p) ∈ ℝ^n$. Denote by $\tilde p = (p_1,\ldots,p_{n-1})$. Then the isometry is defined by

\[π(p) = \frac{1}{\lVert \tilde p \rVert^2 - (p_n-1)^2} \begin{pmatrix}2p_1\\⋮\\2p_{n-1}\\1-\lVert p\rVert^2\end{pmatrix}.\]

convert(::Type{PoincareHalfSpaceTVector}, p::HyperboloidPoint, ::HyperboloidTVector)
convert(::Type{PoincareHalfSpaceTVector}, p::P, X::T) where {P<:AbstractVector, T<:AbstractVector}

convert a HyperboloidTVectorX at p to a PoincareHalfSpaceTVector on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Hyperboloid to the Poincaré half space, cf. convert(::Type{PoincareHalfSpacePoint}, ::HyperboloidPoint).

This is done similarly to the approach there, i.e. by using the Poincaré ball model as an intermediate step.

convert(::Type{PoincareHalfSpaceTVector}, p::PoincareBallPoint, X::PoincareBallTVector)

convert a PoincareBallTVectorX at p to a PoincareHalfSpacePoint on the Hyperbolic manifold $\mathcal H^n$ by computing the push forward $π_*(p)[X]$ of the isometry $π$ that maps from the Poincaré ball to the Poincaré half space, cf. convert(::Type{PoincareHalfSpacePoint}, ::PoincareBallPoint).

The formula reads

\[π_*(p)[X] = \frac{1}{\lVert \tilde p\rVert^2 + (1-p_n)^2} \begin{pmatrix} 2X_1\\ ⋮\\ 2X_{n-1}\\ -2⟨X,p⟩ \end{pmatrix} - \frac{2}{(\lVert \tilde p\rVert^2 + (1-p_n)^2)^2} \begin{pmatrix} 2p_1(⟨X,p⟩-X_n)\\ ⋮\\ 2p_{n-1}(⟨X,p⟩-X_n)\\ (\lVert p \rVert^2-1)(⟨X,p⟩-X_n) \end{pmatrix}\]

where $\tilde p = \begin{pmatrix}p_1\\⋮\\p_{n-1}\end{pmatrix}$.

convert(
    ::Type{Tuple{PoincareHalfSpacePoint,PoincareHalfSpaceTVector}},
    (p,X)::Tuple{HyperboloidPoint,HyperboloidTVector}
)
convert(
    ::Type{Tuple{PoincareHalfSpacePoint,PoincareHalfSpaceTVector}},
    (p, X)::Tuple{P,T},
) where {P<:AbstractVector, T <: AbstractVector}

Convert a HyperboloidPointp and a HyperboloidTVectorX to a PoincareHalfSpacePoint and a PoincareHalfSpaceTVector simultaneously, see convert(::Type{PoincareHalfSpacePoint}, ::HyperboloidPoint) and convert(::Type{PoincareHalfSpaceTVector}, ::Tuple{HyperboloidPoint,HyperboloidTVector}) for the formulae.

convert(
    ::Type{Tuple{PoincareHalfSpacePoint,PoincareHalfSpaceTVector}},
    (p,X)::Tuple{PoincareBallPoint,PoincareBallTVector}
)

Convert a PoincareBallPointp and a PoincareBallTVectorX to a PoincareHalfSpacePoint and a PoincareHalfSpaceTVector simultaneously, see convert(::Type{PoincareHalfSpacePoint}, ::PoincareBallPoint) and convert(::Type{PoincareHalfSpaceTVector}, ::PoincareBallPoint,::PoincareBallTVector) for the formulae.

distance(::Hyperbolic, p::PoincareHalfSpacePoint, q::PoincareHalfSpacePoint)

Compute the distance on the Hyperbolic manifold $\mathcal H^n$ represented in the Poincaré half space model. The formula reads

\[d_{\mathcal H^n}(p,q) = \operatorname{acosh}\Bigl( 1 + \frac{\lVert p - q \rVert^2}{2 p_n q_n} \Bigr)\]

inner(
    ::Hyperbolic{n},
    p::PoincareHalfSpacePoint,
    X::PoincareHalfSpaceTVector,
    Y::PoincareHalfSpaceTVector
)

Compute the inner product in the Poincaré half space model. The formula reads

\[g_p(X,Y) = \frac{⟨X,Y⟩}{p_n^2}.\]

project(::Hyperbolic, ::PoincareHalfSpacePoint ::PoincareHalfSpaceTVector)

projction of tangent vectors in the Poincaré half space model is just the identity, since the tangent space consists of all $ℝ^n$.