Positive Numbers
The manifold PositiveNumbers represents positive numbers with hyperbolic geometry. Additionally, there are also short forms for its corresponding PowerManifolds, i.e. PositiveVectors, PositiveMatrices, and PositiveArrays.
Manifolds.PositiveNumbers — TypePositiveNumbers <: AbstractManifold{ℝ}The hyperbolic manifold of positive numbers $H^1$ is a the hyperbolic manifold represented by just positive numbers.
Constructor
PositiveNumbers()Generate the ℝ-valued hyperbolic model represented by positive positive numbers. To use this with arrays (1-element arrays), please use SymmetricPositiveDefinite(1).
Base.exp — Methodexp(M::PositiveNumbers, p, X)Compute the exponential map on the PositiveNumbers M.
\[\exp_p X = p\operatorname{exp}(X/p).\]
Base.log — Methodlog(M::PositiveNumbers, p, q)Compute the logarithmic map on the PositiveNumbers M.
\[\log_p q = p\log\frac{q}{p}.\]
Manifolds.PositiveArrays — MethodPositiveArrays(n₁,n₂,...,nᵢ)Generate the manifold of i-dimensional arrays with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.
Manifolds.PositiveMatrices — MethodPositiveMatrices(m,n)Generate the manifold of matrices with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.
Manifolds.PositiveVectors — MethodPositiveVectors(n)Generate the manifold of vectors with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.
Manifolds.change_metric — Methodchange_metric(M::PositiveNumbers, E::EuclideanMetric, p, X)Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function with respect to the EuclideanMetric g_E, this function changes the representer into the one with respect to the positivity metric of PositiveNumbers M.
For all $Z,Y$ we are looking for the function $c$ on the tangent space at $p$ such that
\[ ⟨Z,Y⟩ = XY = \frac{c(Z)c(Y)}{p^2} = g_p(c(Y),c(Z))\]
and hence $C(X) = pX$.
Manifolds.change_representer — Methodchange_representer(M::PositiveNumbers, E::EuclideanMetric, p, X)Given a tangent vector $X ∈ T_p\mathcal M$ representing a linear function with respect to the EuclideanMetric g_E, this function changes the representer into the one with respect to the positivity metric representation of PositiveNumbers M.
For all tangent vectors $Y$ the result $Z$ has to fulfill
\[ ⟨X,Y⟩ = XY = \frac{ZY}{p^2} = g_p(YZ)\]
and hence $Z = p^2X$
ManifoldsBase.check_point — Methodcheck_point(M::PositiveNumbers, p)Check whether p is a point on the PositiveNumbers M, i.e. $p>0$.
ManifoldsBase.check_vector — Methodcheck_vector(M::PositiveNumbers, p, X; kwargs...)Check whether X is a tangent vector in the tangent space of p on the PositiveNumbers M. For the real-valued case represented by positive numbers, all X are valid, since the tangent space is the whole real line. For the complex-valued case X [...]
ManifoldsBase.distance — Methoddistance(M::PositiveNumbers, p, q)Compute the distance on the PositiveNumbers M, which is
\[d(p,q) = \Bigl\lvert \log \frac{p}{q} \Bigr\rvert = \lvert \log p - \log q\rvert.\]
ManifoldsBase.injectivity_radius — Methodinjectivity_radius(M::PositiveNumbers[, p])Return the injectivity radius on the PositiveNumbers M, i.e. $\infty$.
ManifoldsBase.inner — Methodinner(M::PositiveNumbers, p, X, Y)Compute the inner product of the two tangent vectors X,Y from the tangent plane at p on the PositiveNumbers M, i.e.
\[g_p(X,Y) = \frac{XY}{p^2}.\]
ManifoldsBase.manifold_dimension — Methodmanifold_dimension(M::PositiveNumbers)Return the dimension of the PositiveNumbers M, i.e. of the 1-dimensional hyperbolic space,
\[\dim(H^1) = 1\]
ManifoldsBase.parallel_transport_to — Methodparallel_transport_to(M::PositiveNumbers, p, X, q)Compute the parallel transport of X from the tangent space at p to the tangent space at q on the PositiveNumbers M.
\[\mathcal P_{q\gets p}(X) = X\cdot\frac{q}{p}.\]
ManifoldsBase.project — Methodproject(M::PositiveNumbers, p, X)Project a value X onto the tangent space of the point p on the PositiveNumbers M, which is just the identity.