Positive Numbers

PositiveNumbers <: 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).

exp(M::PositiveNumbers, p, X)

Compute the exponential map on the PositiveNumbers M.

\[\exp_p X = p\operatorname{exp}(X/p).\]

log(M::PositiveNumbers, p, q)

Compute the logarithmic map on the PositiveNumbers M.

\[\log_p q = p\log\frac{q}{p}.\]

riemannian_Hessian(M::SymmetricPositiveDefinite, p, G, H, X)

The Riemannian Hessian can be computed as stated in Eq. (7.3) [Ngu23]. Let $\nabla f(p)$ denote the Euclidean gradient G, $\nabla^2 f(p)[X]$ the Euclidean Hessian H. Then the formula reads

\[ \operatorname{Hess}f(p)[X] = p\bigl(∇^2 f(p)[X]\bigr)p + X\bigl(∇f(p)\bigr)p\]

PositiveArrays(n₁,n₂,...,nᵢ)

Generate the manifold of i-dimensional arrays with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.

PositiveMatrices(m,n)

Generate the manifold of matrices with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.

PositiveVectors(n)

Generate the manifold of vectors with positive entries. This manifold is modeled as a PowerManifold of PositiveNumbers.

manifold_volume(M::PositiveNumbers)

Return volume of PositiveNumbers M, i.e. Inf.

volume_density(M::PositiveNumbers, p, X)

Compute volume density function of PositiveNumbers. The formula reads

\[\theta_p(X) = \exp(X / p)\]

change_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 $C(X) = pX$.

change_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 $Z = p^2X$

check_point(M::PositiveNumbers, p)

Check whether p is a point on the PositiveNumbers M, i.e. $p>0$.

check_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 [...]

distance(M::PositiveNumbers, p, q)

Compute the distance on the PositiveNumbers M, which is

\[d(p,q) = \Bigl\lvert \log \frac{p}{q} \Bigr\rvert = \lvert \log p - \log q\rvert.\]

get_coordinates(::PositiveNumbers, p, X, ::DefaultOrthonormalBasis{ℝ})

Compute the coordinate of vector X which is tangent to p on the PositiveNumbers manifold. The formula is $X / p$.

get_vector(::PositiveNumbers, p, c, ::DefaultOrthonormalBasis{ℝ})

Compute the vector with coordinate c which is tangent to p on the PositiveNumbers manifold. The formula is $p * c$.

injectivity_radius(M::PositiveNumbers[, p])

Return the injectivity radius on the PositiveNumbers M, i.e. $\infty$.

inner(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.

\[g_p(X,Y) = \frac{XY}{p^2}.\]

is_flat(::PositiveNumbers)

Return false. PositiveNumbers is not a flat manifold.

manifold_dimension(M::PositiveNumbers)

Return the dimension of the PositiveNumbers M, i.e. of the 1-dimensional hyperbolic space,

\[\dim(H^1) = 1\]

parallel_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.

\[\mathcal P_{q\gets p}(X) = X\cdot\frac{q}{p}.\]

project(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.