Specific manifold functions

mid_point(M, p, q, x)
mid_point!(M, y, p, q, x)

Compute the mid point between p and q. If there is more than one mid point of (not neccessarily minimizing) geodesics (e.g. on the sphere), the one nearest to x is returned (in place of y).

reflect(M, p, x)
reflect!(M, q, p, x)

reflect the point x from the manifold M at point p, i.e.

\[ \operatorname{refl}_p(x) = \exp_p(-\log_p x).\]

where exp and log denote the exponential and logarithmic map on M. This can also be done in place of q.

reflect(M, f, x)
reflect!(M, q, f, x)

reflect the point x from the manifold M at the point f(x) of the function $f: \mathcal M → \mathcal M$, i.e.,

\[ \operatorname{refl}_f(x) = \operatorname{refl}_{f(x)}(x),\]

Compute the result in q.

see also reflect(M,p,x).

random_point(M::Sphere, :Gaussian[, σ=1.0])

return a random point on the Sphere by projecting a normal distributed vector from within the embedding to the sphere.

random_point(M::Rotations, :Gaussian [, σ=1.0])

return a random point p on the manifold Rotations by generating a (Gaussian) random orthogonal matrix with determinant $+1$. Let $QR = A$ be the QR decomposition of a random matrix $A$, then the formula reads $p = QD$ where $D$ is a diagonal matrix with the signs of the diagonal entries of $R$, i.e.

\[D_{ij}=\begin{cases} \operatorname{sgn}(R_{ij}) & \text{if} \; i=j \\ 0 & \, \text{otherwise.} \end{cases}\]

It can happen that the matrix gets -1 as a determinant. In this case, the first and second columns are swapped.

random_point(M::AbstractManifold, s::Symbol, options...)

generate a random point using a noise model given by s with its additional options just passed on.

random_point(M::AbstractManifold)

generate a random point on a manifold. By default it uses random_point(M,:Gaussian).

random_point(M::Circle, :Uniform)

return a random point on the Circle $\mathbb S^1$ by picking a random element from $[-\pi,\pi)$ uniformly.

random_point(M::Euclidean[,:Gaussian, σ::Float64=1.0])

generate a random point on the Euclidean manifold M, where the optional parameter determines the type of the entries of the resulting point on the Euclidean space d.

random_point(M::ProductManifold, options...)

return a random point x on Grassmannian manifold M by generating a random (Gaussian) matrix with standard deviation σ in matching size, which is orthonormal.

random_point(M::TangentBundle, options...)

generate a random point on the tangent bundle by calling a random_point and a random_tangent with the given options...

random_point(M::TangentSpaceAtPoint, options...)

generate a random point in the the tangent space of M.point with the given options....

random_point(M::AbstractPowerManifold, options...)

generate a random point on the AbstractPowerManifold M given options that are passed on.

random_point(M::SymmetricPositiveDefinite, :Gaussian[, σ=1.0])

generate a random symmetric positive definite matrix on the SymmetricPositiveDefinite manifold M.

random_point(M::FixedRankMatrices, options...)

return a random point on the FixedRankMatrices manifold. The orthogonal matrices are sampled from the Stiefel manifold and the singular values are sampled uniformly at random.

random_point(M::Grassmannian, :Gaussian [, σ=1.0])

return a random point x on Grassmannian manifold M by generating a random (Gaussian) matrix with standard deviation σ in matching size, which is orthonormal.

random_point(M::Stiefel, :Gaussian[, σ=1.0])

return a random (Gaussian) point x on the Stiefel manifold M by generating a (Gaussian) matrix with standard deviation σ and return the orthogonalized version, i.e. return ​​the Q component of the QR decomposition of the random matrix of size $n×k$.

random_tangent(M::SymmetricPositiveDefinite, p, :Rician [,σ = 0.01])

generate a random tangent vector in the tangent space of p on the SymmetricPositiveDefinite manifold M by using a Rician distribution with standard deviation σ.

random_tangent(M::Sphere, p[, :Gaussian, σ=1.0])

return a random tangent vector in the tangent space of p on the Sphere M.

random_tangent(M::Stiefel, p[,type=:Gaussian, σ=1.0])

return a (Gaussian) random vector from the tangent space $T_p\mathrm{St}(n,k)$ with mean zero and standard deviation σ by projecting a random Matrix onto the p.

random_tangent(M::Circle, p [, :Gaussian, σ=1.0])

return a random tangent vector from the tangent space of the point p on the Circle $\mathbb S^1$ by using a normal distribution with mean 0 and standard deviation 1.

random_tangent(M::Grassmann, p[,type=:Gaussian, σ=1.0])

return a (Gaussian) random vector from the tangent space $T_p\mathrm{Gr}(n,k)$ with mean zero and standard deviation σ by projecting a random Matrix onto the p.

random_tangent(M::Hyperbolic, p, :Gaussian [, σ=1.0])

generate a random point on the Hyperbolic manifold by projecting a point from the embedding with respect to the Minkowski metric.

random_tangent(M::Rotations, p[, type=:Gaussian, σ=1.0])

return a random tangent vector in the tangent space $T_x\mathrm{SO}(n)$ of the point x on the Rotations manifold M by generating a random skew-symmetric matrix. The function takes the real upper triangular matrix of a (Gaussian) random matrix $A$ with dimension $n\times n$ and subtracts its transposed matrix. Finally, the matrix is normalized.

random_tangent(M::SymmetricPositiveDefinite, p[, :Gaussian, σ = 1.0])

generate a random tangent vector in the tangent space of the point p on the SymmetricPositiveDefinite manifold M by using a Gaussian distribution with standard deviation σ on an ONB of the tangent space.

random_tangent(M, p, options...)

generate a random tangent vector in the tangent space of p on M. By default this is a :Gaussian distribution.

random_tangent(M::ProductManifold, p)

generate a random tangent vector in the tangent space of the point p on the ProductManifold M.

random_tangent(M::TangentBundle, p, options...)

generate a random tangent vector at p on the tangent bundle by calling random_tangent with the given options... twice.

random_tangent(M::TangentSpaceAtPoint, _, options...)

generate a random tangent vector from the tangent space of M.point with the given options..., which is the same as generating a point in the tangent space at M.point.

random_tangent(M::FixedRankMatrices, p, options...)

generate a random tangent vector in the tangent space of the point p on the FixedRankMatrices manifold M.