# Specific manifold functions

This small section extends the functions available from ManifoldsBase.jl and Manifolds.jl, espcially a few random generators, that are simpler than the functions available.

ManifoldsBase.mid_pointMethod
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).

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Manopt.reflectMethod
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.

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## Simplified random functions

While statistics are available in Manifolds.jl, the following functions provide default random points and vectors on manifolds.

Manopt.random_pointFunction
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.

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Manopt.random_pointFunction
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.

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Manopt.random_pointMethod
random_point(M::AbstractGroupManifold, options...)

On an abstract group manifold, the random point is taken from the internally stored M.manifold.

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Manopt.random_pointMethod
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.

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Manopt.random_pointMethod
random_point(M::AbstractManifold)

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

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Manopt.random_pointMethod
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.

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Manopt.random_pointMethod
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.

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

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

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

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

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Manopt.random_pointMethod
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.

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Manopt.random_pointMethod
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.

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Manopt.random_pointMethod
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 $n×k$.

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Manopt.random_tangentFunction
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.

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Manopt.random_tangentFunction
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.

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Manopt.random_tangentFunction
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.

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Manopt.random_tangentFunction
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.

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

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

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Manopt.random_tangentFunction
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.

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Manopt.random_tangentFunction
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.

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Manopt.random_tangentFunction
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.

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Manopt.random_tangentMethod
random_tangent(M::AbstractGroupManifold, p, options...)

On an abstract group manifold, the random tangent is taken from the internally stored M.manifolds tangent space at p.

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Manopt.random_tangentMethod
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.

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Manopt.random_tangentMethod
random_tangent(M::ProductManifold, p)

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

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

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

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