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Generalized Stiefel

Manifolds.GeneralizedStiefel โ€” Type
GeneralizedStiefel{n,k,๐”ฝ,B} <: AbstractEmbeddedManifold{๐”ฝ,DefaultEmbeddingType}

The Generalized Stiefel manifold consists of all $n\times k$, $n\geq k$ orthonormal matrices w.r.t. an arbitrary scalar product with symmetric positive definite matrix $B\in R^{n ร— n}$, i.e.

\[\operatorname{St}(n,k,B) = \bigl\{ p \in \mathbb F^{n ร— k}\ \big|\ p^{\mathrm{H}} B p = I_k \bigr\},\]

where $๐”ฝ โˆˆ \{โ„, โ„‚\}$, $\cdot^{\mathrm{H}}$ denotes the complex conjugate transpose or Hermitian, and $I_k \in \mathbb R^{k ร— k}$ denotes the $k ร— k$ identity matrix.

In the case $B=I_k$ one gets the usual Stiefel manifold.

The tangent space at a point $p\in\mathcal M=\operatorname{St}(n,k,B)$ is given by

\[T_p\mathcal M = \{ X \in ๐”ฝ^{n ร— k} : p^{\mathrm{H}}BX + X^{\mathrm{H}}Bp=0_n\},\]

where $0_k$ is the $k ร— k$ zero matrix.

This manifold is modeled as an embedded manifold to the Euclidean, i.e. several functions like the zero_tangent_vector are inherited from the embedding.

The manifold is named after Eduard L. Stiefel (1909โ€“1978).

Constructor

GeneralizedStiefel(n, k, B=I_n, F=โ„)

Generate the (real-valued) Generalized Stiefel manifold of $n\times k$ dimensional orthonormal matrices with scalar product B.

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ManifoldsBase.check_manifold_point โ€” Method
check_manifold_point(M::GeneralizedStiefel, p; kwargs...)

Check whether p is a valid point on the GeneralizedStiefel M=$\operatorname{St}(n,k,B)$, i.e. that it has the right AbstractNumbers type and $x^{\mathrm{H}}Bx$ is (approximately) the identity, where $\cdot^{\mathrm{H}}$ is the complex conjugate transpose. The settings for approximately can be set with kwargs....

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ManifoldsBase.check_tangent_vector โ€” Method
check_tangent_vector(M::GeneralizedStiefel, p, X; kwargs...)

Check whether X is a valid tangent vector at p on the GeneralizedStiefel M=$\operatorname{St}(n,k,B)$, i.e. the AbstractNumbers fits, p is a valid point on M and it (approximately) holds that $p^{\mathrm{H}}BX + \overline{X^{\mathrm{H}}Bp} = 0$, where kwargs... is passed to the isapprox.

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ManifoldsBase.inner โ€” Method
inner(M::GeneralizedStiefel, p, X, Y)

Compute the inner product for two tangent vectors X, Y from the tangent space of p on the GeneralizedStiefel manifold M. The formula reads

\[(X, Y)_p = \operatorname{trace}(v^{\mathrm{H}}Bw),\]

i.e. the metric induced by the scalar product B from the embedding, restricted to the tangent space.

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ManifoldsBase.manifold_dimension โ€” Method
manifold_dimension(M::GeneralizedStiefel)

Return the dimension of the GeneralizedStiefel manifold M=$\operatorname{St}(n,k,B,๐”ฝ)$. The dimension is given by

\[\begin{aligned} \dim \mathrm{St}(n, k, B, โ„) &= nk - \frac{1}{2}k(k+1) \\ \dim \mathrm{St}(n, k, B, โ„‚) &= 2nk - k^2\\ \dim \mathrm{St}(n, k, B, โ„) &= 4nk - k(2k-1) \end{aligned}\]
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ManifoldsBase.project โ€” Method
project(M:GeneralizedStiefel, p, X)

Project X onto the tangent space of p to the GeneralizedStiefel manifold M. The formula reads

\[\operatorname{proj}_{\operatorname{St}(n,k)}(p,X) = X - p\operatorname{Sym}(p^{\mathrm{H}}BX),\]

where $\operatorname{Sym}(y)$ is the symmetrization of $y$, e.g. by $\operatorname{Sym}(y) = \frac{y^{\mathrm{H}}+y}{2}$.

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ManifoldsBase.project โ€” Method
project(M::GeneralizedStiefel,p)

Project p from the embedding onto the GeneralizedStiefel M, i.e. compute q as the polar decomposition of $p$ such that $q^{\mathrm{H}}Bq$ is the identity, where $\cdot^{\mathrm{H}}$ denotes the hermitian, i.e. complex conjugate transposed.

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ManifoldsBase.retract โ€” Method
retract(M::GeneralizedStiefel, p, X)
retract(M::GeneralizedStiefel, p, X, ::PolarRetraction)
retract(M::GeneralizedStiefel, p, X, ::ProjectionRetraction)

Compute the SVD-based retraction PolarRetraction on the GeneralizedStiefel manifold M, which in this case is the same as the projection based retraction employing the exponential map in the embedding and projecting the result back to the manifold.

The default retraction for this manifold is the ProjectionRetraction.

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