# Stiefel

Manifolds.StiefelType
Stiefel{n,k,𝔽} <: AbstractEmbeddedManifold{𝔽,DefaultIsometricEmbeddingType}

The Stiefel manifold consists of all $n × k$, $n ≥ k$ unitary matrices, i.e.

$\operatorname{St}(n,k) = \bigl\{ p ∈ 𝔽^{n × k}\ \big|\ p^{\mathrm{H}}p = I_k \bigr\},$

where $𝔽 ∈ \{ℝ, ℂ\}$, $\cdot^{\mathrm{H}}$ denotes the complex conjugate transpose or Hermitian, and $I_k ∈ ℝ^{k × k}$ denotes the $k × k$ identity matrix.

The tangent space at a point $p ∈ \mathcal M$ is given by

$T_p \mathcal M = \{ X ∈ 𝔽^{n × k} : p^{\mathrm{H}}X + X^{\mathrm{H}}p = 0_k\},$

where $0_k$ is the $k × k$ zero matrix.

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

The manifold is named after Eduard L. Stiefel (1909–1978).

Constructor

Stiefel(n, k, field = ℝ)

Generate the (real-valued) Stiefel manifold of $n × k$ dimensional orthonormal matrices.

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Base.expMethod
exp(M::Stiefel, p, X)

Compute the exponential map on the Stiefel{n,k,𝔽}() manifold M emanating from p in tangent direction X.

$\exp_p X = \begin{pmatrix} p\\X \end{pmatrix} \operatorname{Exp} \left( \begin{pmatrix} p^{\mathrm{H}}X & - X^{\mathrm{H}}X\\ I_n & p^{\mathrm{H}}X\end{pmatrix} \right) \begin{pmatrix} \exp( -p^{\mathrm{H}}X) \\ 0_n\end{pmatrix},$

where $\operatorname{Exp}$ denotes matrix exponential, $\cdot^{\mathrm{H}}$ denotes the complex conjugate transpose or Hermitian, and $I_k$ and $0_k$ are the identity matrix and the zero matrix of dimension $k × k$, respectively.

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ManifoldsBase.check_manifold_pointMethod
check_manifold_point(M::Stiefel, p; kwargs...)

Check whether p is a valid point on the Stiefel M=$\operatorname{St}(n,k)$, i.e. that it has the right AbstractNumbers type and $p^{\mathrm{H}}p$ 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_vectorMethod
check_tangent_vector(M::Stiefel, p, X; check_base_point = true, kwargs...)

Checks whether X is a valid tangent vector at p on the Stiefel M=$\operatorname{St}(n,k)$, i.e. the AbstractNumbers fits and it (approximately) holds that $p^{\mathrm{H}}X + X^{\mathrm{H}}p = 0$. The optional parameter check_base_point indicates, whether to call check_manifold_point for p. The settings for approximately can be set with kwargs....

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ManifoldsBase.inverse_retractMethod
inverse_retract(M::Stiefel, p, q, ::PolarInverseRetraction)

Compute the inverse retraction based on a singular value decomposition for two points p, q on the Stiefel manifold M. This follows the folloing approach: From the Polar retraction we know that

$\operatorname{retr}_p^{-1}q = qs - t$

if such a symmetric positive definite $k × k$ matrix exists. Since $qs - t$ is also a tangent vector at $p$ we obtain

$p^{\mathrm{H}}qs + s(p^{\mathrm{H}}q)^{\mathrm{H}} + 2I_k = 0,$

which can either be solved by a Lyapunov approach or a continuous-time algebraic Riccati equation as described in [KanekoFioriTanaka2013]

This implementation follows the Lyapunov approach.

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ManifoldsBase.manifold_dimensionMethod
manifold_dimension(M::Stiefel)

Return the dimension of the Stiefel manifold M=$\operatorname{St}(n,k,𝔽)$. The dimension is given by

\begin{aligned} \dim \mathrm{St}(n, k, ℝ) &= nk - \frac{1}{2}k(k+1)\\ \dim \mathrm{St}(n, k, ℂ) &= 2nk - k^2\\ \dim \mathrm{St}(n, k, ℍ) &= 4nk - k(2k-1) \end{aligned}
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ManifoldsBase.projectMethod
project(M::Stiefel,p)

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

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ManifoldsBase.projectMethod
project(M::Stiefel, p, X)

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

$\operatorname{proj}_{\mathcal M}(p, X) = X - p \operatorname{Sym}(p^{\mathrm{H}}X),$

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

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ManifoldsBase.retractMethod
retract(M::Stiefel, p, X, ::QRRetraction )

Compute the QR-based retraction QRRetraction on the Stiefel manifold M. With $QR = p + X$ the retraction reads

$\operatorname{retr}_p X = QD,$

where $D$ is a $n × k$ matrix with

$D = \operatorname{diag}\bigl(\operatorname{sgn}(R_{ii}+0,5)_{i=1}^k \bigr),$

where $\operatorname{sgn}(p) = \begin{cases} 1 & \text{ for } p > 0,\\ 0 & \text{ for } p = 0,\\ -1& \text{ for } p < 0. \end{cases}$

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## Literature

• H. Sato and T. Iawi, “A complex singular value decomposition algorithm based on the Riemannian Newton method”, in: IEEE 52nd Annual Conference on Decision and Control (CDC), 2013, p. 2972--2978, doi: 10.1109/CDC.2013.6760335
• H. Sato: “Riemannian conjugate gradient method for complex singular value decomposition problem”, in: IEEE 53rd Annual COnference on Decision and Control (CDC), 2014, p. 5849–5854, doi: 10.1109/CDC.2014.7040305