Generalized Stiefel

GeneralizedStiefel{𝔽,T,B} <: AbstractDecoratorManifold{𝔽}

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

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

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

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

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

\[T_p\mathcal M = \{ X \in 𝔽^{n×k} : p^{\mathrm{H}}BX + X^{\mathrm{H}}Bp=0_k\},\]

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 zero_vector are inherited from the embedding.

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

Constructor

GeneralizedStiefel(n, k, B=I_n, F=ℝ; parameter::Symbol=:type)

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

parameter: whether a type parameter should be used to store n and k. By default size is stored in type. Value can either be :field or :type.

rand(::GeneralizedStiefel; vector_at=nothing, σ::Real=1.0)

When vector_at is nothing, return a random (Gaussian) point p on the GeneralizedStiefel manifold M. This generates a (Gaussian) matrix of size $n×k$ with standard deviation σ and returns its (generalized) orthogonalized version, i.e. the projection onto the manifold of the Q component of its QR decomposition.

When vector_at is not nothing, return a (Gaussian) random vector from the tangent space $T_{vector\_at}\mathrm{St}(n,k)$ with mean zero and standard deviation σ by projecting a random Matrix onto the tangent vector at vector_at.

check_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 $⋅^{\mathrm{H}}$ is the complex conjugate transpose. The settings for approximately can be set with kwargs....

check_vector(M::GeneralizedStiefel, p, X; atol, rtol, kwargs...)

Check whether X is a valid tangent vector at p on the GeneralizedStiefel M=$\operatorname{St}(n,k,B)$. This requires that the AbstractNumbers fits, p is a valid point on M and it (approximately) holds that $p^{\mathrm{H}}BX + X^{\mathrm{H}}Bp = 0$, that is, the norm of the left hand side is at most atol or at most rtol times $2\lVert X \rVert_p$, the largest value this norm can attain.

default_retraction_method(M::GeneralizedStiefel)

Return ProjectionRetraction as the default retraction for the GeneralizedStiefel manifold.

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.

inverse_retract(M::GeneralizedStiefel{ℝ}, p, q, ::PolarInverseRetraction)

Compute the inverse of the PolarRetraction on the real GeneralizedStiefel manifold M for q close enough to p,

\[\operatorname{retr}_p^{-1} q = qZ - p,\]

where $Z$ is the symmetric positive definite solution of the Lyapunov equation $p^{\mathrm{T}}BqZ + Zq^{\mathrm{T}}Bp = 2I_k$, see Eqs. (3.4) and (3.5) in [SA23].

inverse_retract(M::GeneralizedStiefel{ℝ}, p, q, ::QRInverseRetraction)

Compute the inverse of the QRRetraction on the real GeneralizedStiefel manifold M for q close enough to p,

\[\operatorname{retr}_p^{-1} q = qR - p,\]

where $R$ is the upper triangular solution with positive diagonal of $p^{\mathrm{T}}BqR + R^{\mathrm{T}}q^{\mathrm{T}}Bp = 2I_k$, see Eqs. (3.7) and (3.8) in [SA23].

is_flat(M::GeneralizedStiefel)

Return true if GeneralizedStiefel M is one-dimensional.

manifold_dimension(M::GeneralizedStiefel)

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

\[\begin{aligned} \dim \mathrm{St}(n, k, B, ℝ) &= nk - \frac{1}{2}k(k+1) \\ \dim \mathrm{St}(n, k, B, ℂ) &= 2nk - k^2\\ \dim \mathrm{St}(n, k, B, ℍ) &= 4nk - k(2k-1) \end{aligned}\]

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 $\operatorname{Sym}(y) = \frac{y^{\mathrm{H}}+y}{2}$.

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 $⋅^{\mathrm{H}}$ denotes the hermitian, i.e. complex conjugate transposed.

retract(M::GeneralizedStiefel, p, X, ::PolarRetraction)

Compute the PolarRetraction on the GeneralizedStiefel manifold M. For real matrices this is the polar decomposition of $p + X$ with respect to $B$,

\[\operatorname{retr}_p X = (p + X)(I_k + X^{\mathrm{T}}BX)^{-1/2},\]

see Eq. (3.3) in [SA23]. For complex matrices it is the projection of $p + X$ onto the manifold.

retract(M::GeneralizedStiefel, p, X, ::ProjectionRetraction)

Compute the projection based retraction on the GeneralizedStiefel manifold M, which employs the exponential map in the embedding and projects the result back to the manifold,

\[\operatorname{retr}_p X = U(U^{\mathrm{H}}BU)^{-1/2}V^{\mathrm{H}},\]

where $UΣV^{\mathrm{H}} = p + X$ is the singular value decomposition.

retract(M::GeneralizedStiefel{ℝ}, p, X, ::CayleyRetraction)

Compute the CayleyRetraction on the real GeneralizedStiefel manifold M, the Cayley transform with respect to $B$,

\[\operatorname{retr}_p X = \Bigl(I_n - \frac{1}{2}W\Bigr)^{-1}\Bigl(I_n + \frac{1}{2}W\Bigr)p, \qquad W = \Bigl(I_n - \frac{1}{2}pp^{\mathrm{T}}B\Bigr)Xp^{\mathrm{T}}B - pX^{\mathrm{T}}\Bigl(I_n - \frac{1}{2}Bpp^{\mathrm{T}}\Bigr)B,\]

see Eq. (3.9) in [SA23].

retract(M::GeneralizedStiefel{ℝ}, p, X, ::QRRetraction)

Compute the QRRetraction on the real GeneralizedStiefel manifold M, the QR decomposition of $p + X$ with respect to $B$,

\[\operatorname{retr}_p X = (p + X)R^{-1},\]

where $R^{\mathrm{T}}R = (p + X)^{\mathrm{T}}B(p + X)$ is the Cholesky decomposition, see Eq. (3.6) in [SA23].