Stiefel

CayleyRetraction <: AbstractRetractionMethod

A retraction based on the Cayley transform, which is realized by using the PadeRetraction{1}.

PadeRetraction{m} <: AbstractRetractionMethod

A retraction based on the Padé approximation of order $m$

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

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

where $0_k$ is the $k × k$ zero matrix and $\overline{\cdot}$ the (elementwise) complex conjugate.

This manifold is modeled as an embedded manifold to the Euclidean, i.e. several functions like the inner product and the zero_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.

uniform_distribution(M::Stiefel{n,k,ℝ}, p)

Uniform distribution on given (real-valued) Stiefel M. Specifically, this is the normalized Haar and Hausdorff measure on M. Generated points will be of similar type as p.

The implementation is based on Section 2.5.1 in ^{[Chikuse2003]}; see also Theorem 2.2.1(iii) in ^{[Chikuse2003]}.

check_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....

check_vector(M::Stiefel, p, X; 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 $p^{\mathrm{H}}X + \overline{X^{\mathrm{H}}p} = 0$, where $\cdot^{\mathrm{H}}$ denotes the Hermitian and $\overline{\cdot}$ the (elementwise) complex conjugate. The settings for approximately can be set with kwargs....

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.

This implementation follows the Lyapunov approach.

inverse_retract(M::Stiefel, p, q, ::QRInverseRetraction)

Compute the inverse retraction based on a qr decomposition for two points p, q on the Stiefel manifold M and return the resulting tangent vector in X. The computation follows Algorithm 1 in ^{[KanekoFioriTanaka2013]}.

manifold_dimension(M::Stiefel)

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

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

representation_size(M::Stiefel)

Returns the representation size of the Stiefel M=$\operatorname{St}(n,k)$, i.e. (n,k), which is the matrix dimensions.

retract(::Stiefel, p, X, ::CayleyRetraction)

Compute the retraction on the Stiefel that is based on the Cayley transform^[Zhu2017]. Using

\[ W_{p,X} = \operatorname{P}_pXp^{\mathrm{H}} - pX^{\mathrm{H}}\operatorname{P_p} \quad\text{where}  \operatorname{P}_p = I - \frac{1}{2}pp^{\mathrm{H}}\]

the formula reads

\[ \operatorname{retr}_pX = \Bigl(I - \frac{1}{2}W_{p,X}\Bigr)^{-1}\Bigl(I + \frac{1}{2}W_{p,X}\Bigr)p.\]

It is implemented as the case $m=1$ of the PadeRetraction.

retract(M::Stiefel, p, X, ::PadeRetraction{m})

Compute the retraction on the Stiefel manifold M based on the Padé approximation of order $m$^{[ZhuDuan2018]}. Let $p_m$ and $q_m$ be defined for any matrix $A ∈ ℝ^{n×x}$ as

\[ p_m(A) = \sum_{k=0}^m \frac{(2m-k)!m!}{(2m)!(m-k)!}\frac{A^k}{k!}\]

and

\[ q_m(A) = \sum_{k=0}^m \frac{(2m-k)!m!}{(2m)!(m-k)!}\frac{(-A)^k}{k!}\]

respectively. Then the Padé approximation (of the matrix exponential $\exp(A)$) reads

\[ r_m(A) = q_m(A)^{-1}p_m(A)\]

Defining further

\[ W_{p,X} = \operatorname{P}_pXp^{\mathrm{H}} - pX^{\mathrm{H}}\operatorname{P_p} \quad\text{where}  \operatorname{P}_p = I - \frac{1}{2}pp^{\mathrm{H}}\]

the retraction reads

\[ \operatorname{retr}_pX = r_m(W_{p,X})p\]

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

Compute the SVD-based retraction PolarRetraction on the Stiefel manifold M. With $USV = p + X$ the retraction reads

\[\operatorname{retr}_p X = U\bar{V}^\mathrm{H}.\]

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

vector_transport_direction(::Stiefel, p, X, d, ::DifferentiatedRetractionVectorTransport{CayleyRetraction})

Compute the vector transport given by the differentiated retraction of the CayleyRetraction, cf. ^[Zhu2017] Equation (17).

The formula reads

\[\operatorname{T}_{p,d}(X) = \Bigl(I - \frac{1}{2}W_{p,d}\Bigr)^{-1}W_{p,X}\Bigl(I - \frac{1}{2}W_{p,d}\Bigr)^{-1}p,\]

with

\[ W_{p,X} = \operatorname{P}_pXp^{\mathrm{H}} - pX^{\mathrm{H}}\operatorname{P_p} \quad\text{where}  \operatorname{P}_p = I - \frac{1}{2}pp^{\mathrm{H}}\]

Since this is the differentiated retraction as a vector transport, the result will be in the tangent space at $q=\operatorname{retr}_p(d)$ using the CayleyRetraction.

vector_transport_direction(M::Stiefel, p, X, d, DifferentiatedRetractionVectorTransport{PolarRetraction})

Compute the vector transport by computing the push forward of retract(::Stiefel, ::Any, ::Any, ::PolarRetraction) Section 3.5 of ^[Zhu2017]:

\[T_{p,d}^{\text{Pol}}(X) = q*Λ + (I-qq^{\mathrm{T}})X(1+d^\mathrm{T}d)^{-\frac{1}{2}},\]

where $q = \operatorname{retr}^{\mathrm{Pol}}_p(d)$, and $Λ$ is the unique solution of the Sylvester equation

\[ Λ(I+d^\mathrm{T}d)^{\frac{1}{2}} + (I + d^\mathrm{T}d)^{\frac{1}{2}} = q^\mathrm{T}X - X^\mathrm{T}q\]

vector_transport_direction(M::Stiefel, p, X, d, DifferentiatedRetractionVectorTransport{QRRetraction})

Compute the vector transport by computing the push forward of the retract(::Stiefel, ::Any, ::Any, ::QRRetraction), See ^{[AbsilMahonySepulchre2008]}, p. 173, or Section 3.5 of ^[Zhu2017].

\[T_{p,d}^{\text{QR}}(X) = q*\rho_{\mathrm{s}}(q^\mathrm{T}XR^{-1}) + (I-qq^{\mathrm{T}})XR^{-1},\]

where $q = \operatorname{retr}^{\mathrm{QR}}_p(d)$, $R$ is the $R$ factor of the QR decomposition of $p + d$, and

\[\bigl( \rho_{\mathrm{s}}(A) \bigr)_{ij} = \begin{cases} A_{ij}&\text{ if } i > j\\ 0 \text{ if } i = j\\ -A_{ji} \text{ if } i < j.\\ \end{cases}\]

vector_transport_to(M::Stiefel, p, X, q, DifferentiatedRetractionVectorTransport{PolarRetraction})

Compute the vector transport by computing the push forward of the retract(M::Stiefel, ::Any, ::Any, ::PolarRetraction), see Section 4 of ^{[HuangGallivanAbsil2015]} or Section 3.5 of ^[Zhu2017]:

\[T_{q\gets p}^{\text{Pol}}(X) = q*Λ + (I-qq^{\mathrm{T}})X(1+d^\mathrm{T}d)^{-\frac{1}{2}},\]

where $d = \bigl( \operatorname{retr}^{\mathrm{Pol}}_p\bigr)^{-1}(q)$, and $Λ$ is the unique solution of the Sylvester equation

\[ Λ(I+d^\mathrm{T}d)^{\frac{1}{2}} + (I + d^\mathrm{T}d)^{\frac{1}{2}} = q^\mathrm{T}X - X^\mathrm{T}q\]

vector_transport_to(M::Stiefel, p, X, q, DifferentiatedRetractionVectorTransport{QRRetraction})

Compute the vector transport by computing the push forward of the retract(M::Stiefel, ::Any, ::Any, ::QRRetraction), see ^{[AbsilMahonySepulchre2008]}, p. 173, or Section 3.5 of ^[Zhu2017].

\[T_{q \gets p}^{\text{QR}}(X) = q*\rho_{\mathrm{s}}(q^\mathrm{T}XR^{-1}) + (I-qq^{\mathrm{T}})XR^{-1},\]

where $d = \bigl(\operatorname{retr}^{\mathrm{QR}}\bigr)^{-1}_p(q)$, $R$ is the $R$ factor of the QR decomposition of $p+X$, and

\[\bigl( \rho_{\mathrm{s}}(A) \bigr)_{ij} = \begin{cases} A_{ij}&\text{ if } i > j\\ 0 \text{ if } i = j\\ -A_{ji} \text{ if } i < j.\\ \end{cases}\]

vector_transport_to(M::Stiefel, p, X, q, ::ProjectionTransport)

Compute a vector transport by projection, i.e. project X from the tangent space at p by projection it onto the tangent space at q.

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.

get_basis(M::Stiefel{n,k,ℝ}, p, B::DefaultOrthonormalBasis) where {n,k}

Create the default basis using the parametrization for any $X ∈ T_p\mathcal M$. Set $p_\bot \in ℝ^{n\times(n-k)}$ the matrix such that the $n\times n$ matrix of the common columns $[p\ p_\bot]$ is an ONB. For any skew symmetric matrix $a ∈ ℝ^{k\times k}$ and any $b ∈ ℝ^{(n-k)\times k}$ the matrix

\[X = pa + p_\bot b ∈ T_p\mathcal M\]

and we can use the $\frac{1}{2}k(k-1) + (n-k)k = nk-\frac{1}{2}k(k+1)$ entries of $a$ and $b$ to specify a basis for the tangent space. using unit vectors for constructing both the upper matrix of $a$ to build a skew symmetric matrix and the matrix b, the default basis is constructed.

Since $[p\ p_\bot]$ is an automorphism on $ℝ^{n\times p}$ the elements of $a$ and $b$ are orthonormal coordinates for the tangent space. To be precise exactly one element in the upper trangular entries of $a$ is set to $1$ its symmetric entry to $-1$ and we normalize with the factor $\frac{1}{\sqrt{2}}$ and for $b$ one can just use unit vectors reshaped to a matrix to obtain orthonormal set of parameters.

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.

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

ApproximateLogarithmicMap <: ApproximateInverseRetraction

An approximate implementation of the logarithmic map, which is an inverse_retraction. See inverse_retract(::MetricManifold{ℝ,Stiefel{n,k,ℝ},CanonicalMetric}, ::Any, ::Any, ::ApproximateLogarithmicMap) where {n,k} for a use case.

Fields

• max_iterations – maximal number of iterations used in the approximation
• tolerance – a tolerance used as a stopping criterion

CanonicalMetric <: AbstractMetric

The Canonical Metric refers to a metric for the Stiefel manifold, see^{[EdelmanAriasSmith1998]}.

q = exp(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, CanonicalMetric}, p, X)
exp!(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, q, CanonicalMetric}, p, X)

Compute the exponential map on the Stiefel(n,k) manifold with respect to the CanonicalMetric.

First, decompose The tangent vector $X$ into its horizontal and vertical component with respect to $p$, i.e.

\[X = pp^{\mathrm{T}}X + (I_n-pp^{\mathrm{T}})X,\]

where $I_n$ is the $n\times n$ identity matrix. We introduce $A=p^{\mathrm{T}}X$ and $QR = (I_n-pp^{\mathrm{T}})X$ the qr decomposition of the vertical component. Then using the matrix exponential $\operatorname{Exp}$ we introduce $B$ and $C$ as

\[\begin{pmatrix} B\\C \end{pmatrix} \coloneqq \operatorname{Exp}\left( \begin{pmatrix} A & -R^{\mathrm{T}}\\ R & 0 \end{pmatrix} \right) \begin{pmatrix}I_k\\0\end{pmatrix}\]

the exponential map reads

\[q = \exp_p X = pC + QB.\]

For more details, see ^{[EdelmanAriasSmith1998][Zimmermann2017]}.

inner(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, X, CanonicalMetric}, p, X, Y)

compute the inner procuct on the Stiefel manifold with respect to the CanonicalMetric. The formula reads

\[g_p(X,Y) = \operatorname{tr}\bigl( X^{\mathrm{T}}(I_n - \frac{1}{2}pp^{\mathrm{T}})Y \bigr).\]

X = inverse_retract(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, CanonicalMetric}, p, q, a::ApproximateLogarithmicMap)
inverse_retract!(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, X, CanonicalMetric}, p, q, a::ApproximateLogarithmicMap)

Compute an approximation to the logarithmic map on the Stiefel(n,k) manifold with respect to the CanonicalMetric using a matrix-algebraic based approach to an iterative inversion of the formula of the exp.

The algorithm is derived in^{[Zimmermann2017]} and it uses the max_iterations and the tolerance field from the ApproximateLogarithmicMap.