# Generalized Grassmann

Manifolds.GeneralizedGrassmann β Type
GeneralizedGrassmann{T,π½,TB<:AbstractMatrix} <: AbstractDecoratorManifold{π½}

The generalized Grassmann manifold $\operatorname{Gr}(n,k,B)$ consists of all subspaces spanned by $k$ linear independent vectors $π½^n$, where $π½ β \{β, β\}$ is either the real- (or complex-) valued vectors. This yields all $k$-dimensional subspaces of $β^n$ for the real-valued case and all $2k$-dimensional subspaces of $β^n$ for the second.

The manifold can be represented as

$$$\operatorname{Gr}(n, k, B) := \bigl\{ \operatorname{span}(p)\ \big|\ p β π½^{nΓk}, p^\mathrm{H}Bp = I_k\},$$$

where $β ^{\mathrm{H}}$ denotes the complex conjugate (or Hermitian) transpose and $I_k$ is the $kΓk$ identity matrix. This means, that the columns of $p$ form an unitary basis of the subspace with respect to the scaled inner product, that is a point on $\operatorname{Gr}(n,k,B)$, and hence the subspace can actually be represented by a whole equivalence class of representers. For $B=I_n$ this simplifies to the Grassmann manifold.

The tangent space at a point (subspace) $p$ is given by

$$$T_x\mathrm{Gr}(n,k,B) = \bigl\{ X β π½^{nΓk} : X^{\mathrm{H}}Bp + p^{\mathrm{H}}BX = 0_{k} \bigr\},$$$

where $0_{k}$ denotes the $kΓk$ zero matrix.

Note that a point $p β \operatorname{Gr}(n,k,B)$ might be represented by different matrices (i.e. matrices with $B$-unitary column vectors that span the same subspace). Different representations of $p$ also lead to different representation matrices for the tangent space $T_p\mathrm{Gr}(n,k,B)$

The manifold is named after Hermann G. GraΓmann (1809-1877).

Constructor

GeneralizedGrassmann(n, k, B=I_n, field=β)

Generate the (real-valued) Generalized Grassmann manifold of $nΓk$ dimensional orthonormal matrices with scalar product B.

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Base.exp β Method
exp(M::GeneralizedGrassmann, p, X)

Compute the exponential map on the GeneralizedGrassmann M $= \mathrm{Gr}(n,k,B)$ starting in p with tangent vector (direction) X. Let $X^{\mathrm{H}}BX = USV$ denote the SVD decomposition of $X^{\mathrm{H}}BX$. Then the exponential map is written using

$$$\exp_p X = p V\cos(S)V^\mathrm{H} + U\sin(S)V^\mathrm{H},$$$

where $β ^{\mathrm{H}}$ denotes the complex conjugate transposed or Hermitian and the cosine and sine are applied element wise to the diagonal entries of $S$.

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Base.log β Method
log(M::GeneralizedGrassmann, p, q)

Compute the logarithmic map on the GeneralizedGrassmann M $= \mathcal M=\mathrm{Gr}(n,k,B)$, i.e. the tangent vector X whose corresponding geodesic starting from p reaches q after time 1 on M. The formula reads

$$$\log_p q = Vβ \operatorname{atan}(S) β U^\mathrm{H},$$$

where $β ^{\mathrm{H}}$ denotes the complex conjugate transposed or Hermitian. The matrices $U$ and $V$ are the unitary matrices, and $S$ is the diagonal matrix containing the singular values of the SVD-decomposition

$$$USV = (q^\mathrm{H}Bp)^{-1} ( q^\mathrm{H} - q^\mathrm{H}Bpp^\mathrm{H}).$$$

In this formula the $\operatorname{atan}$ is meant elementwise.

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Base.rand β Method
rand(::GeneralizedGrassmann; vector_at=nothing, Ο::Real=1.0)

When vector_at is nothing, return a random (Gaussian) point p on the GeneralizedGrassmann manifold M by generating a (Gaussian) matrix with standard deviation Ο and return the (generalized) orthogonalized version, i.e. return the projection onto the manifold of the Q component of the QR decomposition of the random matrix of size $nΓk$.

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 Ο by projecting a random Matrix onto the tangent vector at vector_at.

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ManifoldsBase.change_metric β Method
change_metric(M::GeneralizedGrassmann, ::EuclideanMetric, p X)

Change X to the corresponding vector with respect to the metric of the GeneralizedGrassmann M, i.e. let $B=LL'$ be the Cholesky decomposition of the matrix M.B, then the corresponding vector is $L\X$.

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ManifoldsBase.change_representer β Method
change_representer(M::GeneralizedGrassmann, ::EuclideanMetric, p, X)

Change X to the corresponding representer of a cotangent vector at p with respect to the scaled metric of the GeneralizedGrassmann M, i.e, since

$$$g_p(X,Y) = \operatorname{tr}(Y^{\mathrm{H}}BZ) = \operatorname{tr}(X^{\mathrm{H}}Z) = β¨X,Zβ©$$$

has to hold for all $Z$, where the repreenter X is given, the resulting representer with respect to the metric on the GeneralizedGrassmann is given by $Y = B^{-1}X$.

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ManifoldsBase.check_point β Method
check_point(M::GeneralizedGrassmann, p)

Check whether p is representing a point on the GeneralizedGrassmann M, i.e. its a n-by-k matrix of unitary column vectors with respect to the B inner product and of correct eltype with respect to π½.

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ManifoldsBase.check_vector β Method
check_vector(M::GeneralizedGrassmann, p, X; kwargs...)

Check whether X is a tangent vector in the tangent space of p on the GeneralizedGrassmann M, i.e. that X is of size and type as well as that

$$$p^{\mathrm{H}}BX + \overline{X^{\mathrm{H}}Bp} = 0_k,$$$

where $β ^{\mathrm{H}}$ denotes the complex conjugate transpose or Hermitian, $\overline{β }$ the (elementwise) complex conjugate, and $0_k$ denotes the $kΓk$ zero natrix.

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ManifoldsBase.inner β Method
inner(M::GeneralizedGrassmann, p, X, Y)

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

$$$g_p(X,Y) = \operatorname{tr}(X^{\mathrm{H}}BY),$$$

where $β ^{\mathrm{H}}$ denotes the complex conjugate transposed or Hermitian.

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ManifoldsBase.project β Method
project(M::GeneralizedGrassmann, p, X)

Project the n-by-k X onto the tangent space of p on the GeneralizedGrassmann M, which is computed by

$$$\operatorname{proj_p}(X) = X - pp^{\mathrm{H}}B^\mathrm{T}X,$$$

where $β ^{\mathrm{H}}$ denotes the complex conjugate transposed or Hermitian and $β ^{\mathrm{T}}$ the transpose.

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ManifoldsBase.project β Method
project(M::GeneralizedGrassmann, p)

Project p from the embedding onto the GeneralizedGrassmann M, i.e. compute q as the polar decomposition of $p$ such that $q^{\mathrm{H}}Bq$ is the identity, where $β ^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transpose.

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