(Real) Symplectic Stiefel

SymplecticStiefel{T,𝔽} <: AbstractEmbeddedManifold{𝔽, DefaultIsometricEmbeddingType}

The symplectic Stiefel manifold consists of all $2n×2k, n ≥ k$ matrices satisfying the requirement

\[\mathrm{SpSt}(2n, 2k, ℝ) := \bigl\{ p ∈ ℝ^{2n×2k} \ \big| \ p^{\mathrm{T}}J_{2n}p = J_{2k} \bigr\},\]

where $J_{2n}$ denotes the SymplecticElement

\[J_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}.\]

The symplectic Stiefel tangent space at $p$ can be parametrized as [BZ21]

\[\begin{align*} T_p\mathrm{SpSt}(2n, 2k) &= \{X ∈ ℝ^{2n×2k} ∣ p^{T}J_{2n}X + X^{T}J_{2n}p = 0 \}, \\ &= \{X = pΩ + p^sB \mid Ω ∈ ℝ^{2k×2k}, Ω^+ = -Ω, \\ &\quad\qquad p^s ∈ \mathrm{SpSt}(2n, 2(n- k)), B ∈ ℝ^{2(n-k)×2k}, \}, \end{align*}\]

where $Ω ∈ \mathfrak{sp}(2n,F)$ is Hamiltonian and $p^s$ means the symplectic complement of $p$ s.t. $p^{+}p^{s} = 0$. Here $p^+$ denotes the symplectic_inverse.

You can also use StiefelPoint and StiefelTangentVector with this manifold, they are equivalent to using arrays.

Constructor

SymplecticStiefel(2n::Int, 2k::Int, field::AbstractNumbers=ℝ; parameter::Symbol=:type)

Generate the (real-valued) symplectic Stiefel manifold of $2n×2k$ matrices which span a $2k$ dimensional symplectic subspace of $ℝ^{2n×2k}$. The constructor for the SymplecticStiefel manifold accepts the even column dimension $2n$ and an even number of columns $2k$ for the real symplectic Stiefel manifold with elements $p ∈ ℝ^{2n×2k}$.

exp(::SymplecticStiefel, p, X)
exp!(M::SymplecticStiefel, q, p, X)

Compute the exponential mapping

\[ \exp\colon T\mathrm{SpSt}(2n, 2k) → \mathrm{SpSt}(2n, 2k)\]

at a point $p ∈ \mathrm{SpSt}(2n, 2k)$ in the direction of $X ∈ T_p\mathrm{SpSt}(2n, 2k)$.

The tangent vector $X$ can be written in the form $X = \bar{\Omega}p$ [BZ21], with

\[ \bar{\Omega} = X (p^{\mathrm{T}}p)^{-1}p^{\mathrm{T}} + J_{2n}p(p^{\mathrm{T}}p)^{-1}X^{\mathrm{T}}(I_{2n} - J_{2n}^{\mathrm{T}}p(p^{\mathrm{T}}p)^{-1}p^{\mathrm{T}}J_{2n})J_{2n} ∈ ℝ^{2n×2n},\]

where $J_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}$ denotes the SymplecticElement.

Using this expression for $X$, the exponential mapping can be computed as

\[ \exp_p(X) = \operatorname{Exp}([\bar{\Omega} - \bar{\Omega}^{\mathrm{T}}]) \operatorname{Exp}(\bar{\Omega}^{\mathrm{T}})p,\]

where $\operatorname{Exp}(⋅)$ denotes the matrix exponential.

Computing the above mapping directly however, requires taking matrix exponentials of two $2n×2n$ matrices, which is computationally expensive when $n$ increases. Therefore we instead follow [BZ21] who express the above exponential mapping in a way which only requires taking matrix exponentials of an $8k×8k$ matrix and a $4k×4k$ matrix.

To this end, first define

\[\bar{A} = J_{2k}p^{\mathrm{T}}X(p^{\mathrm{T}}p)^{-1}J_{2k} + (p^{\mathrm{T}}p)^{-1}X^{\mathrm{T}}(p - J_{2n}^{\mathrm{T}}p(p^{\mathrm{T}}p)^{-1}J_{2k}) ∈ ℝ^{2k×2k},\]

and

\[\bar{H} = (I_{2n} - pp^+)J_{2n}X(p^{\mathrm{T}}p)^{-1}J_{2k} ∈ ℝ^{2n×2k}.\]

We then let $\bar{\Delta} = p\bar{A} + \bar{H}$, and define the matrices

\[ γ = \left[\left(I_{2n} - \frac{1}{2}pp^+\right)\bar{\Delta} \quad -p \right] ∈ ℝ^{2n×4k},\]

and

\[ λ = \left[J_{2n}^{\mathrm{T}}pJ_{2k} \quad \left(\bar{\Delta}^+\left(I_{2n} - \frac{1}{2}pp^+\right)\right)^{\mathrm{T}}\right] ∈ ℝ^{2n×4k}.\]

With the above defined matrices it holds that $\bar{\Omega} = λγ^{\mathrm{T}}$. As a last preliminary step, concatenate $γ$ and $λ$ to define the matrices $Γ = [λ \quad -γ] ∈ ℝ^{2n×8k}$ and $Λ = [γ \quad λ] ∈ ℝ^{2n×8k}$.

With these matrix constructions done, we can compute the exponential mapping as

\[ \exp_p(X) = Γ \operatorname{Exp}(ΛΓ^{\mathrm{T}}) \begin{bmatrix} 0_{4k} \\ I_{4k} \end{bmatrix} \operatorname{Exp}(λγ^{\mathrm{T}}) \begin{bmatrix} 0_{2k} \\ I_{2k} \end{bmatrix}.\]

which only requires computing the matrix exponentials of $ΛΓ^{\mathrm{T}} ∈ ℝ^{8k×8k}$ and $λγ^{\mathrm{T}} ∈ ℝ^{4k×4k}$.

inv(::SymplecticStiefel, A)
inv!(::SymplecticStiefel, q, p)

Compute the symplectic inverse $A^+$ of matrix $A ∈ ℝ^{2n×2k}$. Given a matrix

\[A ∈ ℝ^{2n×2k},\quad A = \begin{bmatrix} A_{1, 1} & A_{1, 2} \\ A_{2, 1} & A_{2, 2} \end{bmatrix}, \quad A_{i, j} ∈ ℝ^{2n×2k}\]

the symplectic inverse is defined as:

\[A^{+} := J_{2k}^{\mathrm{T}} A^{\mathrm{T}} J_{2n},\]

where $J_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}$ denotes the SymplecticElement.

The symplectic inverse of a matrix A can be expressed explicitly as:

\[A^{+} = \begin{bmatrix} A_{2, 2}^{\mathrm{T}} & -A_{1, 2}^{\mathrm{T}} \\[1.2mm] -A_{2, 1}^{\mathrm{T}} & A_{1, 1}^{\mathrm{T}} \end{bmatrix}.\]

rand(M::SymplecticStiefel; vector_at=nothing, σ = 1.0)

Generate a random point $p ∈ \mathrm{SpSt}(2n, 2k)$ or a random tangent vector $X ∈ T_p\mathrm{SpSt}(2n, 2k)$ if vector_at is set to a point $p ∈ \mathrm{Sp}(2n)$.

A random point on $\mathrm{SpSt}(2n, 2k)$ is found by first generating a random point on the symplectic manifold $\mathrm{Sp}(2n)$, and then projecting onto the Symplectic Stiefel manifold using the canonical_project $π_{\mathrm{SpSt}(2n, 2k)}$. That is, $p = π_{\mathrm{SpSt}(2n, 2k)}(p_{\mathrm{Sp}})$.

To generate a random tangent vector in $T_p\mathrm{SpSt}(2n, 2k)$ this code exploits the second tangent vector space parametrization of SymplecticStiefel, that any $X ∈ T_p\mathrm{SpSt}(2n, 2k)$ can be written as $X = pΩ_X + p^sB_X$. To generate random tangent vectors at $p$ then, this function sets $B_X = 0$ and generates a random Hamiltonian matrix $Ω_X ∈ \mathfrak{sp}(2n,F)$ with Frobenius norm of σ before returning $X = pΩ_X$.

X = riemannian_gradient(::SymplecticStiefel, f, p, Y; embedding_metric::EuclideanMetric=EuclideanMetric())
riemannian_gradient!(::SymplecticStiefel, f, X, p, Y; embedding_metric::EuclideanMetric=EuclideanMetric())

Compute the riemannian gradient X of f on SymplecticStiefel at a point p, provided that the gradient of the function $\tilde f$, which is f continued into the embedding is given by Y. The metric in the embedding is the Euclidean metric.

The manifold gradient X is computed from Y as

\[ X = Yp^{\mathrm{T}}p + J_{2n}pY^{\mathrm{T}}J_{2n}p,\]

where $J_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}$ denotes the SymplecticElement.

canonical_project(::SymplecticStiefel, p_Sp)
canonical_project!(::SymplecticStiefel, p, p_Sp)

Define the canonical projection from $\mathrm{Sp}(2n, 2n)$ onto $\mathrm{SpSt}(2n, 2k)$, by projecting onto the first $k$ columns and the $n + 1$'th onto the $n + k$'th columns [BZ21].

It is assumed that the point $p$ is on $\mathrm{Sp}(2n, 2n)$.

get_total_space(::SymplecticStiefel)

Return the total space of the SymplecticStiefel manifold, which is the corresponding SymplecticMatrices manifold.

symplectic_inverse_times(::SymplecticStiefel, p, q)
symplectic_inverse_times!(::SymplecticStiefel, A, p, q)

Directly compute the symplectic inverse of $p ∈ \mathrm{SpSt}(2n, 2k)$, multiplied with $q ∈ \mathrm{SpSt}(2n, 2k)$. That is, this function efficiently computes $p^+q = (J_{2k}p^{\mathrm{T}}J_{2n})q ∈ ℝ^{2k×2k}$, where $J_{2n}, J_{2k}$ are the SymplecticElement of sizes $2n×2n$ and $2k×2k$ respectively.

This function performs this common operation without allocating more than a $2k×2k$ matrix to store the result in, or in the case of the in-place function, without allocating memory at all.

check_point(M::SymplecticStiefel, p; kwargs...)

Check whether p is a valid point on the SymplecticStiefel, $\mathrm{SpSt}(2n, 2k)$ manifold, that is $p^{+}p$ is the identity, $(⋅)^+$ denotes the symplectic_inverse.

check_vector(M::SymplecticMatrices, p, X; kwargs...)

Checks whether X is a valid tangent vector at p on the SymplecticStiefel, $\mathrm{SpSt}(2n, 2k)$ manifold.

The check consists of verifying that $H = p^{+}X ∈ 𝔤_{2k}$, where $𝔤$ is the Lie Algebra of the symplectic group $\mathrm{Sp}(2k)$, that is the set of [HamiltonianMatrices])(@ref), where $(⋅)^+$ denotes the symplectic_inverse.

inner(M::SymplecticStiefel, p, X. Y)

Compute the Riemannian inner product $g^{\mathrm{SpSt}}$ at $p ∈ \mathrm{SpSt}$ of tangent vectors $Y, X ∈ T_p\mathrm{SpSt}$. Given by Proposition 3.10 in [BZ21].

\[g^{\mathrm{SpSt}}_p(X, Y) = \operatorname{tr}\Bigl( X^{\mathrm{T}}\bigl( I_{2n} - \frac{1}{2}J_{2n}^{\mathrm{T}} p(p^{\mathrm{T}}p)^{-1}p^{\mathrm{T}}J_{2n} \bigr) Y (p^{\mathrm{T}}p)^{-1}\Bigr).\]

inverse_retract(::SymplecticStiefel, p, q, ::CayleyInverseRetraction)
inverse_retract!(::SymplecticStiefel, X, p, q, ::CayleyInverseRetraction)

Compute the Cayley Inverse Retraction $X = \mathcal{L}_p^{\mathrm{SpSt}}(q)$ such that the Cayley Retraction from $p$ along $X$ lands at $q$, i.e. $\mathcal{R}_p(X) = q$ [BZ21].

For $p, q ∈ \mathrm{SpSt}(2n, 2k, ℝ)$ we can define the inverse cayley retraction as long as the following matrices exist.

\[ U = (I + p^+ q)^{-1} ∈ ℝ^{2k×2k}, \quad V = (I + q^+ p)^{-1} ∈ ℝ^{2k×2k},\]

where $(⋅)^+$ denotes the symplectic_inverse.

THen the inverse retraction reads

\[\mathcal{L}_p^{\mathrm{Sp}}(q) = 2p\bigl(V - U\bigr) + 2\bigl((p + q)U - p\bigr) ∈ T_p\mathrm{Sp}(2n).\]

is_flat(::SymplecticStiefel)

Return false. SymplecticStiefel is not a flat manifold.

manifold_dimension(::SymplecticStiefel)

Returns the dimension of the symplectic Stiefel manifold embedded in $ℝ^{2n×2k}$, i.e. [BZ21]

\[ \operatorname{dim}(\mathrm{SpSt}(2n, 2k)) = (4n - 2k + 1)k.\]

project(::SymplecticStiefel, p, A)
project!(::SymplecticStiefel, Y, p, A)

Given a point $p ∈ \mathrm{SpSt}(2n, 2k)$, project an element $A ∈ ℝ^{2n×2k}$ onto the tangent space $T_p\mathrm{SpSt}(2n, 2k)$ relative to the euclidean metric of the embedding $ℝ^{2n×2k}$.

That is, we find the element $X ∈ T_p\mathrm{SpSt}(2n, 2k)$ which solves the constrained optimization problem

\[ \displaystyle\operatorname{min}_{X ∈ ℝ^{2n×2k}} \frac{1}{2}||X - A||^2, \quad \text{s.t.}\; h(X) := X^{\mathrm{T}} J p + p^{\mathrm{T}} J X = 0,\]

where $h : ℝ^{2n×2k} → \operatorname{skew}(2k)$ defines the restriction of $X$ onto the tangent space $T_p\mathrm{SpSt}(2n, 2k)$.

retract(::SymplecticStiefel, p, X, ::CayleyRetraction)
retract!(::SymplecticStiefel, q, p, X, ::CayleyRetraction)

Compute the Cayley retraction on the Symplectic Stiefel manifold, from p along X (computed inplace of q).

Given a point $p ∈ \mathrm{SpSt}(2n, 2k)$, every tangent vector $X ∈ T_p\mathrm{SpSt}(2n, 2k)$ is of the form $X = \tilde{\Omega}p$, with

\[ \tilde{\Omega} = \left(I_{2n} - \frac{1}{2}pp^+\right)Xp^+ - pX^+\left(I_{2n} - \frac{1}{2}pp^+\right) ∈ ℝ^{2n×2n},\]

as shown in Proposition 3.5 of [BZ21]. Using this representation of $X$, the Cayley retraction on $\mathrm{SpSt}(2n, 2k)$ is defined pointwise as

\[ \mathcal{R}_p(X) = \operatorname{cay}\left(\frac{1}{2}\tilde{\Omega}\right)p.\]

The operator $\operatorname{cay}(A) = (I - A)^{-1}(I + A)$ is the Cayley transform.

However, the computation of an $2n×2n$ matrix inverse in the expression above can be reduced down to inverting a $2k×2k$ matrix due to Proposition 5.2 of [BZ21].

Let $A = p^+X$ and $H = X - pA$. Then an equivalent expression for the Cayley retraction defined pointwise above is

\[ \mathcal{R}_p(X) = -p + (H + 2p)(H^+H/4 - A/2 + I_{2k})^{-1}.\]

This expression is computed inplace of q.