Symplectic
The Symplectic
manifold, denoted $\operatorname{Sp}(2n, \mathbb{F})$, is a closed, embedded, submanifold of $\mathbb{F}^{2n \times 2n}$ that represents transformations into symplectic subspaces which keep the canonical symplectic form over $\mathbb{F}^{2n \times 2n }$ invariant under the standard embedding inner product. The canonical symplectic form is a non-degenerate bilinear and skew symmetric map $\omega\colon \mathbb{F}^{2n} \times \mathbb{F}^{2n} \rightarrow \mathbb{F}$, given by $\omega(x, y) = x^T Q_{2n} y$ for elements $x, y \in \mathbb{F}^{2n}$, with
\[ Q_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}.\]
That means that an element $p \in \operatorname{Sp}(2n)$ must fulfill the requirement that
\[ \omega (p x, p y) = x^T(p^TQp)y = x^TQy = \omega(x, y),\]
leading to the requirement on $p$ that $p^TQp = Q$.
The symplectic manifold also forms a group under matrix multiplication, called the $\textit{symplectic group}$. Since all the symplectic matrices necessarily have determinant one, the symplectic group $\operatorname{Sp}(2n, \mathbb{F})$ is a subgroup of the special linear group, $\operatorname{SL}(2n, \mathbb{F})$. When the underlying field is either $\mathbb{R}$ or $\mathbb{C}$ the symplectic group with a manifold structure constitutes a Lie group, with the Lie Algebra
\[ \mathfrak{sp}(2n,F) = \{H \in \mathbb{F}^{2n \times 2n} \;|\; Q H + H^{T} Q = 0\}.\]
This set is also known as the Hamiltonian matrices, which have the property that $(QH)^T = QH$ and are commonly used in physics.
Manifolds.ExtendedSymplecticMetric
— TypeExtendedSymplecticMetric <: AbstractMetric
The extension of the RealSymplecticMetric
at a point p \in \operatorname{Sp}(2n)
as an inner product over the embedding space $ℝ^{2n \times 2n}$, i.e.
\[ \langle x, y \rangle_{p} = \langle p^{-1}x, p^{-1}\rangle_{\operatorname{Fr}} = \operatorname{tr}(x^T(pp^T)^{-1}y), \;\forall\; x, y \in ℝ^{2n \times 2n}.\]
Manifolds.RealSymplecticMetric
— TypeRealSymplecticMetric <: RiemannianMetric
The canonical Riemannian metric on the symplectic manifold, defined pointwise for $p \in \operatorname{Sp}(2n)$ by [Fiori2011]
\[\begin{align*} & g_p \colon T_p\operatorname{Sp}(2n) \times T_p\operatorname{Sp}(2n) \rightarrow ℝ, \\ & g_p(Z_1, Z_2) = \operatorname{tr}((p^{-1}Z_1)^T (p^{-1}Z_2)). \end{align*}\]
This metric is also the default metric for the Symplectic
manifold.
Manifolds.Symplectic
— TypeSymplectic{n, 𝔽} <: AbstractEmbeddedManifold{𝔽, DefaultIsometricEmbeddingType}
The symplectic manifold consists of all $2n \times 2n$ matrices which preserve the canonical symplectic form over $𝔽^{2n × 2n} \times 𝔽^{2n × 2n}$,
\[ \omega\colon 𝔽^{2n × 2n} \times 𝔽^{2n × 2n} \rightarrow 𝔽, \quad \omega(x, y) = p^T Q_{2n} q, \; x, y \in 𝔽^{2n × 2n},\]
where
\[Q_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}.\]
That is, the symplectic manifold consists of
\[\operatorname{Sp}(2n, ℝ) = \bigl\{ p ∈ ℝ^{2n × 2n} \, \big| \, p^TQ_{2n}p = Q_{2n} \bigr\},\]
with $0_n$ and $I_n$ denoting the $n × n$ zero-matrix and indentity matrix in $ℝ^{n \times n}$ respectively.
The tangent space at a point $p$ is given by [BendokatZimmermann2021]
\[\begin{align*} T_p\operatorname{Sp}(2n) &= \{X \in \mathbb{R}^{2n \times 2n} \;|\; p^{T}Q_{2n}X + X^{T}Q_{2n}p = 0 \}, \\ &= \{X = pQS \;|\; S ∈ R^{2n × 2n}, S^T = S \}. \end{align*}\]
Constructor
Symplectic(2n, field=ℝ) -> Symplectic{div(2n, 2), field}()
Generate the (real-valued) symplectic manifold of $2n \times 2n$ symplectic matrices. The constructor for the Symplectic
manifold accepts the even column/row embedding dimension $2n$ for the real symplectic manifold, $ℝ^{2n × 2n}$.
Manifolds.SymplecticMatrix
— TypeSymplecticMatrix{T}
A lightweight structure to represent the action of the matrix representation of the canonical symplectic form,
\[Q_{2n}(λ) = λ \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix} \quad \in ℝ^{2n \times 2n},\]
such that the canonical symplectic form is represented by
\[\omega_{2n}(x, y) = x^TQ_{2n}(1)y, \quad x, y \in ℝ^{2n}.\]
The entire matrix is however not instantiated in memory, instead a scalar $λ$ of type T
is stored, which is used to keep track of scaling and transpose operations applied to each SymplecticMatrix
. For example, given Q = SymplecticMatrix(1.0)
represented as 1.0*[0 I; -I 0]
, the adjoint Q'
returns SymplecticMatrix(-1.0) = (-1.0)*[0 I; -I 0]
.
Base.exp
— Methodexp(M::Symplectic, p, X)
exp!(M::Symplectic, q, p, X)
The Exponential mapping on the Symplectic manifold with the RealSymplecticMetric
Riemannian metric.
For the point $p \in \operatorname{Sp}(2n)$ the exponential mapping along the tangent vector $X \in T_p\operatorname{Sp}(2n)$ is computed as [WangSunFiori2018]
\[ \operatorname{exp}_p(X) = p \operatorname{Exp}((p^{-1}X)^T) \operatorname{Exp}(p^{-1}X - (p^{-1}X)^T),\]
where $\operatorname{Exp}(\cdot)$ denotes the matrix exponential.
Base.inv
— Methodinv(::Symplectic, A)
inv!(::Symplectic, A)
Compute the symplectic inverse $A^+$ of matrix $A ∈ ℝ^{2n × 2n}$. Given a matrix
\[A ∈ ℝ^{2n × 2n},\quad A = \begin{bmatrix} A_{1,1} & A_{1,2} \\ A_{2,1} & A_{2, 2} \end{bmatrix}\]
the symplectic inverse is defined as:
\[A^{+} := Q_{2n}^T A^T Q_{2n},\]
where
\[Q_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}.\]
The symplectic inverse of A can be expressed explicitly as:
\[A^{+} = \begin{bmatrix} A_{2, 2}^T & -A_{1, 2}^T \\[1.2mm] -A_{2, 1}^T & A_{1, 1}^T \end{bmatrix}.\]
Base.rand
— Methodrand(::SymplecticStiefel; vector_at=nothing,
hamiltonian_norm = (vector_at === nothing ? 1/2 : 1.0))
Generate a random point on $\operatorname{Sp}(2n)$ or a random tangent vector $X \in T_p\operatorname{Sp}(2n)$ if vector_at
is set to a point $p \in \operatorname{Sp}(2n)$.
A random point on $\operatorname{Sp}(2n)$ is constructed by generating a random Hamiltonian matrix $Ω \in \mathfrak{sp}(2n,F)$ with norm hamiltonian_norm
, and then transforming it to a symplectic matrix by applying the Cayley transform
\[ \operatorname{cay}\colon \mathfrak{sp}(2n,F) \rightarrow \operatorname{Sp}(2n), \; \Omega \mapsto (I - \Omega)^{-1}(I + \Omega).\]
To generate a random tangent vector in $T_p\operatorname{Sp}(2n)$, this code employs the second tangent vector space parametrization of Symplectic. It first generates a random symmetric matrix $S$ by S = randn(2n, 2n)
and then symmetrizes it as S = S + S'
. Then $S$ is normalized to have Frobenius norm of hamiltonian_norm
and X = pQS
is returned, where Q
is the SymplecticMatrix
.
Manifolds.change_representer
— Methodchange_representer(::Symplectic, ::EuclideanMetric, p, X)
change_representer!(::Symplectic, Y, ::EuclideanMetric, p, X)
Compute the representation of a tangent vector $ξ ∈ T_p\operatorname{Sp}(2n, ℝ)$ s.t.
\[ g_p(c_p(ξ), η) = ⟨ξ, η⟩^{\text{Euc}} \;∀\; η ∈ T_p\operatorname{Sp}(2n, ℝ).\]
with the conversion function
\[ c_p : T_p\operatorname{Sp}(2n, ℝ) \rightarrow T_p\operatorname{Sp}(2n, ℝ), \quad c_p(ξ) = \frac{1}{2} pp^T ξ + \frac{1}{2} pQ ξ^T pQ.\]
Each of the terms $c_p^1(ξ) = p p^T ξ$ and $c_p^2(ξ) = pQ ξ^T pQ$ from the above definition of $c_p(η)$ are themselves metric compatible in the sense that
\[ c_p^i : T_p\operatorname{Sp}(2n, ℝ) \rightarrow \mathbb{R}^{2n \times 2n}\quad g_p^i(c_p(ξ), η) = ⟨ξ, η⟩^{\text{Euc}} \;∀\; η ∈ T_p\operatorname{Sp}(2n, ℝ),\]
for $i \in {1, 2}$. However the range of each function alone is not confined to $T_p\operatorname{Sp}(2n, ℝ)$, but the convex combination
\[ c_p(ξ) = \frac{1}{2}c_p^1(ξ) + \frac{1}{2}c_p^2(ξ)\]
does have the correct range $T_p\operatorname{Sp}(2n, ℝ)$.
Manifolds.change_representer
— Methodchange_representer(MetMan::MetricManifold{𝔽, Euclidean{Tuple{m, n}, 𝔽}, ExtendedSymplecticMetric},
EucMet::EuclideanMetric, p, X)
change_representer!(MetMan::MetricManifold{𝔽, Euclidean{Tuple{m, n}, 𝔽}, ExtendedSymplecticMetric},
Y, EucMet::EuclideanMetric, p, X)
Change the representation of a matrix $ξ ∈ \mathbb{R}^{2n \times 2n}$ into the inner product space $(ℝ^{2n \times 2n}, g_p)$ where the inner product is given by $g_p(ξ, η) = \langle p^{-1}ξ, p^{-1}η \rangle = \operatorname{tr}(ξ^T(pp^T)^{-1}η)$, as the extension of the RealSymplecticMetric
onto the entire embedding space.
By changing the representation we mean to apply a mapping
\[ c_p : \mathbb{R}^{2n \times 2n} \rightarrow \mathbb{R}^{2n \times 2n},\]
defined by requiring that it satisfy the metric compatibility condition
\[ g_p(c_p(ξ), η) = ⟨p^{-1}c_p(ξ), p^{-1}η⟩ = ⟨ξ, η⟩^{\text{Euc}} \;∀\; η ∈ T_p\operatorname{Sp}(2n, ℝ).\]
In this case, we compute the mapping
\[ c_p(ξ) = pp^T ξ.\]
Manifolds.gradient
— Methodgradient(M::Symplectic, f, p, backend::RiemannianProjectionBackend;
extended_metric=true)
gradient!(M::Symplectic, f, p, backend::RiemannianProjectionBackend;
extended_metric=true)
Compute the manifold gradient $\text{grad}f(p)$ of a scalar function $f \colon \operatorname{Sp}(2n) \rightarrow ℝ$ at $p \in \operatorname{Sp}(2n)$.
The element $\text{grad}f(p)$ is found as the Riesz representer of the differential $\text{D}f(p) \colon T_p\operatorname{Sp}(2n) \rightarrow ℝ$ w.r.t. the Riemannian metric inner product at $p$ [Fiori2011]. That is, $\text{grad}f(p) \in T_p\operatorname{Sp}(2n)$ solves the relation
\[ g_p(\text{grad}f(p), X) = \text{D}f(p) \quad\forall\; X \in T_p\operatorname{Sp}(2n).\]
The default behaviour is to first change the representation of the Euclidean gradient from the Euclidean metric to the RealSymplecticMetric
at $p$, and then we projecting the result onto the correct tangent tangent space $T_p\operatorname{Sp}(2n, ℝ)$ w.r.t the Riemannian metric $g_p$ extended to the entire embedding space.
Arguments:
extended_metric = true
: Iftrue
, compute the gradient $\text{grad}f(p)$ by first changing the representer of the Euclidean gradient of a smooth extension of $f$, $∇f(p)$, w.r.t. theRealSymplecticMetric
at $p$ extended to the entire embedding space, before projecting onto the correct tangent vector space w.r.t. the same extended metric $g_p$. Iffalse
, compute the gradient by first projecting $∇f(p)$ onto the tangent vector space, before changing the representer in the tangent vector space to comply with theRealSymplecticMetric
.
Manifolds.project_normal!
— Methodproject_normal!(::MetricManifold{𝔽,Euclidean,ExtendedSymplecticMetric}, Y, p, X)
Project onto the normal of the tangent space $(T_p\operatorname{Sp}(2n))^{\perp_g}$ at a point $p ∈ \operatorname{Sp}(2n)$, relative to the riemannian metric $g$ RealSymplecticMetric
. That is,
\[(T_p\operatorname{Sp}(2n))^{\perp_g} = \{Y \in \mathbb{R}^{2n \times 2n} : g_p(Y, X) = 0 \;\forall\; X \in T_p\operatorname{Sp}(2n)\}.\]
The closed form projection operator onto the normal space is given by [GaoSonAbsilStykel2021]
\[\operatorname{P}^{(T_p\operatorname{Sp}(2n))\perp}_{g_p}(X) = pQ\operatorname{skew}(p^TQ^TX),\]
where $\operatorname{skew}(A) = \frac{1}{2}(A - A^T)$. This function is not exported.
Manifolds.symplectic_inverse_times
— Methodsymplectic_inverse_times(::Symplectic, p, q)
symplectic_inverse_times!(::Symplectic, A, p, q)
Directly compute the symplectic inverse of $p \in \operatorname{Sp}(2n)$, multiplied with $q \in \operatorname{Sp}(2n)$. That is, this function efficiently computes $p^+q = (Q_{2n}p^TQ_{2n})q \in ℝ^{2n \times 2n}$, where $Q_{2n}$ is the SymplecticMatrix
of size $2n \times 2n$.
ManifoldsBase.check_point
— Methodcheck_point(M::Symplectic, p; kwargs...)
Check whether p
is a valid point on the Symplectic
M
=$\operatorname{Sp}(2n)$, i.e. that it has the right AbstractNumbers
type and $p^{+}p$ is (approximately) the identity, where $A^{+} = Q_{2n}^TA^TQ_{2n}$ is the symplectic inverse, with
\[Q_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}.\]
The tolerance can be set with kwargs...
(e.g. atol = 1.0e-14
).
ManifoldsBase.check_vector
— Methodcheck_vector(M::Symplectic, p, X; kwargs...)
Checks whether X
is a valid tangent vector at p
on the Symplectic
M
=$\operatorname{Sp}(2n)$, i.e. the AbstractNumbers
fits and it (approximately) holds that $p^{T}Q_{2n}X + X^{T}Q_{2n}p = 0$, where
\[Q_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix}.\]
The tolerance can be set with kwargs...
(e.g. atol = 1.0e-14
).
ManifoldsBase.distance
— Methoddistance(M::Symplectic, p, q)
Compute an approximate geodesic distance between two Symplectic matrices $p, q \in \operatorname{Sp}(2n)$, as done in [WangSunFiori2018].
\[ \operatorname{dist}(p, q) ≈ ||\operatorname{Log}(p^+q)||_{\operatorname{Fr}},\]
where the $\operatorname{Log}(\cdot)$ operator is the matrix logarithm.
This approximation is justified by first recalling the Baker-Campbell-Hausdorf formula,
\[\operatorname{Log}(\operatorname{Exp}(A)\operatorname{Exp}(B)) = A + B + \frac{1}{2}[A, B] + \frac{1}{12}[A, [A, B]] + \frac{1}{12}[B, [B, A]] + \ldots \;.\]
Then we write the expression for the exponential map from $p$ to $q$ as
\[ q = \operatorname{exp}_p(X) = p \operatorname{Exp}((p^{+}X)^T) \operatorname{Exp}([p^{+}X - (p^{+}X)^T]), X \in T_p\operatorname{Sp},\]
and with the geodesic distance between $p$ and $q$ given by $\operatorname{dist}(p, q) = ||X||_p = ||p^+X||_{\operatorname{Fr}}$ we see that
\[ \begin{align*} ||\operatorname{Log}(p^+q)||_{\operatorname{Fr}} &= ||\operatorname{Log}\left( \operatorname{Exp}((p^{+}X)^T) \operatorname{Exp}(p^{+}X - (p^{+}X)^T) \right)||_{\operatorname{Fr}} \\ &= ||p^{+}X + \frac{1}{2}[(p^{+}X)^T, p^{+}X - (p^{+}X)^T] + \ldots ||_{\operatorname{Fr}} \\ &≈ ||p^{+}X||_{\operatorname{Fr}} = \operatorname{dist}(p, q). \end{align*}\]
ManifoldsBase.inner
— Methodinner(::Symplectic{n, ℝ}, p, X, Y)
Compute the canonical Riemannian inner product RealSymplecticMetric
\[ g_p(X, Y) = \operatorname{tr}((p^{-1}X)^T (p^{-1}Y))\]
between the two tangent vectors $X, Y \in T_p\operatorname{Sp}(2n)$.
ManifoldsBase.inverse_retract
— Methodinverse_retract(M::Symplectic, p, q, ::CayleyInverseRetraction)
Compute the Cayley Inverse Retraction $X = \mathcal{L}_p^{\operatorname{Sp}}(q)$ such that the Cayley Retraction from $p$ along $X$ lands at $q$, i.e. $\mathcal{R}_p(X) = q$ [BendokatZimmermann2021].
First, recall the definition the standard symplectic matrix
\[Q = \begin{bmatrix} 0 & I \\ -I & 0 \end{bmatrix}\]
as well as the symplectic inverse of a matrix $A$, $A^{+} = Q^T A^T Q$.
For $p, q ∈ \operatorname{Sp}(2n, ℝ)$ then, we can then define the inverse cayley retraction as long as the following matrices exist.
\[ U = (I + p^+ q)^{-1}, \quad V = (I + q^+ p)^{-1}.\]
If that is the case, the inverse cayley retration at $p$ applied to $q$ is
\[\mathcal{L}_p^{\operatorname{Sp}}(q) = 2p\bigl(V - U\bigr) + 2\bigl((p + q)U - p\bigr) ∈ T_p\operatorname{Sp}(2n).\]
ManifoldsBase.manifold_dimension
— Methodmanifold_dimension(::Symplectic{n})
Returns the dimension of the symplectic manifold embedded in $ℝ^{2n \times 2n}$, i.e.
\[ \operatorname{dim}(\operatorname{Sp}(2n)) = (2n + 1)n.\]
ManifoldsBase.project!
— Methodproject!(::MetricManifold{𝔽,Euclidean,ExtendedSymplecticMetric}, Y, p, X) where {𝔽}
Compute the projection of $X ∈ R^{2n × 2n}$ onto $T_p\operatorname{Sp}(2n, ℝ)$ w.r.t. the Riemannian metric $g$ RealSymplecticMetric
. The closed form projection mapping is given by [GaoSonAbsilStykel2021]
\[ \operatorname{P}^{T_p\operatorname{Sp}(2n)}_{g_p}(X) = pQ\operatorname{sym}(p^TQ^TX),\]
where $\operatorname{sym}(A) = \frac{1}{2}(A + A^T)$. This function is not exported.
ManifoldsBase.project
— Methodproject(::Symplectic, p, A)
project!(::Symplectic, Y, p, A)
Given a point $p \in \operatorname{Sp}(2n)$, project an element $A \in \mathbb{R}^{2n \times 2n}$ onto the tangent space $T_p\operatorname{Sp}(2n)$ relative to the euclidean metric of the embedding $\mathbb{R}^{2n \times 2n}$.
That is, we find the element $X \in T_p\operatorname{SpSt}(2n, 2k)$ which solves the constrained optimization problem
\[ \operatorname{min}_{X \in \mathbb{R}^{2n \times 2n}} \frac{1}{2}||X - A||^2, \quad \text{s.t.}\; h(X) \colon= X^T Q p + p^T Q X = 0,\]
where $h\colon\mathbb{R}^{2n \times 2n} \rightarrow \operatorname{skew}(2n)$ defines the restriction of $X$ onto the tangent space $T_p\operatorname{SpSt}(2n, 2k)$.
ManifoldsBase.retract
— Methodretract(::Symplectic, p, X, ::CayleyRetraction)
retract!(::Symplectic, q, p, X, ::CayleyRetraction)
Compute the Cayley retraction on $p ∈ \operatorname{Sp}(2n, ℝ)$ in the direction of tangent vector $X ∈ T_p\operatorname{Sp}(2n, ℝ)$, as defined in by Birtea et al in proposition 2 [BirteaCaşuComănescu2020].
Using the symplectic inverse of a matrix $A \in ℝ^{2n \times 2n}$, $A^{+} := Q_{2n}^T A^T Q_{2n}$ where
\[Q_{2n} = \begin{bmatrix} 0_n & I_n \\ -I_n & 0_n \end{bmatrix},\]
the retraction $\mathcal{R}\colon T\operatorname{Sp}(2n) \rightarrow \operatorname{Sp}(2n)$ is defined pointwise as
\[\begin{align*} \mathcal{R}_p(X) &= p \operatorname{cay}\left(\frac{1}{2}p^{+}X\right), \\ &= p \operatorname{exp}_{1/1}(p^{+}X), \\ &= p (2I - p^{+}X)^{-1}(2I + p^{+}X). \end{align*}\]
Here $\operatorname{exp}_{1/1}(z) = (2 - z)^{-1}(2 + z)$ denotes the Padé (1, 1) approximation to $\operatorname{exp}(z)$.
Literature
- WangSunFiori2018
Wang, Jing and Sun, Huafei and Fiori, Simone: A Riemannian-steepest-descent approach for optimization on the real symplectic group, Mathematical Methods in the Applied Sciences, 41(11) pp. 4273-4286, 2018 doi 10.1002/mma.4890
- Fiori2011
Simone Fiori: Solving minimal-distance problems over the manifold of real-symplectic matrices, SIAM Journal on Matrix Analysis and Applications 32(3), pp. 938-968, 2011. doi 10.1137/100817115.
- GaoSonAbsilStykel2021
Gao, Bin and Son, Nguyen Thanh and Absil, P-A and Stykel, Tatjana: Riemannian optimization on the symplectic Stiefel manifold, SIAM Journal on Optimization 31(2), pp. 1546-1575, 2021. doi 10.1137/20M1348522
- BendokatZimmermann2021
Bendokat, Thomas and Zimmermann, Ralf: The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications arXiv preprint arXiv:2108.12447, 2021.
- BirteaCaşuComănescu2020
Birtea, Petre and Caşu, Ioan and Comănescu, Dan: Optimization on the real symplectic group, Monatshefte f{"u}r Mathematik, Springer, 2020. doi 10.1007/s00605-020-01369-9