Symplectic Stiefel Manifold Overview

• Symplectic Stiefel manifold is defined as the set of 2n×2k real matrices whose columns form a symplectic basis satisfying UᵀJ₍₂ₙ₎U = J₍₂ₖ₎, providing a foundation for symplectic geometry.
• Its tangent space structure and various retraction methods, including block-parameterizations and Cayley-type maps, enable efficient Riemannian optimization for constrained problems.
• This manifold is central to applications in scientific computing, quantum control, and structure-preserving model reduction in Hamiltonian systems.

The symplectic Stiefel manifold is a foundational geometric object that encodes partial symplectic frames and admits a rich differential, Riemannian, and algebraic structure. It serves as the feasible set for a wide variety of constrained matrix optimization problems, particularly in scientific computing, quantum mechanics, geometric model reduction, and control of Hamiltonian systems. The manifold, denoted $\SpSt(2n,2k)$, consists of all $2n \times 2k$ real matrices whose columns form a symplectic basis: $P^T J_{2n} P = J_{2k}$, where $$J_{2m}=\begin{pmatrix} 0 & I_m \ -I_m & 0 \end{pmatrix}$$ is the canonical symplectic form. Its structure generalizes the role of the (real or unitary) Stiefel manifold in orthogonal geometry but replaces orthogonality with symplecticity, yielding a homogeneous space for the real symplectic group.

1. Definition, Embedding, and Basic Properties

The real symplectic Stiefel manifold is defined as

$\SpSt(2n,2k) = \{U \in \mathbb{R}^{2n \times 2k} ~|~ U^T J_{2n} U = J_{2k}\}, \qquad k \leq n.$

Here, $J_{2m}$ is the standard symplectic matrix. The columns of $U$ form a basis for a $2k$-dimensional symplectic subspace of $\mathbb{R}^{2n}$, preserving the canonical two-form. This is a real, smooth, embedded submanifold of $\mathbb{R}^{2n\times2k}$ of codimension $k(2k-1)$; thus,

$\dim \SpSt(2n,2k) = 4nk - k(2k-1).$

When $k=n$, $\SpSt(2n,2n)$ coincides with the (real, noncompact) symplectic group $\Sp(2n)$.

The manifold admits a homogeneous space structure: $\SpSt(2n,2k) \cong \Sp(2n,\mathbb{R}) / \Sp(2n-2k,\mathbb{R}),$ and further, in the language of generalized Stiefel manifolds, can be viewed as $L\backslash H$ with $L = \Sp(2k,\mathbb{R})$, $H = \Sp(2n,\mathbb{R})$ (Sedano-Mendoza, 2019, Bendokat et al., 2021).

2. Tangent Space Structure and Parametrizations

The tangent space at $U\in\SpSt(2n,2k)$ is characterized by differentiating the constraint: $T_U \SpSt(2n,2k) = \{Z\in\mathbb{R}^{2n\times2k}\mid Z^T J_{2n} U + U^T J_{2n} Z=0\}.$ Several equivalent formulations exist:

• Block-parametrization: Each $Z$ in the tangent space decomposes as

$Z = UJ_{2k} W + J_{2n} U_{\perp} K,$

where $W \in \Sym(2k)$, $K\in\mathbb{R}^{(2n-2k)\times 2k}$, and $U_{\perp}$ is a $J$-orthogonal complement of $U$ (Gao et al., 2020, Gao et al., 2021, Gao et al., 2022).

• Hamiltonian form: Alternatively,

$T_U\SpSt = \{S J_{2k} U : S \in \Sym(2n)\}.$

• Symplectic inverse characterization: $U^+ := J_{2k}^T U^T J_{2n}$, and $T_U\SpSt = \{\Delta : U^+ \Delta \in \mathfrak{sp}(2k,\mathbb{R}) \}$ (Bendokat et al., 2021).

These structures facilitate the derivation of projection operators, gradients, and feasible step mappings for Riemannian optimization.

3. Riemannian and Pseudo-Riemannian Metrics

Multiple Riemannian metrics have been studied on $\SpSt(2n,2k)$:

• Euclidean (Frobenius) metric: The restriction of the ambient inner product, frequently employed for numerical tractability. Projection onto the tangent space in the Euclidean metric involves solving a Lyapunov equation to compute the skew-symmetric multiplier. The Riemannian gradient is then given by

$$\operatorname{grad} f(X) = \nabla \bar f(X) - J_{2n} X \Omega_X,$$

where $\Omega_X$ solves

$X^T X \Omega_X + \Omega_X X^T X = 2 \skew(X^T J_{2n} \nabla \bar f(X)).$

(Gao et al., 2021, Gao et al., 2022)

• Canonical-like and right-invariant metrics: Generalizations of the canonical metric on classical Stiefel manifolds. For $X\in\SpSt$,

$$g_\rho(\xi,\eta) = \mathrm{tr}(\xi^T H_X \eta), \quad H_X = I_{2n} + \frac{\rho}{2}XX^T - J_{2n} X (X^T X)^{-1} X^T J_{2n}^T.$$

(Son et al., 2021, Gao et al., 2020). The metric is right-invariant under the action of $\Sp(2k,\mathbb{R})$ (Bendokat et al., 2021).

• Pseudo-Riemannian metric: Inherited from the ambient symplectic group, with signature induced by the Killing form. For $U\in\SpSt(2n,2k)$,

$h_U^{\SpSt}(\Delta_1,\Delta_2) = \operatorname{tr}(\Delta_1^+ (I-\tfrac{1}{2} UU^+) \Delta_2)$

(Bendokat et al., 2021, Sedano-Mendoza, 2019).

Recent developments introduce weighted "tractable" metrics $$g_X^W(\xi,\eta) = \operatorname{tr}(\xi^T W(X) \eta)$$, parameterized by a positive-definite weighting function $W(X)$, facilitating preconditioned second-order schemes (Gao et al., 20 Jun 2024).

4. Geodesics, Retractions, and Second-Order Geometry

Closed-form expressions for geodesics are rare except in full-rank group or simplified settings. For general metrics and points, geodesic flows require integrating ODEs on the group with vertical/horizontal splitting. In $\SpSt(2n,2k)$, several practical alternatives are in regular use:

• Cayley-type retractions: Rational mappings that approximate the exponential map while preserving symplecticity. General forms include

$$R_X(Z) = (I + \frac{1}{2}A_X J_{2n})^{-1} (I - \frac{1}{2}A_X J_{2n}) X,$$

with $A_X$ constructed from the search direction (Son et al., 2021, Gao et al., 2020, Gao et al., 2022, Jensen et al., 12 Apr 2024).

• SR decomposition-based retraction: Uses a symplectic-QR-like decomposition, ensuring the updated iterate remains on the manifold and the retraction is globally defined on an open neighborhood. For $X+Z = SR$ with $S \in \SpSt, R$ upper-triangular, $R_X^{\mathrm{SR}}(Z) = S$ (Gao et al., 2022).
• Quasi-geodesic retractions: Explicit curve formulas with analytically tractable derivatives, yielding globally defined step mappings (Gao et al., 2020, Gao et al., 2022).

Second-order geometry has now reached explicit operator-valued formulas for the Riemannian Hessian under general tractable metrics. In the weighted Euclidean case,

$\Hess^M f(X)[Z] = P_X(M^{-1} D^2 \bar f(X)[Z] + D_Z P_X(M^{-1} \nabla \bar f(X)))$

(Gao et al., 20 Jun 2024, Jensen et al., 12 Apr 2024).

This enables the deployment of Newton and trust-region methods, with projections and saddle-point solvers for the Newton step.

5. Algebraic, Homogeneous, and Symmetry Properties

$\SpSt(2n,2k)$ can be viewed as a homogeneous space for the symplectic group: $\SpSt(2n,2k) \cong \Sp(2n,\mathbb{R})/\Sp(2n-2k,\mathbb{R}),$

with

$St_\omega(k,2n) \cong L\backslash H, \quad L = \Sp(2k,\mathbb{R}),$

and the full isometry group (identity component) is

$\frac{\Sp(2(n-k),\mathbb{R}) \times \Sp(2n,\mathbb{R})}{\{\pm I,\pm I\}}$

(Sedano-Mendoza, 2019). The tangent space at the identity reflects the reductive decomposition of the symplectic Lie algebra and underpins geodesic flow and curvature computations.

Affine and connection properties follow from the reductive homogeneous structure, with the canonical connection associated to the symplectic group quotient and encoded by a nonassociative algebra $m = g \oplus p$.

6. Riemannian Optimization Methods

Optimization problems with symplecticity constraints formulate as unconstrained smooth optimization on $\SpSt(2n,2k)$:

• Nearest symplectic matrix: $\min_{U \in \SpSt(2n,2k)} \|U - A\|_F^2$
• Symplectic eigenvalue problem: $\min_{X \in \SpSt(2n,2k)} \operatorname{tr}(X^T M X)$
• Symplectic model reduction: $\min_{X \in \SpSt(2n,2k)} \|A - X X^+ A\|_F^2$

Principal optimization methods include Riemannian gradient descent, conjugate gradients, Barzilai–Borwein steps with Armijo line-search, and, most recently, second-order Newton and trust-region strategies exploiting explicit Riemannian Hessian expressions and robust, globally defined retractions (Son et al., 2021, Gao et al., 2022, Jensen et al., 12 Apr 2024, Gao et al., 20 Jun 2024).

Empirical results demonstrate major efficiency and accuracy gains from using second-order information, especially via low-rank Cayley and SR-based retractions. Trust-region and Newton methods exhibit rapid convergence once the gradient is sufficiently small, with global convergence guaranteed by suitable switch and line-search mechanisms (Gao et al., 20 Jun 2024, Jensen et al., 12 Apr 2024).

7. Applications, Extensions, and Related Manifolds

Symplectic Stiefel manifolds underpin a variety of applications:

• Quantum computation and geometric quantum control, e.g., in optimal design of symplectic gates (Gao et al., 2022)
• Structure-preserving model order reduction in Hamiltonian dynamical systems, via proper symplectic decomposition (Bendokat et al., 2021)
• Symplectic eigenvalue problems, generalizing Williamson’s theorem; finding symplectic eigenvectors of SPD matrices (Son et al., 2021)
• Numerical linear algebra and Riemannian optimization with constraints beyond orthogonality (Gao et al., 2020, Gao et al., 2021)

Extensions include symplectic Grassmannians ($\SpGr$) of symplectic subspaces, infinite-dimensional symplectic Stiefel manifolds of weighted embeddings associated with contact manifolds and coadjoint orbit theory, and closely related quotient spaces (Haller et al., 2019).

Recent advances have introduced explicit metrics, tractable Hessian and projection computations, and robust Newton-type optimization frameworks, establishing $\SpSt(2n,2k)$ as a central object for both geometric analysis and computational applications in symplectic geometry (Gao et al., 20 Jun 2024, Jensen et al., 12 Apr 2024, Gao et al., 2022).