Cayley-Based Retractions

• Cayley-based retractions are specialized maps that parameterize the real symplectic Stiefel manifold by approximating exponential maps while preserving symplectic structure.
• They achieve first-order accuracy and facilitate efficient projections from the tangent space to the manifold, which is vital for Riemannian optimization.
• With a computational cost of O(p^3 + n p^2) and inherent stability conditions, they offer a practical balance of accuracy and efficiency for geometric algorithms.

A Cayley-based retraction is a specific class of retraction map used for locally parametrizing points on matrix manifolds, most prominently the real symplectic Stiefel manifold, in computational geometric and optimization contexts. Retractions approximate exponential maps, enabling efficient first-order embeddings between a tangent space and the manifold, and are pivotal in Riemannian algorithms where movements off the manifold must be projected back. On the symplectic Stiefel manifold, the Cayley retraction stands out due to its closed-form inverse and computational efficiency, balancing accuracy and stability while preserving symplectic structure (Zimmermann, 7 May 2026).

1. Mathematical Setting and Definition

The Cayley-based retraction is defined on the real symplectic Stiefel manifold

$$SpSt(2n,2p) = \{ Y \in \mathbb{R}^{2n \times 2p} \mid Y^T J_n Y = J_p \}, \;\; J_n = \begin{pmatrix}0 & I_n \ -I_n & 0\end{pmatrix},$$

where $J_n$ is the standard symplectic matrix and $J_p$ its $2p$-dimensional analog. For a base point $Y \in SpSt(2n,2p)$ and tangent vector $S \in T_Y SpSt$, define the $2p \times 2p$ Hamiltonian matrix

$A := Y^+S = J_p^T Y^T J_n S \in sp(2p),$

where the Moore–Penrose pseudoinverse $Y^+$ is used. $sp(2p)$ denotes the Lie algebra of Hamiltonian matrices satisfying $J_n$0.

The Cayley map for Hamiltonian matrices is

$J_n$1

The Cayley retraction from the tangent bundle to the manifold is then

$J_n$2

This mapping provides a computationally tractable surrogate to the Riemannian exponential map with first-order accuracy (Zimmermann, 7 May 2026).

2. Closed-Form Inverse and Domain

One of the distinguishing features of the Cayley retraction on $J_n$3 is its closed-form inverse. Given $J_n$4 in a neighborhood of $J_n$5 such that $J_n$6 and $J_n$7, the inverse, sometimes called the Cayley logarithm, is

$J_n$8

The tangent vector is then recovered as

$J_n$9

Domain conditions for invertibility require

$J_p$0

ensuring the relevant linear systems are solvable. This feature enables practical implementations requiring mapping between ambient and tangent representations (Zimmermann, 7 May 2026).

3. Structural and Accuracy Properties

The Cayley-based retraction is characterized by several important mathematical properties:

• First-order accuracy: The expansion

$J_p$1

shows that it correctly recovers the tangent step to first order, matching the exponential map up to $J_p$2.

• Symplectic preservation: For $J_p$3,

$J_p$4

so symplectic structure is maintained. The cross-term vanishes by the algebraic properties of $J_p$5 and $J_p$6.

• Hamiltonicity and orthogonality: The construction guarantees $J_p$7, with $J_p$8 symmetric, $J_p$9, and $2p$0, ensuring the relevant invariances and projection properties.

These properties make the Cayley retraction particularly suitable for geometric optimization and integration schemes where structure preservation is critical (Zimmermann, 7 May 2026).

4. Computational Complexity and Numerical Stability

The cost profile of the Cayley retraction is determined by several dominating operations:

• Computing $2p$1: $2p$2.
• Forming $2p$3: $2p$4, due to matrix inversion or solving two linear systems.
• Multiplying $2p$5: $2p$6.
• Computing the orthogonal complement $2p$7: $2p$8.

Thus, overall complexity per evaluation is

$2p$9

for both the forward and inverse maps. Stability requires maintaining distance from singularities (eigenvalue $Y \in SpSt(2n,2p)$0 of $Y \in SpSt(2n,2p)$1 or $Y \in SpSt(2n,2p)$2 of $Y \in SpSt(2n,2p)$3), which in practice is managed by monitoring $Y \in SpSt(2n,2p)$4 or the minimal singular value of $Y \in SpSt(2n,2p)$5. The Cayley map is efficient for moderate step sizes due to its avoidance of computationally intensive decompositions (Zimmermann, 7 May 2026).

5. Comparison with Polar-Factor Retraction

Zimmermann (2024) introduced a contrasting retraction, the symplectic polar-factor (or "polar-light") retraction, defined as

$Y \in SpSt(2n,2p)$6

where $Y \in SpSt(2n,2p)$7 is a skew-Hamiltonian symplectifier obtained from a Schur decomposition and matrix square root. Comparing the Cayley and polar-factor retractions:

Retraction	Cost per Step	Residual $Y \in SpSt(2n,2p)$8
Cayley	$Y \in SpSt(2n,2p)$9	$S \in T_Y SpSt$0
Polar-factor	$S \in T_Y SpSt$1 (larger constant)	$S \in T_Y SpSt$2

In practical scenarios ($S \in T_Y SpSt$3), Cayley retraction is about three times faster and delivers slightly better symplectic residuals for moderate step sizes. The polar-factor’s use of the exact matrix exponential confers better robustness for larger steps, at the expense of higher per-iteration cost (Zimmermann, 7 May 2026).

6. Implementation Details

MATLAB-style pseudocode for the Cayley retraction and its inverse, aligning with Oviedo–Herrera (2023) and Gao et al. (2020), is as follows:

$S \in T_Y SpSt$5 and for the inverse: $S \in T_Y SpSt$6

A plausible implication is that these routines are immediately applicable in manifold optimization libraries where efficient, structure-preserving updates are required.

7. References and Further Directions

Relevant contributions to the theory and implementation of Cayley retractions on $S \in T_Y SpSt$4 are provided by Gao-Son-Absil-Stykel (SIAM J. Optim. 2021), Bendokat–Zimmermann (Bendokat et al., 2021), and Oviedo–Herrera (2023). Recent work by Zimmermann has advanced alternatives, particularly the polar-factor retraction, and benchmarks these schemes for cost and accuracy (Zimmermann, 7 May 2026). Further exploration is warranted in the direction of large-step stability, hybrid schemes, and the incorporation of such retractions into large-scale optimization and numerical integration routines.