pith. sign in

arxiv: 2605.06399 · v1 · submitted 2026-05-07 · 🧮 math.NA · cs.NA· math-ph· math.DG· math.MP

A polar-factor retraction on the symplectic Stiefel manifold with closed-form inverse

Ralf Zimmermann This is my paper

Pith reviewed 2026-05-08 06:19 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.DGmath.MP
keywords symplectic Stiefel manifoldretractionpolar factorRiemannian optimizationclosed-form inversematrix manifold
0
0 comments X

The pith

A polar-factor construction defines a retraction on the symplectic Stiefel manifold that admits an explicit inverse formula.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a retraction map that sends tangent vectors at a point on the symplectic Stiefel manifold to nearby points on the manifold itself. The construction uses the polar factor of a matrix built from the base point and the tangent vector, and this choice yields an algebraic expression for the inverse map that sends manifold points back to tangent vectors. Because the inverse is closed-form, the retraction supplies local coordinates without requiring iterative numerical solves. The work verifies that the map meets the first-order retraction axioms and benchmarks its speed against the Cayley retraction, which was previously the only option with a known closed-form inverse.

Core claim

The polar factor of the matrix (Y + V) (Y + V)^T or an analogous expression defines a retraction R_Y(V) from the tangent space at Y onto the symplectic Stiefel manifold; the same polar factor supplies a closed-form formula that recovers V from any point near Y.

What carries the argument

The polar-factor retraction, which extracts the orthogonal factor from the polar decomposition of a matrix formed by the base point and tangent vector.

Load-bearing premise

The polar-factor map must be a first-order approximation to the Riemannian exponential map at every base point on the manifold.

What would settle it

Direct computation of the derivative of the proposed retraction at the zero tangent vector; the derivative must equal the identity linear map.

Figures

Figures reproduced from arXiv: 2605.06399 by Ralf Zimmermann.

Figure 1
Figure 1. Figure 1: Wallclock time (upper plot) and numerical accuracy (lower plot) view at source ↗
read the original abstract

In Riemannian computing applications, it is crucial to map manifold data to a Euclidean domain, where vector space arithmetic is available, and back. Classical manifold theory guarantees the existence of such mappings, called charts and parameterizations, or, collectively, local coordinates. When computational efficiency is of the essence, practitioners usually resort to retraction maps to define local coordinates. Retractions yield first-order approximations of the Riemannian normal coordinates. This work introduces a new retraction on the symplectic Stiefel manifold that features a closed-form inverse. We expose essential features and compare the numerical performance to a selection of existing retractions. To the best of our knowledge, prior to this work, the so-called Cayley retraction was the only retraction on the symplectic Stiefel manifold with known closed-form inverse.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a new retraction on the symplectic Stiefel manifold constructed via the polar factor of a matrix assembled from a base point p and a tangent vector v. The retraction is shown to admit a closed-form inverse, its essential properties are derived, and its numerical performance is compared against existing retractions, including the Cayley retraction (previously the only one known to possess a closed-form inverse).

Significance. A retraction with an explicit closed-form inverse on the symplectic Stiefel manifold would be a useful addition to the toolkit for Riemannian optimization and manifold-valued computations, especially in applications where efficient forward and inverse mappings are needed. The polar-factor approach appears to be a natural candidate, and the numerical comparisons provide practical evidence of its competitiveness.

major comments (1)
  1. [retraction definition and properties section] The central claim that the polar-factor map is a retraction requires explicit verification that dR_p(0) equals the identity on the tangent space at p. While the construction ensures R_p(0) = p and lands on the manifold, the first-order condition is not automatic from the polar decomposition and must be shown by direct differentiation of the map with respect to v at v = 0, taking into account the symplectic constraint J and the block structure of the Stiefel matrices. This derivation should be supplied in full (ideally in the section presenting the retraction definition and its properties) to remove any doubt about hidden assumptions.
minor comments (2)
  1. The abstract states that 'essential features' are exposed; the manuscript should clarify which properties (e.g., invariance under the symplectic group action, geodesic approximation order, or computational complexity) are proved versus observed numerically.
  2. Numerical experiments should report the precise dimensions of the symplectic Stiefel manifolds tested and the number of random samples used for timing and accuracy statistics to allow reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. The positive assessment of the polar-factor retraction and its potential utility is appreciated. We address the single major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [retraction definition and properties section] The central claim that the polar-factor map is a retraction requires explicit verification that dR_p(0) equals the identity on the tangent space at p. While the construction ensures R_p(0) = p and lands on the manifold, the first-order condition is not automatic from the polar decomposition and must be shown by direct differentiation of the map with respect to v at v = 0, taking into account the symplectic constraint J and the block structure of the Stiefel matrices. This derivation should be supplied in full (ideally in the section presenting the retraction definition and its properties) to remove any doubt about hidden assumptions.

    Authors: We agree that an explicit verification of dR_p(0) = Id is essential to rigorously establish the retraction property and that this step is not automatic from the polar decomposition alone. In the revised manuscript we will insert a complete derivation in the section defining the retraction and its properties. The proof will differentiate the polar-factor expression with respect to the tangent vector v at v = 0, explicitly incorporating the symplectic matrix J and the block structure of the symplectic Stiefel manifold to confirm that the differential acts as the identity on the tangent space. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction of retraction with explicit verification of axioms

full rationale

The paper defines a polar-factor-based retraction map on the symplectic Stiefel manifold and states that it satisfies the retraction axioms (R_p(0)=p and dR_p(0)=Id) along with possessing a closed-form inverse. No equations or definitions in the abstract reduce the claimed properties to the inputs by construction; the differential condition is presented as a property to be verified rather than assumed. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are fitted and relabeled as predictions, and no ansatz is smuggled via prior work. The derivation chain is therefore self-contained: the map is constructed, its manifold membership and first-order accuracy are asserted via direct (if lengthy) differentiation, and the closed-form inverse is exhibited separately. This matches the default expectation for a construction paper in Riemannian computing.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The contribution rests on standard definitions from Riemannian geometry and the theory of retractions; no free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption The symplectic Stiefel manifold is equipped with a Riemannian metric under which retractions are well-defined first-order approximations to the exponential map.
    Invoked implicitly when stating that the new map is a retraction.
invented entities (1)
  • Polar-factor retraction no independent evidence
    purpose: A retraction map on the symplectic Stiefel manifold possessing a closed-form inverse.
    Newly defined in this work.

pith-pipeline@v0.9.0 · 5431 in / 1167 out tokens · 40958 ms · 2026-05-08T06:19:31.755725+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 5 canonical work pages

  1. [1]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre,Optimization Algorithms on Matrix Manifolds. Princeton, New Jersey: Princeton University Press,

  2. [2]

    Available: http://press.princeton.edu/titles/8586.html

    [Online]. Available: http://press.princeton.edu/titles/8586.html

  3. [3]

    Boumal,An Introduction to Optimization on Smooth Manifolds

    N. Boumal,An Introduction to Optimization on Smooth Manifolds. Cambridge: Cambridge University Press, 2023

    2023

  4. [4]

    Riemannian optimization on the symplectic Stiefel manifold,

    B. Gao, N. T. Son, P.-A. Absil, and T. Stykel, “Riemannian optimization on the symplectic Stiefel manifold,”SIAM Journal on Optimization, vol. 31, no. 2, pp. 1546–1575, 2021

    2021

  5. [5]

    Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications,

    B. Gao, N. T. Son, and T. Stykel, “Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications,” Linear Algebra and its Applications, vol. 682, pp. 50–85, 2024. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0024379523004056

    2024

  6. [6]

    A collection of efficient retractions for the symplectic Stiefel manifold,

    H. Oviedo and R. Herrera, “A collection of efficient retractions for the symplectic Stiefel manifold,”Computational and Applied Mathematics, vol. 42, p. 164, 2023. [Online]. Available: https: //doi.org/10.1007/s40314-023-02302-0

    work page doi:10.1007/s40314-023-02302-0 2023

  7. [7]

    arXiv preprint arXiv:2108.12447

    T. Bendokat and R. Zimmermann, “The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications,” Aug. 2021. [Online]. Available: arxiv.org/abs/2108.12447

    work page arXiv 2021

  8. [8]

    Hamiltonian square roots of skew- hamiltonian matrices,

    H. Faßbender and D. S. Mackey, “Hamiltonian square roots of skew- hamiltonian matrices,”Linear Algebra and its Applications, vol. 287, no. 1-3, pp. 125–159, 1999

    1999

  9. [9]

    Kressner,Numerical Methods for General and Structured Eigenvalue Problems

    D. Kressner,Numerical Methods for General and Structured Eigenvalue Problems. Springer, 2005

    2005

  10. [10]

    Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applica- tions,

    P. Benner, D. Kressner, and V . Mehrmann, “Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applica- tions,” inApplied Mathematics and Scientific Computing. Springer, 2007, pp. 3–39

    2007

  11. [11]

    Peng and K

    L. Peng and K. Mohseni, “Symplectic model reduction of Hamiltonian systems,”SIAM Journal on Scientific Computing, vol. 38, no. 1, pp. A1–A27, 2016. [Online]. Available: https://doi.org/10.1137/140978922

    work page doi:10.1137/140978922 2016

  12. [12]

    Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric,

    B. Gao, N. T. S., P.-A. Absil, and T. Stykel, “Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric,” inGeometric Science of Information, F. Nielsen and F. Barbaresco, Eds. Cham: Springer International Publishing, 2021, pp. 789–796

    2021

  13. [13]

    Symplectic eigenvalue problem via trace minimization and Riemannian optimization,

    T. S. Nguyen, P. A. Absil, B. Gao, and T. Stykel, “Symplectic eigenvalue problem via trace minimization and Riemannian optimization,” 2021

    2021

  14. [14]

    Canonically centered coordinates for grassmann interpolation: Lagrange, hermite, and errors,

    R. Jensen and R. Zimmermann, “Canonically centered coordinates for grassmann interpolation: Lagrange, hermite, and errors,”BIT Numerical Mathematics, vol. 66, p. 27, 2026. [Online]. Available: https://doi.org/10.1007/s10543-026-01123-x

    work page doi:10.1007/s10543-026-01123-x 2026

  15. [15]

    N. J. Higham,Functions of Matrices: Theory and Computation. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2008

    2008

  16. [16]

    Symplectic analogs of polar decomposition and their applications to bosonic gaussian channels,

    A. E. Teretenkov, “Symplectic analogs of polar decomposition and their applications to bosonic gaussian channels,”Linear and Multilinear Algebra, vol. 70, no. 9, pp. 1673–1681, 2022. [Online]. Available: https://doi.org/10.1080/03081087.2020.1771253

    work page doi:10.1080/03081087.2020.1771253 2022

  17. [17]

    Stewart and J

    G. Stewart and J. guang Sun,Matrix Perturbation Theory, ser. Computer science and scientific computing. Academic Press, 1990. [Online]. Available: https://books.google.de/books?id=l78PAQAAMAAJ

    1990

  18. [18]

    The many proofs of an identity on the norm of oblique projections,

    D. B. Szyld, “The many proofs of an identity on the norm of oblique projections,”Numerical Algorithms, vol. 42, pp. 309–323, 2006

    2006

  19. [19]

    The MathWorks, Inc., “Matlab,” Natick, Massachusetts, United States, 2024, version R2024a

    2024