An new polar factor retraction on the Stiefel manifold with closed-form inverse

Ralf Zimmermann; Rasmus Jensen

REVIEW 3 major objections 6 minor 17 references

An new polar factor retraction on the Stiefel manifold with closed-form inverse

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The polar-light retraction gives the Stiefel manifold its first second-order retraction with a closed-form inverse, enabling explicit round-trip coordinates on the manifold of orthogonal frames.

desk verdict A sound, incremental retraction with a genuinely closed-form inverse, but the paper's own text contradicts its 'first' claim and the chart domain is vague. read the letter →

arxiv 2602.19923 v2 pith:I7DXBJXF submitted 2026-02-23 math.NA cs.NAmath.DG

Rasmus Jensen , Ralf Zimmermann This is my paper

classification math.NAcs.NAmath.DG MSC 15A1615B1053Z5065D0565F60

keywords Stiefelmanifoldretractionpolardecompositionclosed-forminverseEuclideanmetricmatrixlogarithminterpolationRiemanniancomputing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces the polar-light retraction, a map from tangent vectors to the Stiefel manifold of orthogonal frames built from a polar decomposition of a small block. The central claim is that this retraction is the first that is second-order accurate under the Euclidean metric and also has a closed-form inverse. If true, it provides an explicit, non-iterative logarithm-style map for the Stiefel manifold, replacing the Sylvester or Lyapunov equation solvers previously needed for the polar factor retraction. Numerical experiments suggest the new retraction tracks Riemannian geodesics more closely than the classical polar factor retraction, especially in higher dimensions.

What carries the argument

The polar-light retraction: a map that exponentiates only the small p×p skew-symmetric block $A = \hat U^T \xi$ of a tangent vector, forms $\hat U \exp(A) + (I - \hat U\hat U^T)\xi$, and normalizes with $\left(I + \xi^T(I - \hat U\hat U^T)\xi\right)^{-1/2}$ to land on the Stiefel manifold. The inverse uses the SVD $M S R^T = \hat U^T U$ and the matrix logarithm of the orthogonal Procrustes rotation $M R^T$ to recover $A$ and $B$, making the inverse closed-form.

What would settle it

Take $p=1$ (the sphere) with $U_0 = e_1$ and $U_1 = -e_1$. Then $\hat U^T U = -1$, and the inverse formula (3.7) calls logm on a scalar with no principal logarithm, so the closed-form inverse fails. More generally, any pair of Stiefel points whose Procrustes rotation $M R^T$ has a $-1$ eigenvalue breaks the inverse formula.

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Extended reading notes

Core claim

The paper defines a coordinate chart centered at any Stiefel point $\hat U$ via the forward map $\phi_{\hat U}(\xi) = \left(\hat U \exp(\hat U^T \xi) + (I - \hat U\hat U^T)\xi\right)\left(I + \xi^T(I - \hat U\hat U^T)\xi\right)^{-1/2}$ and its inverse $\psi_{\hat U}(U)$, which is computable in closed form from the SVD of $\hat U^T U$. The authors claim that $\phi$ is second-order accurate under the Euclidean metric ($\beta=1$ in the metric family of the paper) and that $\psi$ is the exact inverse on a neighborhood of the base point. This makes the pair the first Stiefel retraction with both second-order accuracy and a closed-form inverse, giving explicit local coordinates for the manifold.

Load-bearing premise

The closed-form inverse is only valid on the uncharacterized neighborhood where the Procrustes rotation between the two Stiefel points has no eigenvalue exactly -1, so the principal matrix logarithm is well-defined.

Editorial extensions

If this is right

Computing the inverse retraction no longer requires solving a Sylvester or Lyapunov matrix equation; the cost is comparable to evaluating the retraction itself.
Riemannian barycenters and interpolation on the Stiefel manifold can be computed with explicit, non-iterative round-trip maps under the Euclidean metric.
The explicit inverse reveals how the skew-symmetric A-block and rectangular B-block enter the local coordinates, which may simplify error analysis in manifold optimization.
Because only p×p matrix functions appear, the method scales well when n >> p, with large matrices handled by simple multiplications.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same small-block polar decomposition construction could likely be adapted to other reductive homogeneous spaces, giving explicit retraction/log pairs beyond the Stiefel manifold.
Because the inverse is explicit, tasks that repeatedly need the logarithm, such as intrinsic statistics or manifold regression, may become practical for larger p than before.
The paper's numerical results suggest the advantage over the polar factor retraction grows with p; a systematic study of the error versus p and distance would clarify when the extra forward cost is worthwhile.
Replacing expm/logm with Cayley approximations, as the authors suggest, would preserve second order only if the square-root factor is adjusted; the paper notes this but does not test it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Jensen and Zimmermann give a new retraction on the Stiefel manifold, the polar-light retraction, built by replacing the upper p×p block in the classical polar factor retraction with expm(A). The clean part is the inverse: an explicit SVD-based formula that is genuinely closed-form, and the Taylor expansion in Lemma 3.3 shows second-order accuracy under the Euclidean metric. The code is on GitHub, so the numerical comparison with the polar factor retraction is reproducible. That part is solid and useful.

The soft spot is the novelty claim. The abstract says the Cayley retraction 'features a closed-form inverse,' while the introduction says the authors are 'not aware of any published work on computing the inverse' for Cayley and that quasi-geodesics are the only ones with a closed-form inverse. Both cannot be true. This contradiction sits directly on top of the paper's central assertion that the new retraction is the first with second-order accuracy under the Euclidean metric and a closed-form inverse. The authors do not systematically survey the retraction literature to rule out existing examples; they assert it. That needs to be either proven or softened.

A second soft spot, smaller, is the domain of the inverse chart. The logm in (3.7) requires the orthogonal factor M R^T to have no negative real eigenvalues. The paper defines B as a 'suitable neighborhood' but never characterizes it. For a local chart that's acceptable, but it limits the practical claim of an 'efficient logarithm' for arbitrary pairs (U0,U1). The numerical experiments only compare against the polar factor retraction, not against Cayley or quasi-geodesics, so the 'closer to geodesics' conclusion is narrower than the abstract implies.

Overall: the mathematics is correct and the closed-form inverse is the real contribution. The paper is worth a serious referee, but the authors need to clean up the contradictory statements about the Cayley retraction and be more careful about what 'first' means. I'd send it to review with the expectation of a revision.

Referee Report

3 major / 6 minor

Summary. The paper introduces a new retraction on the compact Stiefel manifold, called the polar-light retraction. For a base point Û and tangent vector ξ, the forward map is φ_Û(ξ) = (Û expm(Ûᵀξ) + (I − ÛÛᵀ)ξ)(I + ξᵀ(I − ÛÛᵀ)ξ)^{-1/2}. The authors prove that this map is a second-order retraction under the Euclidean metric, and they give an explicit inverse formula ψ_Û(U) based on a singular value decomposition and a matrix logarithm. Numerical experiments compare the retraction and its inverse with the classical polar factor retraction in terms of accuracy and runtime. The paper's central claim is that this is the first Stiefel retraction that is simultaneously second-order accurate under the Euclidean metric and has a closed-form inverse.

Significance. The mathematical development is largely sound and transparent: the Taylor expansion in Lemma 3.3 correctly matches the Riemannian exponential second derivative for the Euclidean metric, and the inverse identities in Lemma 3.2 are algebraically verified. Providing reproducible Python code is a clear strength. If the novelty claim is properly substantiated and the domain of the inverse is characterized, the polar-light retraction would be a useful tool for manifold interpolation, barycenter computation, and other applications requiring an explicit inverse retraction. However, the significance depends critically on two issues: the paper's own text contradicts itself about the Cayley retraction's inverse, and the inverse formula's domain of validity is left unspecified.

major comments (3)

[Abstract and §1] The abstract (as printed at the top) states that the Cayley retraction 'is second-order accurate under the canonical metric and features a closed-form inverse,' while §1 (lines 30–35) states that for the Cayley retraction 'we are not aware of any published work on computing the inverse' and that quasi-geodesics 'are the only ones on the list that feature a closed form inverse.' These statements are mutually contradictory. Because the paper's main novelty claim is being 'the first' with these properties under the Euclidean metric, this contradiction is load-bearing. The authors must correct the abstract, clarify whether the novelty is relative to the Euclidean metric only, and provide a more systematic literature check on existing retractions with closed-form inverses.
[Lemma 3.1 and Eqs. (3.5)–(3.7)] The inverse formula is not well-defined on the entire Stiefel manifold. It requires ÛᵀU to have full rank, otherwise the SVD factor S is singular and S^{-1} is undefined. Additionally, the matrix M Rᵀ = ÛᵀU(UᵀÛÛᵀU)^{-1/2} is only in O(p), not necessarily SO(p), and the principal matrix logarithm is real only when this matrix has no negative real eigenvalues. The paper merely states that B is 'a relative open, path-connected neighborhood' without characterizing it. The experiments use endpoints at distance π/2, where ÛᵀU can be singular in low-dimensional examples. The authors must specify conditions under which (3.5)–(3.7) produce a real tangent vector and discuss branch selection for the logarithm.
[Lemma 3.1, proof of det(R)=+1] The argument that det(R)=det(U₁(U₁ᵀU₁)^{-1/2})=+1 uses a continuous path from E to U inside B. However, if the path passes through a point where U₁ is singular, the polar factor R is not defined. B must be chosen inside the open set {U : det(U₁)≠0}, and this restriction should be stated explicitly. The current text gives the impression that B can be any small neighborhood, but the chart construction requires avoiding both singular U₁ and orthogonal factors with negative real eigenvalues.

minor comments (6)

[Title] The title contains a grammatical error: 'An new' should be 'A new'.
[Lemma 3.2] Typo: 'paramaterization' should be 'parametrization' or 'parameterization'.
[Section 4] Typo: 'Forbenius' should be 'Frobenius'.
[References] Reference [9] lists the same URL twice; this should be cleaned up.
[Section 4] The numerical experiments are presented as single runs without error bars or repeated trials. Since the claims about accuracy are empirical, a few repetitions or confidence intervals would strengthen the presentation.
[Section 5] The term 'closed-form' is used for formulas involving SVD and matrix logarithm, which are iterative in floating-point arithmetic. The authors should clarify that 'closed-form' means 'no iterative solve of a matrix equation' in this context.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: polar-light retraction and its inverse are derived algebraically, not fitted or imported from self-citations.

full rationale

The derivation chain is self-contained. Lemma 3.1 defines a chart via the polar decomposition of the upper block U1, and Lemma 3.2 verifies the inverse map φ_E by direct substitution using U1^T U1 + U2^T U2 = I_p; the inverse is thus constructed and checked, not assumed. The second-order accuracy claim in Lemma 3.3 is proven by expanding φ_Û(tξ) in t and comparing with the Taylor expansion of the Riemannian exponential; the comparison uses the known β-metric exponential formula cited from [17], but that formula is independent of φ and does not contain the retraction as an input. No parameters are fitted, and the numerical experiments are endpoint-matched comparisons, not predictions of fitted quantities. The paper contains an internal inconsistency about whether the Cayley retraction has a closed-form inverse (Abstract vs. Introduction), but that is a support/novelty issue, not circularity: it does not reduce the derivation to its inputs. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities or free parameters; it relies on standard matrix analysis and a local chart domain assumption. The only domain-specific premise is the local well-definedness of the logarithm on the chart neighborhood.

assumptions (3)

standard math Matrix exponential maps skew-symmetric matrices surjectively onto SO(p) and logm is a local inverse
Used to define chart Ψ_E and inverse φ_E in Lemma 3.1/3.2.
domain assumption The Riemannian exponential under the one-parameter metric family has the form in eq. (3.8), with D^2 Exp as stated
Taken from [17]; used to verify second-order property in Lemma 3.3.
domain assumption The neighborhood B is chosen so that U1 is invertible and the principal logarithm is well-defined
The chart is only local; the inverse formula requires a unique real logm of the orthogonal factor M R^T.

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Pith. "Pith review of An new polar factor retraction on the Stiefel manifold with closed-form inverse." pith.science (2026). https://pith.science/paper/I7DXBJXF

@misc{pith2026260219923,
  author       = {Pith},
  title        = {Pith review of: An new polar factor retraction on the Stiefel manifold with closed-form inverse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7DXBJXF}},
  note         = {Machine review of arXiv:2602.19923}
}

read the original abstract

Retractions are the workhorses in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order accurate under the Euclidean metric and features a closed-form inverse that can be efficiently computed. A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction, quasi--geodesics and the Cayley retraction. The Cayley retraction is second--order accurate under the canonical metric and features a closed-form inverse. The new retraction is the first one with the corresponding features under the Euclidean metric. We present numerical experiments which illustrates the properties of the new retraction, as well as compare it to numerous of the currently available alternatives. In addition, we examine the performance of the retraction when used for interpolation and for computing a Riemannian barycenter.

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Works this paper leans on

17 extracted references · 5 canonical work pages

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Absil, R

P.-A. Absil, R. Mahony, and R. Sepulchre , Optimization Algorithms on Matrix Manifolds , Princeton University Press, Princeton, New Jersey, 2008, http://press.princeton.edu/titles/8586.html

2008

[2] [2]

Bendokat and R

T. Bendokat and R. Zimmermann , Efficient quasi-geodesics on the S tiefel manifold , in Geometric Science of Information, F. Nielsen and F. Barbaresco, eds., Lecture Notes in Computer Science, Springer, 2021, pp. 763--771, https://doi.org/10.1007/978-3-030-80209-7_82

work page doi:10.1007/978-3-030-80209-7_82 2021

[3] [3]

Edelman, T

A. Edelman, T. A. Arias, and S. T. Smith , The geometry of algorithms with orthogonality constraints , SIAM Journal on Matrix Analysis and Applications, 20 (1998), pp. 303--353, http://dx.doi.org/10.1137/S0895479895290954

doi:10.1137/s0895479895290954 1998

[4] [4]

E. S. Gawlik and M. Leok , High-order retractions on matrix manifolds using projected polynomials , SIAM Journal on Matrix Analysis and Applications, 39 (2018), pp. 801--828, https://doi.org/10.1137/17M1130459

work page doi:10.1137/17m1130459 2018

[5] [5]

Godement , Introduction to the Theory of Lie Groups , Universitext, Springer International Publishing, 2017

R. Godement , Introduction to the Theory of Lie Groups , Universitext, Springer International Publishing, 2017

2017

[6] [6]

G. H. Golub and C. F. Van Loan , Matrix Computations , The John Hopkins University Press, Baltimore, 4th ed., 2013

2013

[7] [7]

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

2008

[8] [8]

H \"u per, I

K. H \"u per, I. Markina, and F. Silva Leite , A L agrangian approach to extremal curves on S tiefel manifolds , Journal of Geometrical Mechanics, 13 (2021), pp. 55--72

2021