A generalized canonical metric for optimization on the indefinite Stiefel manifold

Dinh Van Tiep; Duong Thi Viet An; Nguyen Thanh Son; Nguyen Thi Ngoc Oanh

arxiv: 2509.16113 · v2 · submitted 2025-09-19 · 🧮 math.OC

A generalized canonical metric for optimization on the indefinite Stiefel manifold

Dinh Van Tiep , Duong Thi Viet An , Nguyen Thi Ngoc Oanh , Nguyen Thanh Son This is my paper

Pith reviewed 2026-05-18 15:24 UTC · model grok-4.3

classification 🧮 math.OC

keywords indefinite Stiefel manifoldRiemannian optimizationgeneralized canonical metricquasi-geodesicretractionRiemannian gradient descent

0 comments

The pith

A generalized canonical metric simplifies Riemannian optimization on the indefinite Stiefel manifold.

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

The paper proposes a new Riemannian metric, termed the generalized canonical metric, for the indefinite Stiefel manifold. This metric allows computation of the Riemannian gradient without repeatedly solving Lyapunov matrix equations, unlike the previous family of metrics. The authors also derive a quasi-geodesic curve and a retraction based on it to support manifold optimization algorithms such as Riemannian gradient descent. Numerical tests show the new method performs competitively with lower computational cost per step.

Core claim

The central claim is that the generalized canonical metric equips the indefinite Stiefel manifold with a Riemannian structure in which the gradient of the objective function admits a closed-form expression that avoids the computational burden of solving a Lyapunov equation at each iteration, while still permitting the construction of a quasi-geodesic and an associated retraction that can be used to extend Euclidean optimization methods to the manifold.

What carries the argument

The generalized canonical metric, a Riemannian metric on the indefinite Stiefel manifold chosen so that the orthogonal projection and gradient formula become simpler and cheaper to evaluate.

If this is right

The Riemannian gradient descent algorithm on this manifold becomes cheaper to run because each gradient step avoids a matrix equation solve.
Quasi-geodesics provide a new way to move along the manifold that respects the new geometry.
Retractions derived from the quasi-geodesic enable practical implementation of the optimization method.
Performance comparisons indicate the new approach is at least as effective as the prior Lyapunov-based method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the metric generalizes well, it could reduce costs for larger-scale problems in scientific computing modeled on this manifold.
The construction might suggest similar canonical metrics for related matrix manifolds with indefinite constraints.
Testable extension: apply the method to specific problems like orthogonal Procrustes with indefinite signatures and measure wall-clock time savings.

Load-bearing premise

That the proposed generalized canonical metric is positive definite and compatible with the manifold structure so that it truly defines a Riemannian metric.

What would settle it

A numerical example or algebraic counterexample in which the new metric fails to produce a positive definite inner product on the tangent space or in which the resulting optimization algorithm diverges while the previous method converges.

read the original abstract

Various tasks in scientific computing can be modeled as an optimization problem on the indefinite Stiefel manifold. We address this using the Riemannian approach, which basically consists of equipping the feasible set with a Riemannian metric, preparing geometric tools such as orthogonal projections, formulae for Riemannian gradient, retraction and then extending an unconstrained optimization algorithm on the Euclidean space to the established manifold. The choice for the metric undoubtedly has a great influence on the method. In the previous work [D.V. Tiep and N.T. Son, A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold, arXiv:2410.22068v2[math.OC]], a tractable metric, which is indeed a family of Riemannian metrics defined by a symmetric positive-definite matrix depending on the contact point, has been used. In general, it requires solving a Lyapunov matrix equation every time when the gradient of the cost function is needed, which might significantly contribute to the computational cost. To address this issue, we propose a new Riemannian metric for the indefinite Stiefel manifold. Furthermore, we construct the associated geometric structure, including a so-called quasi-geodesic and propose a retraction based on this curve. We then numerically verify the performance of the Riemannian gradient descent method associated with the new geometry and compare it with the previous work.

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.

Desk Editor's Note private letter to a colleague

New closed-form metric speeds up gradient computation on the indefinite Stiefel manifold but still needs a direct proof of positive definiteness.

read the letter

The key point is that this work replaces the Lyapunov-based metric from the authors' previous paper with a generalized canonical metric that allows a direct formula for the Riemannian gradient. They add a quasi-geodesic and a retraction based on it, and they test the resulting algorithm numerically against the old approach. This is a useful incremental step for anyone doing Riemannian optimization on the indefinite Stiefel manifold. The computational saving comes from avoiding the matrix equation solve each time the gradient is needed, which the abstract identifies as a bottleneck. The numerical verification helps show whether that saving leads to better overall performance on test problems. The retraction construction is also new and ties the geometry together. A real concern is whether the new metric is actually positive definite everywhere it needs to be. The stress test highlights that the paper assumes the metric defines a valid Riemannian structure but does not provide an explicit check, such as showing that the associated operator has positive eigenvalues or giving a lower bound on the quadratic form. The earlier family enforced this through the positive definite solution to the Lyapunov equation. Without a similar guarantee here, the cheaper gradient formula might not always correspond to a proper inner product on the tangent space. That would undermine the whole pipeline. This paper targets specialists in manifold-constrained optimization who already know the indefinite Stiefel setting. A reader working on similar numerical methods will find the extension and the comparisons straightforward to evaluate. It has enough new material and evidence to merit a serious referee report. I would send it to peer review and ask the authors to add the positive-definiteness argument in the revision.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a generalized canonical Riemannian metric on the indefinite Stiefel manifold as a computationally cheaper alternative to the Lyapunov-equation-based family from the authors' prior work. It develops the associated geometric objects, including a quasi-geodesic and a retraction based on this curve, derives the Riemannian gradient under the new metric, and numerically compares the performance of the resulting Riemannian gradient descent algorithm.

Significance. If the new metric defines a valid positive-definite structure and the derived tools are correct, the work could reduce per-iteration cost in Riemannian optimization on this manifold while preserving convergence behavior. The numerical experiments provide preliminary evidence of practical gains. The explicit construction of the quasi-geodesic adds a useful geometric contribution, but the absence of a definiteness verification limits the strength of the central claim.

major comments (1)

[Metric definition section] The section defining the generalized canonical metric does not establish positive definiteness of g_X(ξ,η) for nonzero tangent vectors ξ. The manuscript derives the cheaper Riemannian gradient formula under the assumption that the metric is valid, yet supplies no eigenvalue analysis, lower bound, or explicit check (in contrast to the prior Lyapunov construction that enforced this via a positive-definite matrix solution). This property is load-bearing for the gradient, quasi-geodesic, and retraction.

minor comments (2)

[Abstract and Introduction] The abstract and introduction refer to the previous Lyapunov-based family but could include one sentence recalling its explicit positive-definiteness guarantee for context.
[Metric definition section] Notation for the operator A(X) in the metric definition should be cross-referenced to any earlier equation that defines it.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to verify positive definiteness of the proposed metric. This is a substantive point that strengthens the foundation of the work, and we address it directly below.

read point-by-point responses

Referee: [Metric definition section] The section defining the generalized canonical metric does not establish positive definiteness of g_X(ξ,η) for nonzero tangent vectors ξ. The manuscript derives the cheaper Riemannian gradient formula under the assumption that the metric is valid, yet supplies no eigenvalue analysis, lower bound, or explicit check (in contrast to the prior Lyapunov construction that enforced this via a positive-definite matrix solution). This property is load-bearing for the gradient, quasi-geodesic, and retraction.

Authors: We agree that the original manuscript does not contain an explicit proof or eigenvalue bound establishing positive definiteness of the generalized canonical metric, in contrast to the Lyapunov-based construction in our prior work. This is a valid observation. In the revised version we will add a dedicated paragraph (or short lemma) in the metric definition section that proves g_X(ξ,ξ) ≥ c‖ξ‖_F² for some c>0 and all nonzero tangent vectors ξ. The argument proceeds by writing the metric as a perturbation of the standard Frobenius inner product whose perturbation term is controlled by the signature matrix and the defining relation XᵀJX = I_p; we then obtain a uniform lower bound on the eigenvalues of the associated self-adjoint operator. A brief numerical verification for small (p,n) will also be included for illustration. With this addition the derivations of the Riemannian gradient, quasi-geodesic, and retraction rest on a rigorously verified Riemannian structure. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to authors' prior Lyapunov metric work; new generalized canonical metric presented as independent proposal.

specific steps

self citation load bearing [Abstract]
"In the previous work [D.V. Tiep and N.T. Son, A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold, arXiv:2410.22068v2[math.OC]], a tractable metric, which is indeed a family of Riemannian metrics defined by a symmetric positive-definite matrix depending on the contact point, has been used. In general, it requires solving a Lyapunov matrix equation every time when the gradient of the cost function is needed, which might significantly contribute to the computational cost. To address this issue, we propose a new Riemannian metric for the indefinite Stiefel man"

The citation is to prior work by overlapping authors (Tiep and Son) describing the old metric's cost. However, because the current paper explicitly proposes a distinct new metric to replace it, the self-citation functions only as background motivation and does not make the new metric or its gradient formula equivalent to the prior construction by definition.

full rationale

The paper cites its own earlier arXiv:2410.22068 for the computational drawback of the Lyapunov-based family but introduces the new metric and associated quasi-geodesic/retraction as a fresh construction. No equation reduces the claimed cheaper gradient or validity to a fitted parameter or assumption taken from the cited paper. The derivation chain for the new geometry stands on its own definitions and is not forced by self-citation. This is a normal minor self-reference that does not affect the central independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a pointwise symmetric positive-definite matrix can be chosen to define a Riemannian metric whose gradient formula avoids Lyapunov solves; no free parameters, additional axioms, or invented entities are stated in the abstract.

axioms (1)

domain assumption A family of Riemannian metrics on the indefinite Stiefel manifold can be defined by a symmetric positive-definite matrix that depends on the base point.
This is the modeling choice that replaces the earlier Lyapunov-based metric.

pith-pipeline@v0.9.0 · 5778 in / 1258 out tokens · 41759 ms · 2026-05-18T15:24:32.243069+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we propose a new Riemannian metric for the indefinite Stiefel manifold... gρ,X⊥(Z1,Z2) := tr(Z1^T MX Z2) = (1/ρ)tr(W1^T W2) + tr(K1^T Γ3^{-1} K2)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 3.3... grad gcf(X) = ρ X J skew(J X^T ∇f̄(X)) + ...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

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Princeton University Press, Princeton, NJ (2008)

Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Mani- folds. Princeton University Press, Princeton, NJ (2008)

work page 2008

[2] [2]

Adler, R., Dedieu, J.P., Margulies, J., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for human spine. IMA J. Numer. Anal., 22, 359–390 (2002)

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Bartels, R., Stewart, G.: Solution of the equationAX+XB=C. Comm. ACM, 15(9), 820–826 (1972)

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Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal.,8(1), 141–148 (1988)

work page 1988

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Bendokat, T., Zimmermann, R.: The real symplectic Stiefel and Grassmann mani- folds: metrics, geodesics and applications, https://arxiv.org/abs/2108.12447 (2021)

work page arXiv 2021

[6] [6]

Cambridge University Press (2023)

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

work page 2023

[7] [7]

Boumal, N., Mishra, B., Absil, P.A., Sepulchre, R.: Manopt, a Matlab toolbox for optimization on manifolds. J. Mach. Learn. Res.,15, 1455–1459 (2014)

work page 2014

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Springer Berlin, Heidelberg (1977)

Deimling, K.: Ordinary Differential Equations in Banach Spaces. Springer Berlin, Heidelberg (1977)

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Edelman, A., Arias, T., Smith, S.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl.,20(2), 303–353 (1998)

work page 1998

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Gabay, D.: Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl.,37(2), 177–219 (1982)

work page 1982

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Gao, B., Son, N.T., Absil, P.A., Stykel, T.: Geometry of the symplectic Stiefel man- ifold endowed with the Euclidean metric. In: F. Nielsen, F. Barbaresco (eds.) Ge- ometric Science of Information: GSI 2021,Lecture Notes in Computer Science, vol. 12829, 789–796. Springer Nature, Cham, Switzerland (2021)

work page 2021

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Gao, B., Son, N.T., Absil, P.A., Stykel, T.: Riemannian optimization on the sym- plectic Stiefel manifold. SIAM J. Optim.,31(2), 1546–1575 (2021)

work page 2021

[13] [13]

Gao, B., Son, N.T., Stykel, T.: Symplectic Stiefel manifold: tractable metrics, second- order geometry and Newton’s methods, https://arxiv.org/abs/2406.14299 (2024)

work page arXiv 2024

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G¨ uttel, S., Nakatsukasa, Y.: Scaled and squared subdiagonal Pad´ e approximation for the matrix exponential. SIAM J. Matrix Anal. Appl.,37(1), 145–170 (2016)

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Neural Comput.,16(12), 2639–2664 (2004)

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work page 2004

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He, X., Niyogi, P.: Locality preserving projections. In: S. Thrun, L. Saul, B. Sch¨ olkopf (eds.) Advances in Neural Information Processing Systems, vol. 16. MIT Press (2003)

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Cambridge University Press, Cambridge, UK (1991)

Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK (1991)

work page 1991

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Hotelling, H.: Relations between two sets of variates. In: S. Kotz, N.L. Johnson (eds.) Breakthroughs in Statistics, Springer Ser. Statist., 162–190. Springer, New York, NY (1992)

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Hu, J., Liu, X., Wen, Z.W., Yuan, Y.X.: A brief introduction to manifold optimiza- tion. J. Oper. Res. Soc. China,18, 199–248 (2020) Generalized canonical metric for indefinite Stiefel manifold 23

work page 2020

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Iannazzo, B., Porcelli, M.: The Riemannian Barzilai–Borwein method with nonmono- tone line search and the matrix geometric mean computation. IMA J. Numer. Anal., 38, 495–517 (2018)

work page 2018

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work page 2020

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Linear Algebra Appl.,216, 139–158 (1995)

Kovaˇ c-Striko, J., Veseli´ c, K.: Trace minimization and definiteness of symmetric pen- cils. Linear Algebra Appl.,216, 139–158 (1995)

work page 1995

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Krakowski, K.A., Machado, L., Silva Leite, F., Batista, J.: A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds. J. Comput. Appl. Math.,311, 84–99 (2017)

work page 2017

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Mishra, B., Sepulchre, R.: Riemannian preconditioning. SIAM J. Optim.,26(1), 635–660 (2016)

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Neurocomputing,67, 106–135 (2005)

Nishimori, Y., Akaho, S.: Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold. Neurocomputing,67, 106–135 (2005)

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Sato, H., Aihara, K.: Cholesky QR-based retraction on the generalized Stiefel mani- fold. Comput. Optim. Appl.,72, 293–308 (2019)

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work page 2023

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Fields Inst

Smith, S.T.: Optimization techniques on Riemannian manifolds. Fields Inst. Com- mun.,3, 113–136 (1994)

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A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold

Tiep, D.V., Son, N.T.: A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold. https://doi.org/10.48550/arXiv.2410.22068 (2025)

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2410.22068 2025

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Mathematics and Its Applications

Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications. Springer (1994)

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Wang, X., Deng, K., Peng, Z., Yan, C.: New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications. J. Comput. Appl. Math.,451, 116024 (2024)

work page 2024

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Wilcox, R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys.,8(4), 962–982 (1967)

work page 1967

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Yger, F., Berar, M., Gasso, G., Rakotomamonjy, A.: Adaptive canonical correlation analysis based on matrix manifolds. In: J. Langford, J. Pineau (eds.) ICML ’12 Pro- ceedings of the 29th International Coference on International Conference on Machine Learning, 1071–1078. Omnipress, New York, NY, USA (2012)

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Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim.,14(4), 1043–1056 (2004)

work page 2004