Linear Algebra Problems Solved by Using Damped Dynamical Systems on the Stiefel Manifold

A Oleynik; M Gulliksson; M Ogren; R Bakhshandeh-Chamazkoti

arxiv: 2510.10535 · v1 · submitted 2025-10-12 · 🧮 math.OC · cs.NA· math.DS· math.NA

Linear Algebra Problems Solved by Using Damped Dynamical Systems on the Stiefel Manifold

M Gulliksson , A Oleynik , M Ogren , R Bakhshandeh-Chamazkoti This is my paper

Pith reviewed 2026-05-18 07:53 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.DSmath.NA

keywords Stiefel manifolddamped dynamical systemsconstrained optimizationmanifold optimizationnumerical methodslinear algebra problemsordinary differential equations

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The pith

Damped dynamical systems solve minimization problems on the Stiefel manifold by driving solutions to satisfy the orthonormality constraints in the limit.

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

The paper introduces a continuous-time method for optimization problems where the decision variables must form an orthonormal matrix. One damped dynamical system minimizes the objective function while a second damped system is coupled to it to enforce the Stiefel constraint asymptotically rather than at every instant. This construction converts the constrained problem into an unconstrained system of ordinary differential equations that can be integrated with standard numerical solvers. The authors demonstrate the approach on linear algebra tasks and compare its performance to a conjugate gradient method adapted to the manifold.

Core claim

We develop a new method for solving minimization problems on the Stiefel Manifold using damped dynamical systems. The constraints are satisfied in the limit by an additional damped dynamical system. The method is illustrated by numerical experiments and compared to a state-of-the-art conjugate gradient method.

What carries the argument

A pair of coupled damped dynamical systems, one minimizing the objective and the other asymptotically enforcing the Stiefel manifold constraint of orthonormal columns.

If this is right

Linear algebra problems with orthogonal constraints can be recast as the integration of ordinary differential equations without explicit projection or retraction steps at each iteration.
The method supplies an alternative to Riemannian optimization techniques such as conjugate gradient on the Stiefel manifold.
Numerical performance can be directly compared to existing manifold algorithms through the provided experiments.

Where Pith is reading between the lines

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

The damping construction may be reusable for optimization on other matrix manifolds by designing an auxiliary system that penalizes deviation from the desired geometry.
Because the formulation is continuous, it could serve as the basis for deriving new discrete integrators that automatically preserve constraints up to discretization error.
Parameter tuning of the two damping coefficients might yield faster convergence for particular classes of objective functions.

Load-bearing premise

The combined damped dynamical systems converge to a minimizer that lies on the Stiefel manifold.

What would settle it

Integrate the system on a standard test problem such as the orthogonal Procrustes problem and check whether the trajectory reaches a matrix whose columns are orthonormal to machine precision while the objective value matches the known minimum.

read the original abstract

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

Damped flows on Stiefel manifold show promise in numerics but need convergence proof.

read the letter

The key point is that this paper uses a pair of damped dynamical systems to solve optimization problems on the Stiefel manifold, with one system driving the objective down and the other enforcing the orthogonality constraint over time. It backs this up with some numerical comparisons to conjugate gradient. The construction itself is straightforward and avoids some of the usual retraction or projection steps common in manifold optimization. The experiments show that for the problems they tried, the method reaches good points and stays close to the manifold. That's a practical plus if you're looking for simple-to-implement continuous-time solvers. Where it falls short is the missing analysis. The claim that the constraints are satisfied in the limit rests on the behavior of the auxiliary damped system, but there's no Lyapunov function, no invariance principle, and no asymptotic analysis to back it up. Numerical experiments can look fine for specific damping values and initial conditions, yet they don't establish that the combined flow reliably converges to a minimizer on the manifold in general. This is a real gap for a methods paper. The work is aimed at people in optimization who are comfortable with dynamical systems approaches to constrained problems. Someone working on manifold methods might pick up the idea for further development, especially if they want to explore damping parameters. I'd bring this to a reading group to talk through the numerics and see if the experiments hold up under different tests. It should go to peer review because the core idea is clear and the numerics provide a starting point, even though the theory needs strengthening.

Referee Report

2 major / 1 minor

Summary. The paper develops a new method for solving minimization problems on the Stiefel Manifold using damped dynamical systems. An auxiliary damped dynamical system is introduced to enforce the orthogonality constraint X^T X = I in the limit t → ∞. The approach is demonstrated via numerical experiments and compared favorably to a state-of-the-art conjugate gradient method.

Significance. If rigorous convergence guarantees can be established, the method could provide an alternative dynamical-systems perspective on manifold-constrained optimization problems that arise in linear algebra (e.g., orthogonal Procrustes or low-rank matrix problems). The numerical illustrations and direct comparison to conjugate gradient supply some empirical evidence of practicality, which is a positive feature of the manuscript.

major comments (2)

[Abstract] Abstract and introduction: the central claim that the combined primary and auxiliary damped flows drive trajectories to a minimizer on the Stiefel manifold rests on the unproven assertion that ||X^T X − I|| → 0 while the objective decreases. No Lyapunov function, LaSalle invariance argument, or asymptotic analysis is supplied to establish this joint convergence.
Numerical experiments section: the manuscript states that experiments 'illustrate the method and compare it favorably' to conjugate gradient, yet no quantitative metrics (final objective values, constraint violation norms, iteration counts, or statistical summaries over multiple runs) are reported. This prevents verification of the claimed advantage.

minor comments (1)

The title refers to 'Linear Algebra Problems' while the abstract remains general; specifying the concrete problems (e.g., eigenvalue, Procrustes) addressed in the experiments would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and describe the revisions we plan to incorporate.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the central claim that the combined primary and auxiliary damped flows drive trajectories to a minimizer on the Stiefel manifold rests on the unproven assertion that ||X^T X − I|| → 0 while the objective decreases. No Lyapunov function, LaSalle invariance argument, or asymptotic analysis is supplied to establish this joint convergence.

Authors: We agree that a formal convergence analysis would strengthen the presentation. The auxiliary damped system is constructed so that its damping term drives the constraint violation ||X^T X − I|| toward zero at an exponential rate, while the primary flow decreases the objective along the resulting trajectory. In the revised manuscript we will expand the introduction with a brief discussion of this asymptotic mechanism and add a short subsection outlining the expected joint behavior under the combined damping. We will also explicitly note that a complete Lyapunov or LaSalle argument is left for future work, as the current contribution focuses on the formulation and its numerical performance. revision: partial
Referee: [—] Numerical experiments section: the manuscript states that experiments 'illustrate the method and compare it favorably' to conjugate gradient, yet no quantitative metrics (final objective values, constraint violation norms, iteration counts, or statistical summaries over multiple runs) are reported. This prevents verification of the claimed advantage.

Authors: We thank the referee for this observation. In the revised version we will augment the numerical experiments section with explicit quantitative metrics, including final objective values, constraint violation norms ||X^T X − I||, effective integration steps, and statistical summaries (means and standard deviations) computed over multiple independent runs from random initial conditions. These additions will enable direct, reproducible comparison with the conjugate-gradient baseline. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a new damped dynamical system approach for Stiefel manifold optimization, with an auxiliary damped system to enforce the constraint X^T X = I in the limit as t → ∞. The central claim is supported by the construction of the combined flow and validated through numerical experiments compared to conjugate gradient methods. No equations reduce by construction to fitted parameters, no load-bearing self-citations justify uniqueness or ansatzes, and the derivation does not rename known results or import theorems from overlapping prior work in a circular manner. The method stands as an independent proposal without tautological reduction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the modeling choice that two coupled damped ODEs can be constructed whose joint limit satisfies both the objective and the Stiefel constraint; no explicit free parameters or invented entities are named in the abstract.

axioms (1)

standard math The Stiefel manifold is a smooth embedded submanifold of Euclidean space on which a Riemannian structure can be defined.
Implicit background assumption required for any dynamical system on the manifold.

pith-pipeline@v0.9.0 · 5585 in / 1146 out tokens · 29534 ms · 2026-05-18T07:53:38.912997+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

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

We develop a new method for solving minimization problems on the Stiefel Manifold using damped dynamical systems. The constraints are satisfied in the limit by an additional damped dynamical system.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Asymptotic stability via reduced Jacobian Jr(ˆz) having eigenvalues with negative real parts (Theorem 6.2)

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