Low-rank solutions to a class of parametrized systems using Riemannian optimization

Marco Sutti; Tommaso Vanzan

arxiv: 2604.07136 · v1 · submitted 2026-04-08 · 🧮 math.NA · cs.NA

Low-rank solutions to a class of parametrized systems using Riemannian optimization

Marco Sutti , Tommaso Vanzan This is my paper

Pith reviewed 2026-05-10 17:21 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords low-rank approximationRiemannian optimizationparametrized systemsmatrix manifoldnumerical methodsnonlinear equations

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The pith

Parametrized systems admit low-rank solutions obtained by Riemannian optimization over fixed-rank matrix manifolds.

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

The paper shows how to collect solutions of a parametrized equation across many parameter values into a single matrix and recover an accurate low-rank version of that matrix without solving the equation separately for each value. The method works by rewriting the original system as the first-order optimality condition of a matrix optimization problem, then minimizing that objective directly on the manifold of matrices with fixed rank using standard Riemannian algorithms such as conjugate gradient or trust-region methods. The same construction applies to both linear and nonlinear cases once mild structural assumptions on the parametrization are met, and the authors supply preconditioners together with a rank-compression step to keep the rank under control when nonlinearities are present. Theoretical bounds establish that the solution matrix stays close to low rank whenever the underlying parametrization satisfies the stated conditions, which in turn produces the reported computational savings.

Core claim

The ensemble of solutions to A(ξ)x(ξ) + g(x(ξ)) = b(ξ) can be assembled into a matrix X whose low-rank factorization is recovered by minimizing a suitable scalar objective over the manifold of fixed-rank matrices; Riemannian optimization then yields accurate low-rank approximations for both linear and nonlinear instances whenever the solution matrix admits such structure under the given mild assumptions on the parametrization.

What carries the argument

Riemannian optimization on the manifold of fixed-rank matrices, applied to the reformulation of the parametrized system as the first-order optimality condition of a matrix-valued optimization problem.

If this is right

The same Riemannian procedure covers both linear and nonlinear parametrized systems once the structural assumptions hold.
Preconditioning and rank-compression steps keep computational cost low even when nonlinear terms would otherwise increase the rank.
Theoretical conditions guarantee that the low-rank approximation remains accurate for systems whose parametrization satisfies the stated mild assumptions.
Significant savings appear relative to solving each parameter instance independently.

Where Pith is reading between the lines

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

The approach could be combined with existing low-rank tensor techniques when the parameter domain is multi-dimensional.
If the rank-compression step proves stable, the framework might serve as a building block for reduced-order models in parametric PDEs.
Testing the method on problems where the nonlinearity couples all parameters strongly would expose the practical limits of the rank-control mechanism.

Load-bearing premise

The parametrized system can be expressed as the gradient condition of an optimization problem whose decision variable is a matrix, and the matrix of solutions across parameters admits accurate low-rank approximations under mild structural assumptions.

What would settle it

Solve a concrete parametrized system whose solution matrix is known to be full rank by both the Riemannian method and by independent solves for each parameter; the central claim fails if the low-rank output deviates from the independent solutions by more than the error bounds claimed in the theory.

Figures

Figures reproduced from arXiv: 2604.07136 by Marco Sutti, Tommaso Vanzan.

**Figure 1.** Figure 1: Evolution of the norm of the Riemannian gradient, of the residual and of the matrix rank during the rank-adaptive algorithm with RCG (top row) and RTR (bottom row) as inner solvers. Parameters: m = 16129, n = 5000, p = 5, initial rank 5 and rup = 5. time, despite requiring more iterations. At the same time, both methods achieve the same approximation errors, which are comparable to those of the best approx… view at source ↗

read the original abstract

We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret the parametrized system as the first-order optimality condition of an optimization problem set over the space of real matrices, which is then minimized over the manifold of fixed-rank matrices. This formulation enables the use of Riemannian optimization techniques, including conjugate gradient and trust-region methods, and covers both linear and nonlinear instances under mild assumptions on the structure of the parametrized system. We further provide a theoretical analysis establishing conditions under which the solution matrix admits accurate low-rank approximations, extending existing results from linear to nonlinear problems. To enhance computational efficiency and robustness, we discuss tailored preconditioning strategies and a rank-compression mechanism to control the rank growth induced by nonlinearities. Numerical experiments demonstrate that the proposed approach achieves significant computational savings compared to solving each system independently, as well as highlight the potential of Riemannian optimization methods for low-rank approximations in large-scale parametrized nonlinear problems.

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

The paper recasts parametrized systems as a matrix optimization problem and minimizes it on the fixed-rank manifold with Riemannian methods, extending low-rank ideas to nonlinear cases with a compression step.

read the letter

The main takeaway is that this work turns the ensemble of solutions to A(ξ)x(ξ) + g(x(ξ)) = b(ξ) into a matrix whose low-rank version is found by Riemannian conjugate gradient or trust-region methods on the fixed-rank manifold. The approach is presented as non-intrusive and applicable to both linear and nonlinear problems under mild structural assumptions, with added rank compression to limit growth from nonlinear terms and some preconditioning discussion. Numerical tests are claimed to show savings over solving each instance separately. What is actually new is the specific combination of the matrix reformulation, the manifold optimization for this ensemble setting, and the extension of low-rank approximation theory plus the compression mechanism to nonlinear instances. The paper does a reasonable job laying out the framework in plain terms and pointing to existing Riemannian solvers that can be reused. The practical angle on controlling rank growth is a useful addition for people who have seen rank blow up in nonlinear parametric problems. The soft spots are in the details that are hard to judge from the description alone. The theoretical conditions for accurate low-rank approximations are stated to extend from linear cases, but how mild those assumptions remain once nonlinearities are present is not obvious without the proofs or counter-examples. The experiments report savings and accuracy, yet the baselines, problem sizes, number of parameters, and how rank r was chosen are not specified here, so it is difficult to know how general the gains are. Choosing r in advance remains a free parameter, and the preconditioners are tailored but their robustness across problem classes is unclear. This is for readers already working on low-rank methods for uncertainty quantification or parametric PDEs who know manifold optimization and want to see it applied to ensemble solves. Someone in that niche could extract the reformulation and the compression idea quickly. It deserves a serious referee because the central claims are concrete enough to check with code and the computational motivation is real, even if revisions will likely be needed on the theory and experiment sections.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a computational framework for low-rank approximations to ensembles of solutions of parametrized systems A(ξ)x(ξ) + g(x(ξ)) = b(ξ). It reinterprets the system as the first-order optimality condition of a matrix-valued optimization problem, restricts the search to the fixed-rank manifold, and applies Riemannian optimization methods (conjugate gradient and trust-region). The work claims to cover both linear and nonlinear cases under mild structural assumptions, provides a theoretical analysis extending existing linear low-rank results, introduces preconditioning and rank-compression techniques, and reports numerical savings relative to independent solves.

Significance. If the reformulation, theoretical conditions, and numerical results hold, the framework could provide a practical route to efficient parametric studies by exploiting low-rank structure across multiple parameter instances via manifold optimization. The extension of low-rank approximability results from linear to nonlinear problems is a potentially useful contribution, and the emphasis on tailored preconditioners and rank control addresses robustness issues that often arise in nonlinear settings. The approach builds on established Riemannian tools, which is a strength when the central claims are substantiated.

major comments (2)

[Theoretical analysis] The theoretical analysis establishing conditions for accurate low-rank approximations in the nonlinear case is central to the broad applicability claim, yet the manuscript provides only a high-level statement of 'mild structural assumptions' without an explicit statement of those assumptions or a proof outline that shows how they extend the linear case (e.g., via the same singular-value decay arguments). This needs to be made precise in the relevant theory section.
[Numerical experiments] The numerical experiments section reports significant computational savings and accuracy, but lacks concrete details on problem dimensions, number of parameter samples, chosen rank r, specific Riemannian solver parameters, and direct comparison baselines (independent solves with the same linear/nonlinear solver). Without these, the efficiency claim cannot be fully assessed or reproduced.

minor comments (3)

[Abstract and §1] The abstract and introduction should clarify the precise form of the matrix-valued objective whose critical points recover the original parametrized system, including any auxiliary variables introduced.
[Throughout] Notation for the solution matrix X(ξ) and the parameter-dependent operators should be introduced consistently before the manifold optimization is described.
[Method description] The rank-compression mechanism is mentioned as a robustness tool; a short algorithmic description or pseudocode would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the framework's potential, and constructive feedback on the theoretical and experimental sections. We address each major comment below and will incorporate the suggested clarifications into the revised manuscript.

read point-by-point responses

Referee: [Theoretical analysis] The theoretical analysis establishing conditions for accurate low-rank approximations in the nonlinear case is central to the broad applicability claim, yet the manuscript provides only a high-level statement of 'mild structural assumptions' without an explicit statement of those assumptions or a proof outline that shows how they extend the linear case (e.g., via the same singular-value decay arguments). This needs to be made precise in the relevant theory section.

Authors: We agree that the assumptions and extension of the proof should be stated more explicitly to support the broad applicability claim. In the revised manuscript we will add a dedicated paragraph in the theory section that lists the precise structural assumptions (Lipschitz continuity of the nonlinearity g with a uniform constant, uniform boundedness and invertibility of A(ξ), and suitable regularity of the right-hand side b(ξ)) and provides a concise outline of how the singular-value decay argument is extended from the linear case: the nonlinear perturbation is controlled via a Lipschitz estimate that preserves the decay rate up to a controllable additive term. This will make the conditions fully transparent without altering the existing high-level statement. revision: yes
Referee: [Numerical experiments] The numerical experiments section reports significant computational savings and accuracy, but lacks concrete details on problem dimensions, number of parameter samples, chosen rank r, specific Riemannian solver parameters, and direct comparison baselines (independent solves with the same linear/nonlinear solver). Without these, the efficiency claim cannot be fully assessed or reproduced.

Authors: We acknowledge that the current presentation of the numerical results omits several implementation details required for reproducibility and quantitative assessment. In the revised manuscript we will expand the experiments section with a table (or dedicated subsection) that reports the matrix dimensions, the number of parameter samples, the chosen ranks r, the specific tolerances and iteration limits used for the Riemannian conjugate-gradient and trust-region solvers, and direct wall-clock and iteration counts for the proposed method versus independent solves performed with the identical underlying linear or nonlinear solver. These additions will allow readers to verify the reported savings. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central derivation reinterprets the parametrized system A(ξ)x(ξ) + g(x(ξ)) = b(ξ) as the first-order optimality condition of a matrix-valued optimization problem, then restricts minimization to the fixed-rank manifold and applies standard Riemannian methods (conjugate gradient, trust-region). This is a constructive reformulation resting on external manifold geometry and optimality conditions rather than any quantity defined from the method's own outputs. The theoretical analysis extends existing linear low-rank approximation results to the nonlinear case under stated mild structural assumptions, without reducing to self-defined parameters, fitted predictions, or load-bearing self-citations. Numerical experiments are presented only as validation of computational savings and accuracy, not as the source of the claims. No step in the chain collapses to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability to recast the parametrized equation as an optimality condition and on the existence of low-rank structure that Riemannian methods can exploit; no explicit free parameters or invented entities are stated in the abstract.

free parameters (1)

fixed rank r
The target rank must be chosen by the user; its value directly controls approximation quality and computational cost.

axioms (1)

domain assumption The parametrized system A(ξ)x(ξ) + g(x(ξ)) = b(ξ) can be expressed as the first-order optimality condition of an optimization problem over real matrices.
This reformulation is the enabling step that allows the problem to be moved onto the fixed-rank manifold.

pith-pipeline@v0.9.0 · 5490 in / 1423 out tokens · 52370 ms · 2026-05-10T17:21:04.729614+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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[ABG07]P.-A. Absil,C. G. Baker, andK. A. Gallivan, Trust-Region Methods on Rieman- nian Manifolds,Found. of Comput. Math.7(2007), 303–330.https://doi.org/10.1007/ s10208-005-0179-9. [AMS08]P.-A. Absil,R. Mahony, andR. Sepulchre,Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ,

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Babuka, F

[BNT10]I. Babuška,F. Nobile, andR. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data,SIAM Review52no. 2 (2010), 317–355.https: //doi.org/10.1137/050645142. [BGW15]P. Benner,S. Gugercin, andK. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems,SIAM review5...

work page doi:10.1137/050645142 2010

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[CdSG21]A. Carr,E. de Sturler, andS. Gugercin, Preconditioning parametrized linear systems, SIAM Journal on Scientific Computing43no. 3 (2021), A2242–A2267. [CS10]S. ChaturantabutandD. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation,SIAM Journal on Scientific Computing32no. 5 (2010), 2737–2764. [CGV26]G. Ciaramella,M. J. Gander...

work page 2021

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CohenandR

[CD15]A. CohenandR. DeVore, Approximation of high-dimensional parametric PDEs,Acta Numer- ica24(2015), 1–159. [CJS22]S. Correnty,E. Jarlebring, andD. B. Szyld, Preconditioned Chebyshev BiCG for param- eterized linear systems,arXiv preprint arXiv:2212.04295(2022). [DL93]R. DeVoreandG. Lorentz,Constructive Approximation, Springer Berlin Heidelberg,

work page arXiv 2015

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Drineas,M

[DMM08]P. Drineas,M. W. Mahoney, andS. Muthukrishnan, Relative-Error CUR Matrix De- compositions,SIAM Journal on Matrix Analysis and Applications30no. 2 (2008), 844–881. https://doi.org/10.1137/07070471X. [EAS98]A. Edelman,T. A. Arias, andS. T. Smith, The Geometry of Algorithms with Orthogonality Constraints,SIAM J. Matrix Anal. Appl.20no. 2 (1998), 303–3...

work page doi:10.1137/07070471x 2008

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HeinkenschlossandD

[HK25]M. HeinkenschlossandD. P. Kouri, Optimization problems governed by systems of PDEs with uncertainties,Acta Numerica34(2025), 491–577. [HS52]M. R. HestenesandE. Stiefel, Methods of conjugate gradients for solving linear systems,J. Res. Natl. Bur. Stand.49no. 6 (1952), 409–436. [HRS16]J. S. Hesthaven,G. Rozza, andB. Stamm,Certified reduced basis metho...

work page 2025

[7] [7]

Hiptmair, Operator preconditioning,Computers & Mathematics with Applications52no

[Hip06]R. Hiptmair, Operator preconditioning,Computers & Mathematics with Applications52no. 5 (2006), 699–706. [KF24]A. KayaandM. A. Freitag, Low-rank solutions to the stochastic Helmholtz equation,Journal of Computational and Applied Mathematics448(2024), 115925. [KS11]B. N. KhoromskijandC. Schwab, Tensor-structured Galerkin approximation of parametric a...

work page 2006

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[MD09]M. W. MahoneyandP. Drineas, CUR matrix decompositions for improved data analysis, Proceedings of the National Academy of Sciences106no. 3 (2009), 697–702.https://doi.org/ 10.1073/pnas.0803205106. [MW11]K.-A. MardalandR. Winther, Preconditioningdiscretizationsofsystemsofpartialdifferential equations,Numerical Linear Algebra with Applications18no. 1 (...

work page doi:10.1073/pnas.0803205106 2009

[9] [9]

PieracciniandT

[PV25]S. PieracciniandT. Vanzan, An adaptive importance sampling algorithm for risk-averse op- timization,Journal of Computational Physics(2025), 114548. [PR69]E. PolakandG. Ribière, Notesurlaconvergencedeméthodesdedirectionsconjuguées,ESAIM Math. Model. Numer. Anal.3no. R1 (1969), 35–43. [PE09]C. E. PowellandH. C. Elman, Block-diagonal preconditioning fo...

work page doi:10.1093/imanum/drn014 2025

[10] [10]

RingandB

34 [RW12]W. RingandB. Wirth, Optimization methods on Riemannian manifolds and their application to shape space,SIAM J. Optim.22no. 2 (2012), 596–627.https://doi.org/10.1137/11082885X. [RAH13]C. R. J. Roger A. Horn,Matrix Analysis, 2nd ed., Cambridge University Press,

work page doi:10.1137/11082885x 2012

[11] [11]

RosseelandS

[RV10]E. RosseelandS. Vandewalle, Iterative solvers for the stochastic finite element method,SIAM Journal on Scientific Computing32no. 1 (2010), 372–397. [Sat16]H. Sato, A Dai–Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions,Comput. Optim. Appl.64no. 1 (2016), 101–118.https://doi.org/10.1007/ s10589-015-9801-1. [Sat21]H. Sato,...

work page 2010

[12] [12]

Sato, Riemannian Optimization and Its Applications, SpringerBriefs in Electrical and Com- puter Engineering, Springer, 2021, https://doi.org/10.1007/978-3-030-62391-3

https://doi.org/10.1007/978-3-030-62391-3. [Sat22]H. Sato, Riemannian Conjugate Gradient Methods: General Framework and Specific Algorithms with Convergence Analyses,SIAM J. Optim.32no. 4 (2022), 2690–2717.https://doi.org/10. 1137/21M1464178. [SI15]H. SatoandT. Iwai, A new, globally convergent riemannian conjugate gradient method,Opti- mization64no. 4 (20...

work page doi:10.1007/978-3-030-62391-3 2022

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[Smi94]S. T. Smith, Optimization Techniques on Riemannian Manifolds, inFields Institute Communi- cations,3, American Mathematical Society, Providence, RI, 1994, pp. 113–146. [Ste83]T. Steihaug, The conjugate gradient method and trust regions in large scale optimization,SIAM J. Numer. Anal.20no. 3 (1983), 626–637.https://doi.org/10.1137/0720042. [SV24]M. S...

work page doi:10.1137/0720042 1994