arxiv: 2605.10573 · v1 · submitted 2026-05-11 · 🧮 math.OC · cs.MS· math.DG

Recognition: 2 theorem links

· Lean Theorem

A Riemannian quasi-Newton algorithm for optimization with Euclidean bounds

Mateusz Baran, Patryk Przybysz, Ronny Bergmann

Pith reviewed 2026-05-12 04:43 UTC · model grok-4.3

classification 🧮 math.OC cs.MSmath.DG

keywords Riemannian optimizationlimited-memory BFGSquasi-Newton methodsEuclidean boundsmanifold optimizationblind source separationprincipal component analysiscovariance estimation

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The pith

A Riemannian limited-memory BFGS method handles Euclidean bounds on manifolds by adapting the generalized Cauchy point strategy to tangent spaces.

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

The paper develops a Riemannian limited-memory BFGS algorithm for optimization problems that combine manifold variables with Euclidean bounds. It integrates tangent-space quasi-Newton updates with a Riemannian version of the generalized Cauchy point from classical L-BFGS-B, allowing the method to respect both the manifold geometry and the bounds. This mixed setting appears in covariance estimation, blind source separation, and signal processing applications. The algorithm is implemented in a standard library and tested on benchmarks where it shows only minor slowdowns versus pure Euclidean L-BFGS-B while delivering large speedups over interior-point methods on the mixed problems.

Core claim

The central claim is that a limited-memory quasi-Newton update performed in the tangent space, paired with a Riemannian adaptation of the generalized Cauchy point strategy, yields a practical solver for manifold optimization problems that also carry Euclidean bounds. On pure Euclidean test problems the method performs close to classical L-BFGS-B; on two mixed manifold-bound problems (amplitude-limited blind source separation with Gaussianity penalization and bounded-variance maximum likelihood common principal components analysis) it outperforms existing solvers by several orders of magnitude.

What carries the argument

The central mechanism is the combination of tangent-space limited-memory BFGS updates with a Riemannian adaptation of the generalized Cauchy point strategy from L-BFGS-B.

If this is right

The algorithm matches classical L-BFGS-B performance with only minor reduction when all variables are Euclidean.
It outperforms interior-point methods on standard benchmark problems.
It delivers several orders of magnitude faster solves on amplitude-limited blind source separation with Gaussianity penalization.
It provides superior performance for bounded-variance maximum likelihood common principal components analysis.

Where Pith is reading between the lines

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

The same adaptation pattern could be applied to other quasi-Newton or first-order methods on manifolds that also need inequality constraints.
Applications listed in the abstract, such as neuroimaging and EEG classification, become more practical once mixed manifold-bound problems can be solved at this speed.
The generic framework may transfer to other manifold optimization libraries without major redesign.

Load-bearing premise

The Riemannian adaptation of the generalized Cauchy point strategy from L-BFGS-B can be combined with tangent-space limited-memory BFGS updates without introducing instability or losing the quasi-Newton convergence properties on the manifold.

What would settle it

Numerical runs on the amplitude-limited blind source separation or bounded-variance common principal components analysis problems in which the proposed algorithm fails to produce the reported orders-of-magnitude speedups over interior-point methods.

Figures

Figures reproduced from arXiv: 2605.10573 by Mateusz Baran, Patryk Przybysz, Ronny Bergmann.

**Figure 1.** Figure 1: Performance plot where W is the unmixing matrix, S are the reconstructed independent components, and λ is a nongaussianity penalty parameter. Gradients were computed using automatic differentiation with Zygote.jl [39]. Four solvers were considered: a simple gradient descent procedure with projection, Riemannian L-BFGS-B, Riemannian augmented Lagrangian method (ALM) [19] and Riemannian exact penalty method … view at source ↗

**Figure 2.** Figure 2: Objective history of different solvers for the amplitude-limited blind source separation problem (top plot: [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original abstract

We propose a Riemannian limited-memory BFGS method for optimization problems with Euclidean bounds. The method combines a limited-memory quasi-Newton update in the tangent space with a Riemannian adaptation of the generalized Cauchy point strategy from classical L-BFGS-B, enabling efficient handling of Euclidean bounds while exploiting the geometric structure of the optimization domain. This setting is important in several applications, including covariance matrix estimation with bounded variance, neuroimaging, EEG signal classification, and other signal processing or computer-vision tasks that couple manifold variables with constrained Euclidean parameters. We provide a generic algorithmic framework and an implementation of the algorithm in the Manopt.jl library. Numerical experiments on benchmark problems indicate only minor reduction in performance on Euclidean problems compared to the classical L-BFGS-B method, while outperforming interior-point methods. Furthermore, the algorithm was tested on two mixed manifold and bounded Euclidean problems: amplitude-limited blind source separation with Gaussianity penalization and bounded-variance maximum likelihood common principal components analysis. The proposed method outperforms existing methods by several orders of magnitude.

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

A practical Riemannian L-BFGS with bound handling that fills a gap for mixed problems, though the theoretical backing could be tighter.

read the letter

The paper's main contribution is a limited-memory BFGS method on Riemannian manifolds that also respects Euclidean bounds on some variables. They do this by performing the quasi-Newton updates in the tangent space and using a Riemannian version of the generalized Cauchy point from L-BFGS-B to find a feasible step when bounds are present. This combination looks new as a packaged framework, and the authors supply a Manopt.jl implementation, which is a real plus for anyone who wants to try it out. The numerical results on pure Euclidean problems show only minor slowdown compared to the standard method, and it beats interior-point approaches. On the two application cases – amplitude-limited blind source separation with a Gaussianity penalty and bounded-variance maximum likelihood common principal components analysis – the method is reported to be much faster than alternatives, by several orders of magnitude in some cases. That kind of performance on real mixed problems is what makes the work useful. One area that needs scrutiny is the behavior when bounds become active. The stress-test note raises a fair point: without a clear argument that the tangent-space update preserves positive definiteness or guarantees sufficient descent after retraction and bound projection, it's possible the good results are tied to the specific test instances rather than holding in general. The abstract and framework description don't include a convergence proof, so the paper leans heavily on the experiments. If the full manuscript has more analysis or at least extensive diagnostics showing the properties hold in practice, that would strengthen it. This kind of paper is for researchers and practitioners in optimization who encounter manifold variables alongside simple bound constraints, such as in signal processing or neuroimaging pipelines. A reader who needs a solver for those settings would find the implementation and benchmark results valuable. It is worth sending for peer review because the algorithmic construction is concrete, the code is available, and the applications demonstrate relevance, even if the theoretical side could use more development.

Referee Report

2 major / 2 minor

Summary. The paper proposes a Riemannian limited-memory BFGS algorithm for optimization problems involving both manifold constraints and Euclidean bounds. It adapts the generalized Cauchy point strategy from classical L-BFGS-B to the Riemannian setting while performing limited-memory quasi-Newton updates entirely in the tangent space, provides a generic framework and Manopt.jl implementation, and reports that the method shows only minor performance loss versus classical L-BFGS-B on pure Euclidean problems while outperforming interior-point methods and delivering orders-of-magnitude speedups on two mixed problems (amplitude-limited blind source separation with Gaussianity penalization and bounded-variance maximum-likelihood common principal components analysis).

Significance. If the central algorithmic construction is shown to preserve descent and positive-definiteness properties, the work would provide a practical tool for mixed manifold-Euclidean problems arising in signal processing, neuroimaging, and covariance estimation. The open-source implementation in Manopt.jl is a clear strength that supports reproducibility.

major comments (2)

[Algorithmic framework (§3–4)] The algorithmic description (presumably §3–4) does not supply a proof or even a sufficient-descent argument that the tangent-space L-BFGS update remains positive definite after the Riemannian retraction and the bound-projection step when Euclidean bounds become active. This property is load-bearing for the claim that the method reliably outperforms existing solvers by orders of magnitude on the two mixed-manifold test problems.
[Numerical experiments] The numerical-experiments section (and the abstract) asserts “outperformance by several orders of magnitude” on the amplitude-limited BSS and bounded-variance ML-CPCA problems, yet supplies neither quantitative tables, problem dimensions, iteration counts, CPU times, nor error-bar statistics. Without these data the central empirical claim cannot be verified.

minor comments (2)

[Abstract] The abstract would be strengthened by a single sentence referencing the specific tables or figures that quantify the claimed speed-ups.
[Notation and preliminaries] Notation for the Riemannian retraction and the tangent-space limited-memory update should be introduced once and used consistently; several symbols appear to be defined only locally.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below and indicate the corresponding revisions to the manuscript.

read point-by-point responses

Referee: [Algorithmic framework (§3–4)] The algorithmic description (presumably §3–4) does not supply a proof or even a sufficient-descent argument that the tangent-space L-BFGS update remains positive definite after the Riemannian retraction and the bound-projection step when Euclidean bounds become active. This property is load-bearing for the claim that the method reliably outperforms existing solvers by orders of magnitude on the two mixed-manifold test problems.

Authors: We agree that an explicit argument establishing positive definiteness and sufficient descent after the retraction and bound-projection steps would strengthen the presentation. In the revised manuscript we will add a concise proof sketch (or sufficient-descent lemma) in §3 showing that the limited-memory BFGS update performed entirely in the tangent space preserves the standard curvature-pair conditions of Euclidean L-BFGS, that the generalized Cauchy-point computation and subsequent projection onto the tangent-space bound set remain first-order consistent with the Euclidean L-BFGS-B construction, and that the retraction (being a local diffeomorphism) does not alter the descent property at the current iterate. This addition directly supports the reliability claims for the mixed-manifold test problems. revision: yes
Referee: [Numerical experiments] The numerical-experiments section (and the abstract) asserts “outperformance by several orders of magnitude” on the amplitude-limited BSS and bounded-variance ML-CPCA problems, yet supplies neither quantitative tables, problem dimensions, iteration counts, CPU times, nor error-bar statistics. Without these data the central empirical claim cannot be verified.

Authors: We acknowledge that the current numerical section lacks the quantitative detail needed for independent verification. The revised manuscript will include new tables (and an expanded appendix) reporting, for each benchmark and application: problem dimensions, iteration counts, CPU times, function/gradient evaluations, and error bars (standard deviation over repeated runs with different random seeds). These data will be presented both for the pure-Euclidean comparisons and for the two mixed manifold-plus-bound problems, allowing direct assessment of the claimed speed-ups. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic construction with independent empirical validation

full rationale

The paper proposes a hybrid Riemannian L-BFGS algorithm with a generalized Cauchy point adaptation for Euclidean bounds. Its central contribution is an implementable framework (with Manopt.jl code) whose correctness and performance are assessed via direct numerical comparison on benchmark and application problems. No equations are presented that define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness or convergence claim is justified solely by self-citation. The outperformance statements rest on explicit test instances rather than on any reduction to the algorithm's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper contributes an algorithmic framework that rests on standard Riemannian manifold assumptions and classical L-BFGS-B components; no new free parameters, invented entities, or non-standard axioms are introduced in the abstract.

axioms (1)

domain assumption The feasible set is the product of a Riemannian manifold and a Euclidean box constraint set.
This is the problem class for which the algorithm and its Cauchy-point adaptation are defined.

pith-pipeline@v0.9.0 · 5482 in / 1179 out tokens · 42349 ms · 2026-05-12T04:43:50.344676+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
The method combines a limited-memory quasi-Newton update in the tangent space with a Riemannian adaptation of the generalized Cauchy point strategy from classical L-BFGS-B
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
We construct limited memory approximations to both the Hessian Hk and its inverse Bk simultaneously

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