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REVIEW 2 major objections 3 minor 1 cited by

On Relatively Smooth Optimization over Riemannian Manifolds

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper shows that Bregman-type methods can minimize relatively smooth functions on Riemannian submanifolds with O(1/ε²) iteration complexity.

desk verdict The abstract describes a plausible Riemannian extension of relatively smooth Bregman methods with O(1/eps^2) guarantees, but the supplied full text is a different paper, so the proofs cannot be checked; the retraction/projection transfer condition is the key thing to look for. read the letter →

arxiv 2508.03048 v2 pith:J34YUZJ6 submitted 2025-08-05 math.OC

Chang He , Jiaxiang Li , Bo Jiang , Shiqian Ma , Shuzhong Zhang This is my paper
classification math.OC MSC 90C3090C4865K1090C15
keywords relativelysmoothoptimizationRiemannianembeddedsubmanifoldsBregmangradientmethodretraction-basedupdateprojection-basediterationcomplexitystochasticsamplenonlineareigenvalueproblem
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 tackles a gap in relatively smooth optimization: the existing Bregman-gradient machinery was built for Euclidean space, yet many low-rank and eigenvalue problems live on curved manifolds. It introduces two first-order methods for objectives that are relatively smooth in the ambient Euclidean space while constrained to an embedded Riemannian submanifold. The retraction-based method handles nonsmooth objectives by solving a convex tangent-space subproblem, and the projection-based method handles smooth objectives through an unconstrained ambient subproblem. The paper claims an iteration complexity of $\mathcal{O}(1/\epsilon^2)$ for reaching an $\epsilon$-approximate Riemannian stationary point and $\mathcal{O}(1/\epsilon^4)$ sample complexity for stochastic variants on compact manifolds. If correct, these would be the first nonasymptotic complexity guarantees of this kind.

What carries the argument

The central object is a Riemannian Bregman gradient step, in which the Euclidean Bregman distance $D_h(x,y)=h(x)-h(y)-\langle\nabla h(y),x-y\rangle$ is placed inside a tangent-space or ambient-space subproblem. The retraction-based method sets $x_{k+1}=R_{x_k}(v_k)$ with $v_k$ solving a convex problem over $T_{x_k}\mathcal{M}$, while the projection-based method solves an unconstrained ambient problem and then projects. These steps are designed so that the relative-smoothness descent inequality $f(y)\le f(x)+\langle\nabla f(x),y-x\rangle+L\,D_h(y,x)$ transfers from Euclidean space to manifold iterates, producing a sufficient decrease per iteration that yields the claimed $\mathcal{O}(1/\epsilon^2)$ complexity.

What would settle it

Run the retraction-based update on the unit sphere for $f(x)=\langle x,Ax\rangle^2$ with $h(x)=\frac{1}{4}\|x\|^4+\frac{1}{2}\|x\|^2$ and the step size prescribed by the proof; if any iterate pair violates the Bregman sufficient-decrease inequality, the $\mathcal{O}(1/\epsilon^2)$ bound collapses, whereas checking the inequality across many random starts and step sizes would support it.

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Extended reading notes

Core claim

The paper's central claim is that relative smoothness, an assumption already known to power Bregman-gradient methods in Euclidean space, can be carried over to embedded Riemannian submanifolds without sacrificing the convergence rate. Concretely, it proposes a retraction-based Riemannian Bregman gradient method whose update solves a convex subproblem over the tangent space, and a projection-based variant that solves an unconstrained subproblem in the ambient Euclidean space. When the reference function is $h(x)=\frac{1}{4}\|x\|^4+\frac{1}{2}\|x\|^2$, the retraction subproblem admits a closed-form solution. For both methods the paper proves an iteration complexity of $\mathcal{O}(1/\epsilon^2)$ to reach an $\epsilon$-approximate Riemannian stationary point, and for compact manifolds it develops stochastic variants with sample complexity $\mathcal{O}(1/\epsilon^4)$. Numerical experiments on the nonlinear eigenvalue problem and low-rank quadratic sensing are presented as evidence of practical advantage.

Load-bearing premise

The load-bearing premise is that the Euclidean relative-smoothness inequality continues to hold with the same constants along the retraction- or projection-generated manifold steps; the supplied full text is a different manuscript, so the proof of this transfer cannot be checked here.

Editorial extensions

If this is right

  • Relatively smooth nonconvex objectives on embedded submanifolds become solvable by a first-order method with $\mathcal{O}(1/\epsilon^2)$ iterations, matching the Euclidean rate.
  • The retraction-based method applies to nonsmooth objectives, and its tangent-space subproblem has a closed-form solution for the quartic reference function $h(x)=\frac{1}{4}\|x\|^4+\frac{1}{2}\|x\|^2$.
  • The projection-based method requires only unconstrained ambient subproblems, so it avoids maintaining feasibility inside each subproblem.
  • On compact manifolds, stochastic variants achieve $\mathcal{O}(1/\epsilon^4)$ sample complexity, making the approach usable on large or streaming data.
  • Numerical tests on nonlinear eigenvalue and low-rank quadratic sensing problems show practical gains over existing approaches.

Reading between the lines

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

  • If the transfer assumption holds for one retraction, the same convergence template should extend to other retractions with a comparable Lipschitz-type property; the unit sphere with the exponential-map retraction is a direct test case.
  • The closed-form quartic update is likely to make the method inexpensive on Stiefel and sphere constraints, so the reported eigenvalue and low-rank results may generalize to a broader class of manifold-constrained problems.
  • Should the rates hold, relative smoothness rather than Lipschitz smoothness would become the natural structural assumption for nonsmooth manifold optimization, reshaping how such problems are formulated.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper under review (arXiv:2508.03048) studies optimization over Riemannian embedded submanifolds when the objective is relatively smooth in the ambient Euclidean space. It proposes two first-order methods: a retraction-based Riemannian Bregman gradient method that handles nonsmooth objectives via a tangent-space convex subproblem, and a projection-based method for smooth objectives solving an unconstrained Euclidean subproblem. The abstract claims an iteration complexity of O(1/epsilon^2) for finding an epsilon-approximate Riemannian stationary point, and, for compact manifolds, a sample complexity of O(1/epsilon^4) for stochastic variants. Numerical experiments on the nonlinear eigenvalue problem and low-rank quadratic sensing are reported as supporting evidence. As supplied, the full text is not the manuscript but an unrelated hyperspectral imaging paper, so the present report can assess only the abstract and the claimed results.

Significance. If the claimed rates are correct, this would be the first nonasymptotic complexity guarantee for relatively smooth optimization over embedded Riemannian submanifolds. The closed-form solution of the tangent-space subproblem for the quartic reference function h(x) = 1/4||x||^4 + 1/2||x||^2 is a practically useful contribution, and the stochastic sample-complexity result for compact manifolds would extend relatively smooth Bregman methods to a stochastic regime. These are potentially valuable advances for applications such as eigenvalue problems and low-rank recovery. However, because the manuscript text is unavailable, the internal correctness of the derivations cannot presently be confirmed.

major comments (2)
  1. [Full text (attached)] The attached full text is not the manuscript under review; it is a hyperspectral image super-resolution paper (arXiv:2508.03049). The derivations, precise assumptions, and numerical details behind the claimed O(1/epsilon^2) iteration complexity and O(1/epsilon^4) sample complexity are therefore not accessible to the referee. This is an evidentiary gap rather than a demonstrated error, but it makes verification of the central claims impossible from the supplied record.
  2. [Abstract] The abstract states only that the objective is relatively smooth in the ambient Euclidean space, while the algorithms use retraction- and projection-based Riemannian updates. The load-bearing step is a transfer of the Euclidean Bregman descent inequality to the manifold: for a retraction update x_{k+1} = R_x(u), one must control f(R_x(u)) - f(x) - <grad f(x), u> by D_h(x+u,x) with a constant independent of the curvature and of the second fundamental form. The abstract does not state any such transfer condition, and it is not automatic from ambient relative smoothness. Since all four complexity claims rest on this inequality, the missing condition is a central unverified premise.
minor comments (3)
  1. [Abstract] The abstract says the retraction-based method 'can handle nonsmooth optimization' but does not specify the stationarity measure used for nonsmooth objectives; it would be useful to state whether the epsilon-approximate stationarity notion reduces to the Riemannian gradient norm in the smooth case and to something like a Clarke stationary condition in the nonsmooth case.
  2. [Abstract] The stochastic variants are described only as being 'on compact manifolds' with sample complexity O(1/epsilon^4); the oracle model (e.g., unbiased stochastic gradient with bounded variance) is not stated, and this is needed to assess whether the sample complexity is in the standard sense.
  3. [Abstract] The quartic reference function h(x) = 1/4||x||^4 + 1/2||x||^2 is claimed to yield a closed-form subproblem solution, but the abstract does not indicate whether this closed form holds for the tangent-space constrained subproblem, the ambient projection subproblem, or both; specifying this would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the abstract presents theorem-style complexity claims and a design choice, with no fitted-input or self-citation reduction evident.

full rationale

No circular step can be identified from the supplied material. The abstract describes two Riemannian first-order methods and states complexity bounds of O(1/epsilon^2) and sample complexity O(1/epsilon^4) as theorem-style results derived from relative smoothness assumptions. The quartic reference function h(x) = (1/4)||x||^4 + (1/2)||x||^2 is explicitly introduced as a design choice that yields a closed-form tangent-space subproblem; it is not calibrated to match the claimed convergence rates. The numerical experiments on the nonlinear eigenvalue problem and low-rank quadratic sensing provide external, non-fitting evidence. The attached full text is an unrelated hyperspectral imaging paper, so the actual proof steps cannot be checked; that is an evidentiary gap, not circularity. No load-bearing self-citation, renamed empirical pattern, or fitted parameter presented as a prediction is present in the abstract.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The main assumptions are standard relative-smoothness and manifold retraction properties. The only ad hoc element is the quartic reference function, selected for tractability rather than data fit. No new entities are introduced.

free parameters (1)
  • quartic reference function coefficients (1/4, 1/2) = h(x) = 1/4 ||x||^4 + 1/2 ||x||^2
    These coefficients are chosen by the authors so the tangent-space subproblem admits a closed-form solution; they are not fitted to data but are an ad hoc design choice.
assumptions (5)
  • domain assumption The objective f is relatively smooth with respect to h in the ambient Euclidean space.
    Abstract states the objective is relatively smooth in ambient Euclidean space; this inequality is the engine of the Bregman gradient analysis.
  • domain assumption M is an embedded submanifold and the retraction maps satisfy standard first- and second-order agreement properties.
    Needed to transfer Euclidean Bregman steps to manifold updates; standard in Riemannian first-order methods.
  • domain assumption For stochastic variants, the manifold is compact and stochastic gradients are unbiased with bounded variance.
    Abstract explicitly says the manifold is compact for stochastic variants; bounded-variance is standard but unstated.
  • domain assumption The tangent-space subproblem in the retraction-based method is convex.
    Abstract states the update direction is generated by solving a convex optimization subproblem; convexity is used for the closed-form and complexity analysis.
  • ad hoc to paper The reference function h(x) = 1/4 ||x||^4 + 1/2 ||x||^2 is a legitimate Bregman reference.
    Special form chosen to obtain a closed-form subproblem; not forced by the problem data.

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Cite this review

Pith. "Pith review of On Relatively Smooth Optimization over Riemannian Manifolds." pith.science (2026). https://pith.science/paper/J34YUZJ6

@misc{pith2026250803048,
  author       = {Pith},
  title        = {Pith review of: On Relatively Smooth Optimization over Riemannian Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J34YUZJ6}},
  note         = {Machine review of arXiv:2508.03048}
}
abstract

We study optimization over Riemannian embedded submanifolds, where the objective function is relatively smooth in the ambient Euclidean space. Such problems have broad applications but are still largely unexplored. We introduce two Riemannian first-order methods, namely the retraction-based and projection-based Riemannian Bregman gradient methods, by incorporating the Bregman distance into the update steps. The retraction-based method can handle nonsmooth optimization; at each iteration, the update direction is generated by solving a convex optimization subproblem constrained to the tangent space. We show that when the reference function is of the quartic form $h(x) = \frac{1}{4}\|x\|^4 + \frac{1}{2}\|x\|^2$, the constraint subproblem admits a closed-form solution. The projection-based approach can be applied to smooth Riemannian optimization, which solves an unconstrained subproblem in the ambient Euclidean space. Both methods are shown to achieve an iteration complexity of $\mathcal{O}(1/\epsilon^2)$ for finding an $\epsilon$-approximate Riemannian stationary point. When the manifold is compact, we further develop stochastic variants and establish a sample complexity of $\mathcal{O}(1/\epsilon^4)$. Numerical experiments on the nonlinear eigenvalue problem and low-rank quadratic sensing problem demonstrate the advantages of the proposed methods.

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