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arxiv: 2412.19427 · v2 · pith:TU2K7Y5Pnew · submitted 2024-12-27 · 🧮 math.OC

Riemannian conditional gradient methods for composite optimization problems

Kangming Chen , Ellen H. Fukuda This is my paper

Pith reviewed 2026-05-23 07:38 UTC · model grok-4.3

classification 🧮 math.OC
keywords Riemannian optimizationconditional gradient methodcomposite optimizationconvergence analysismanifold optimizationstep-size strategiessphere manifoldStiefel manifold
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The pith

Riemannian conditional gradient methods achieve O(1/k) convergence for composite optimization on manifolds.

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

The paper develops conditional gradient methods adapted to Riemannian manifolds for minimizing functions that split into a smooth term and a retraction-convex term. It proves that with adaptive or diminishing step sizes the methods converge at a rate of O(1/k), and with Armijo line search they reach epsilon-optimality in O(1/epsilon squared) iterations. This matters because many constrained optimization problems, such as those on spheres or orthogonal matrices, are naturally posed on manifolds, and conditional gradient steps can be cheaper than projected gradient steps when the linear optimization subproblem is tractable. The analysis relies on the retraction to transfer convexity properties from the tangent space back to the manifold. Numerical tests on the sphere and Stiefel manifold confirm the practical behavior of the algorithms.

Core claim

The authors propose Riemannian conditional gradient methods for composite functions on manifolds, where the objective is the sum of a smooth function and a retraction-based convex function. They show convergence rates of O(1/k) using adaptive and diminishing step sizes, and an iteration complexity of O(1/ε²) for the Armijo step-size rule to reach ε-optimality.

What carries the argument

The retraction-based convex function, which allows the conditional gradient linear optimization step to be performed in the tangent space while preserving the convexity needed for the convergence guarantees.

If this is right

  • The methods apply directly to problems on the sphere and Stiefel manifold with the stated rates.
  • Adaptive step sizes yield the same O(1/k) rate as diminishing ones without needing to tune parameters in advance.
  • Armijo search guarantees finite termination to any accuracy in O(1/ε²) steps.
  • The framework extends the Euclidean conditional gradient analysis to the Riemannian setting via retractions.

Where Pith is reading between the lines

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

  • These methods could be useful when the feasible set is a manifold where linear optimization over the convex hull in the tangent space is efficient.
  • The O(1/ε²) complexity for Armijo might be improvable with better step-size rules or acceleration techniques.
  • Numerical validation on two manifolds suggests the approach works in practice, but broader testing on other manifolds would be valuable.

Load-bearing premise

The nonsmooth part of the objective must be convex when composed with the retraction, so that the linear subproblem in the tangent space yields a valid descent direction with the standard conditional gradient analysis.

What would settle it

A counterexample on the sphere where the method with adaptive steps fails to achieve O(1/k) decrease in the objective gap, or an experiment showing that the Armijo variant requires more than order 1/ε² iterations on a Stiefel manifold problem.

read the original abstract

In this paper, we propose Riemannian conditional gradient methods for minimizing composite functions, i.e., those that can be expressed as the sum of a smooth function and a retraction-based convex function. We analyze the convergence of the proposed algorithms, utilizing three types of step-size strategies: adaptive, diminishing, and those based on the Armijo condition. We establish the convergence rate of \(\mathcal{O}(1/k)\) for the adaptive and diminishing step sizes, where \(k\) denotes the number of iterations. Additionally, we derive an iteration complexity of \(\mathcal{O}(1/\epsilon^2)\) for the Armijo step-size strategy to achieve \(\epsilon\)-optimality, where \(\epsilon\) is the optimality tolerance. Finally, the effectiveness of our algorithms is validated through some numerical experiments performed on the sphere and Stiefel manifolds.

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.

Referee Report

0 major / 3 minor

Summary. The paper proposes Riemannian conditional gradient (Frank-Wolfe) methods for composite optimization on manifolds, where the objective is the sum of a smooth function and a retraction-based convex function. Convergence rates of O(1/k) are claimed for adaptive and diminishing step-size rules, together with an O(1/ε²) iteration complexity for an Armijo line-search variant; numerical results on the sphere and Stiefel manifold are provided to illustrate performance.

Significance. If the analyses hold, the manuscript supplies a direct Riemannian extension of the conditional-gradient framework to composite problems, recovering the classical sublinear rates under the three standard step-size regimes. The retraction-based convexity assumption enables a well-defined linear minimization oracle while preserving the necessary properties for the proofs; this is a natural and useful generalization of existing Riemannian Frank-Wolfe work. The numerical validation on standard manifolds adds modest practical support, but the contribution remains incremental rather than transformative.

minor comments (3)
  1. [§2] §2: the precise statement of the retraction-based convexity assumption (Definition 2.3 or equivalent) should be cross-referenced explicitly in the convergence theorems so that readers can immediately verify which properties are used in each proof.
  2. [Theorem on Armijo complexity] The iteration-complexity claim for the Armijo strategy is stated as O(1/ε²) in the abstract; the corresponding theorem should include the explicit dependence on the Lipschitz constant and the retraction curvature bound to make the constant factors transparent.
  3. [Numerical experiments section] Figure captions for the sphere and Stiefel experiments should state the manifold dimension, the specific retraction used, and the initialization strategy so that the plots are reproducible from the text alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the review and the recommendation of minor revision. The report accurately summarizes the paper's contributions on Riemannian conditional gradient methods for composite problems under the retraction-based convexity assumption, along with the stated convergence rates and numerical illustrations.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proposes Riemannian conditional gradient methods for composite objectives (smooth + retraction-based convex) and derives convergence rates O(1/k) for adaptive/diminishing steps and O(1/ε²) for Armijo steps. These rates follow from standard analysis of the linear minimization oracle enabled by the retraction convexity assumption, matching Euclidean Frank-Wolfe results without reduction to fitted parameters, self-definitions, or load-bearing self-citations. No equations equate a claimed result to its own inputs by construction, and the numerical validation on sphere/Stiefel manifolds is independent of the analysis. The central claim remains externally falsifiable via the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, additional axioms, or invented entities are identifiable. The work relies on standard assumptions from Riemannian optimization theory such as manifold smoothness and retraction properties, which are not detailed here.

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Reference graph

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