An Inexact Proximal Framework for Nonsmooth Riemannian Difference-of-Convex Optimization

Bo Jiang; Meng Xu; Xingju Cai; Ya-Feng Liu

REVIEW 3 major objections 5 minor 70 references

An Inexact Proximal Framework for Nonsmooth Riemannian Difference-of-Convex Optimization

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read On the sphere, nonsmooth DC penalties can equal ℓ0 sparsity exactly, and an inexact proximal framework reaches an ε-Riemannian critical point in O(ε^{-2}) outer iterations.

desk verdict Worth engaging: first sphere DC/sparse equivalence results and a sound iRPDC outer framework, but the inner-complexity proofs have a real reversal error and Lemma 3.6 has a false step. read the letter →

arxiv 2509.08561 v1 pith:L5CPSHVR submitted 2025-09-10 math.OC

Bo Jiang , Meng Xu , Xingju Cai , Ya-Feng Liu This is my paper · ORCID

classification math.OC MSC 68Q2568R1068U05

keywords difference-of-convexoptimizationRiemanniannonsmoothinexactproximalgradientsparseprincipalcomponentanalysisℓ0regularizationspheremanifolditerationcomplexity

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

The paper studies minimizing $f(x)+h(x)-g(x)$ over a Riemannian manifold, with $f$ smooth and $h,g$ convex but nonsmooth. It first shows that on the sphere, two DC penalties — the capped-$\ell_1$ penalty and the $\ell_1$-minus-largest-$k$ penalty — are exactly equivalent to $\ell_0$-regularized and $\ell_0$-constrained problems once the penalty parameter is large enough. It then proposes the iRPDC framework, whose new inexactness criterion sets the subproblem tolerance from previous iterates and proves termination at an $\epsilon$-Riemannian critical point in $O(\epsilon^{-2})$ outer iterations. Three instantiations give overall complexities $O(\epsilon^{-4})$, $O(\epsilon^{-3}\log\epsilon^{-1})$, and $O(\epsilon^{-3})$, the last matching the best-known bound while using fewer gradient and retraction evaluations. A careful reader would care because sparse PCA and similar sparse-manifold problems become solvable by continuous DC methods with guarantees tied to the $\ell_0$ models they approximate.

What carries the argument

The workhorse is a majorization of the pullback $F\circ\operatorname{Retr}_x$ on the tangent space (Lemma 4.1), bounding $F(\operatorname{Retr}_x(\eta))$ by $\langle p_x,\eta\rangle + \frac{L_x}{2}\|\eta\|^2 + h(x+\eta)$ with a curvature constant $L_x$ that is uniformly bounded through the retraction constants of Assumption 2.1. Around this sits the inexactness criterion (4.13): it compares the subproblem model $q_j(\eta_j)$ with $q_j(0)$, bounds the norm of the exact solution $\|\eta_j^*\|$ by computable quantities, and sets tolerances from previous iterates, which enables the backtracking line search and the $O(\epsilon^{-2})$ bound. For the sphere equivalence, the corresponding load-bearing object is the local error bound $\operatorname{dist}(x,\mathcal{S}_k) \le \sqrt{2}\,(1+\sqrt{k/n})^{-1/2}(\|x\|_1-|||x|||_k)$, which forces feasibility of DC critical points.

What would settle it

Take $\mathbb{S}^2$, set $f(x)=-a x_1$ with $a>0$, $\sigma=1$, and choose $\upsilon = a + \sqrt{3}$ so the theorem's condition holds; if any Riemannian critical point of (1.4) has a nonzero coordinate with $|x_i| < 1/\upsilon$, Lemma 3.2 and Theorem 3.3 are false. On the complexity side, running iRPDC-AR and recording prox evaluations as a function of $\epsilon$ would falsify the $O(\epsilon^{-3})$ claim if the empirical growth exceeds $C\epsilon^{-3}$ for every constant $C$.

Watch

Extended reading notes

Core claim

The central claim is twofold. For the sphere $\mathbb{S}$, the capped-$\ell_1$ DC model $f(x)+\sigma\Phi_\upsilon(x)$ is equivalent to the $\ell_0$-regularized model $f(x)+\sigma\|x\|_0$ whenever $\upsilon \ge L_f^0/\sigma + \sqrt{n}$, and the DC model $f(x)+\gamma(\|x\|_1-|||x|||_k)$ is equivalent to the $\ell_0$-constrained model with $\|x\|_0 \le k$ whenever $\gamma > nL_f^0/k$. The mechanism is a lower-bound lemma: any Riemannian critical point of the DC model has no nonzero entry smaller than $1/\upsilon$ (or, in the second case, is already $k$-sparse), so the DC objective and the $\ell_0$ objective coincide at stationary points. On the algorithmic side, the iRPDC framework inexactly minimizes a pullback majorization of the objective on each tangent space; under conditions (4.13) it terminates within $O(\epsilon^{-2})$ iterations, and its AR variant achieves $O(\epsilon^{-3})$ total work including proximal evaluations.

Load-bearing premise

The load-bearing premise is Assumption 2.1's uniform global retraction bound — one pair of constants $\iota_1,\iota_2$ controlling stretch and bending at every point of the manifold — because if it fails, the uniform majorization constant and the $O(\epsilon^{-2})$ termination guarantee both collapse.

Editorial extensions

If this is right

Capped-$\ell_1$ and $\ell_1$-minus-largest-$k$ DC penalties can substitute for $\ell_0$ terms on the sphere exactly once the penalty parameter clears the stated threshold, so sparse recovery models can be solved by continuous DC algorithms without relaxation error.
For any nonsmooth Riemannian DC problem satisfying the assumptions, iRPDC reaches an $\epsilon$-Riemannian critical point in $O(\epsilon^{-2})$ outer iterations, and the AR instantiation needs only $O(\epsilon^{-3})$ proximal evaluations overall, matching the best-known bound while cutting gradient and retraction costs to $O(\epsilon^{-2})$.
Setting $g=0$ turns the framework into inexact Riemannian proximal-gradient algorithms of ManPG type that carry overall complexity guarantees, a gap in existing inexact variants.
The previous-iterate-based subproblem tolerance supports a curvature-aware line search and, on sparse PCA benchmarks, yields higher variance with shorter runtime than the OADMM baseline while needing less parameter tuning.

Reading between the lines

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

The exact-equivalence thresholds require the Lipschitz constant $L_f^0$, which is often unknown; one could estimate it adaptively, and the numerical plateaus suggest equivalence may hold well beyond the proved threshold.
The sphere error-bound machinery and the tangent-space projection structure may extend to Stiefel or oblique manifolds with similar normal-space bases, giving finite-parameter equivalences there; the paper explicitly leaves general manifolds open.
The tolerance-from-previous-iterates scheme could transfer to stochastic or online Riemannian DC settings, where the current iterate's tolerance is unavailable, potentially yielding the first complexity guarantees for such stochastic problems.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth your time. It delivers the first finite-threshold equivalence results between DC relaxations and l0 models on the sphere (Theorems 3.3 and 3.7), and it builds an iRPDC framework with O(eps^-2) outer complexity plus a novel inexactness criterion that supports linesearch. The main descent/backtracking proof (Theorem 4.4) is internally coherent, the constants are derived rather than fitted, and the SPCA experiments are substantial and honest: the DC models beat the l1 baseline and OADMM in solution quality, with fair comparison on tuning.

The soft spots are real but uneven. Lemma 3.6, used to show k-sparsity at critical points, contains a false assertion: from the setup you cannot conclude |x_i|^2 >= 1/j* for i <= k. Counterexamples are easy. The lemma may be salvageable by a different argument, but as written the proof has a genuine gap.

The larger issue is the complexity accounting in Section 5.2. From (4.12) and (5.24), the dual tolerance satisfies eps_j = Omega(eps^2) for small eps, so eps_j^{-1} = O(eps^-2). Theorem 5.5's proof writes "eps_j = O(eps^-2)", which is backwards. Correcting the direction still gives O(eps^-3 log eps^-1) total for iRPDC-NFG, so that result is salvageable but needs rewriting. For iRPDC-AR the problem is sharper: the displayed loop with T_i = ceil(16(ell_j/delta_{j,i}+1)^{1/2}) and delta_{j,i}=4^i eps_j/(8L0_h) sums to O(eps_j^{-1/2}) iterations per outer subproblem, not O(eps_j^{-1}) as claimed. Since eps_j^{-1/2} = O(eps^-1), summing over O(eps^-2) outer iterations could still give O(eps^-3) overall, but the paper's own stated rate is different and the counts are never reconciled. Theorem 5.7's headline complexity is therefore not established by the displayed algebra.

One minor concern: Assumption 2.1 requires global retraction constants, which may fail for general submanifolds unless the retraction is very special. I would treat that as a limitation of scope, not a fatal flaw.

Bottom line: this is a serious paper with genuinely new ideas and mostly sound outer analysis. The inner-iteration proofs need a careful rewrite and Lemma 3.6 needs repair. A good referee should send it back for major revision, not desk-reject it. I would bring it to reading group and would cite the equivalence results once they are cleaned up.

Referee Report

3 major / 5 minor

Summary. The paper studies problem (1.1), the minimization over a Riemannian submanifold of f + h - g with nonsmooth convex h and convex DC component g. It makes two main claims: (i) on the sphere, the capped-ℓ1 model (1.4) is exactly equivalent to the ℓ0-regularized model (1.3), and the ℓ1−ℓ[k] model (1.6) is exactly equivalent to the ℓ0-constrained model (1.5), under explicit thresholds on the DC parameters; and (ii) Algorithm 1, an inexact Riemannian proximal DC framework, returns an ǫ-Riemannian critical point in O(ǫ⁻²) outer iterations, and its instantiations iRPDC-NFG, iRPDC-BB, and iRPDC-AR have overall complexities O(ǫ⁻³ log ǫ⁻¹), O(ǫ⁻⁴), and O(ǫ⁻³), respectively. Numerical experiments on sparse PCA compare the DC models with ℓ1-SPCA and the proposed algorithms with OADMM.

Significance. If the proofs are repaired as indicated below, this is a solid contribution to nonsmooth Riemannian DC optimization. The equivalence results on the sphere are, to my knowledge, new in the manifold setting and come with explicit, derived thresholds rather than fitted constants. The inexact framework in Theorem 4.4 is internally coherent, and the practical algorithms provide the first overall complexity guarantees for this problem class with a DC term. The numerical study is carefully executed, with warm starts, parameter sensitivity discussed, and comparisons against a relevant baseline. The main weaknesses are local but load-bearing: a false intermediate claim in Lemma 3.6 and a reversed tolerance bound plus an unreconciled inner-iteration count in the complexity proofs of Theorems 5.5 and 5.7.

major comments (3)

[§3, Lemma 3.6] The proof asserts that after defining j* as the largest index with |x(j)|>0, one has |x_i|² ≥ 1/j* for 1≤i≤k. This does not follow from the definition of j*. For example, on S⁵ with k=3 and squared entries (0.215,0.215,0.19,0.19,0.19), one has j*=5 but |x_3|²=0.19<1/5. The conclusion of the lemma is nevertheless recoverable: because the first k entries are the k largest among j* nonzero entries, their average squared norm is at least the overall average 1/j*, so ∑_{i=1}^k |x_i|² ≥ k/j*. This is the bound actually needed for the tail estimate, and the rest of the proof then goes through. The proof must be corrected, since Lemma 3.6 underpins Theorem 3.7.
[§5.2, Theorem 5.5 proof] The proof states that “ε_j=O(ǫ⁻²) (by (4.12) and (5.24))” and then applies Lemma 5.4. This is the wrong direction of the tolerance bound. From (4.12), ǫ_j=Θ(ǫ), and from the first argument of the minimum in (5.24), ε_j ≥ cβ1ℓ_jǫ_j²/(2L0_h), so ε_j=Ω(ǫ²) and hence ε_j⁻¹=O(ǫ⁻²). Lemma 5.4 gives O(ε_j⁻¹/² log ε_j⁻¹) inner iterations, which is O(ǫ⁻¹ log ǫ⁻¹) per outer subproblem. The claimed O(ǫ⁻³ log ǫ⁻¹) total is therefore salvageable, but only after rewriting the displayed inequality direction; the proof as written does not establish the bound it invokes.
[§5.2, Theorem 5.7 and Algorithm 4] The complexity accounting for iRPDC-AR is internally inconsistent. Algorithm 4 runs i=0,...,⌈log4(2L0_hℓ_j⁻¹ε_j⁻¹)⌉ stages, each with T_i=⌈16(ℓ_j/δ_{j,i}+1)^{1/2}⌉ and δ_{j,i}=4^i ε_j/(8L0_h). Summing the displayed loop gives O(ε_j⁻¹/²) inner iterations per outer subproblem, not O(ε_j⁻¹) as claimed after (5.32). Since ε_j=Ω(ǫ²), the loop's own accounting gives O(ǫ⁻¹) inner iterations per outer and O(ǫ⁻³) overall, consistent with the theorem's conclusion. However, the proof invokes an O(ε_j⁻¹) rate, which with ε_j=Ω(ǫ²) would give O(ǫ⁻⁴) total. The paper must reconcile these counts; as written, the headline O(ǫ⁻³) complexity of iRPDC-AR is not established by the displayed argument.

minor comments (5)

[Throughout] There are several typos: “formulaton” in the abstract and Introduction, “Riemannain” in Lemma 3.2, and “yileds” in Lemma 3.5. These should be corrected.
[§6.2.2] In the summary paragraph comparing the two DC models, the text refers to capped-ℓ1-SPCA as equation (6.3), but (6.3) is the ℓ1-SPCA baseline; the reference should be to (6.4).
[Assumption 2.1 and §6.2.1] The paper assumes global retraction constants ι1, ι2 satisfying (2.3) for all x and η. It should state which common manifolds and retractions satisfy this assumption and whether the constants are computable. In the numerical experiments, Lmin=10⁻¹⁰L and Lmax=10¹⁰L are used without specifying how L, ι1, ι2, or L0_f are determined in practice.
[§5.1, after (5.19)] The text refers to “Proposition 5.6,” but no Proposition 5.6 appears in the paper; the reference should be to the relevant result, presumably Proposition 5.1 or a related bound on ‖λ‖.
[Theorem 3.7 proof] The proof invokes Lemmas 5 and 9 of [42] without stating them. A brief indication of how the error bound (3.4) is combined with those lemmas would improve readability and self-containedness.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: equivalence thresholds and complexity bounds are derived from stated assumptions, not fitted or self-referential.

full rationale

The paper's two headline results are not circular. Theorem 3.3's threshold υ ≥ L0_f/σ + √n is derived from the critical-point equation (3.1) and the Lipschitz bound (3.3) in Lemma 3.2, so the equality Φυ(x̄)=‖x̄‖0 is a conclusion, not an input. Theorem 3.7 similarly proves the error bound (3.4) in Lemma 3.5 and the sparsity of critical points in Lemma 3.6; the invocation of [42, Lemmas 5 & 9] and [19, Proposition 9.1.2] imports published lemmas, and none of those lemmas is the target equivalence. Theorem 4.4's O(ε^{-2}) bound follows by summing the descent inequality (4.16) with the inexactness conditions (4.13), whose constants ρ, c, κ, β1, β2, Lmin, Lmax are user-specified with explicit constraints; the tolerance εj is defined from ε in (4.12), not fitted to data. The practical subproblem rates for NFG, BB, and AR are quoted from standard convex-optimization references [47, 7, 34], and the paper's own constants only determine the thresholds at which those rates apply. The numerical section uses the algorithms on SPCA benchmarks and does not repackage any numerical result as a theorem. I therefore find no step where a claimed prediction reduces by construction to an input. A separate correctness concern exists in the proof of Theorems 5.5 and 5.7: the text says 'εj=O(ε^{-2})' where the displayed definition (5.24) gives εj=Ω(ε²), and the AR loop's displayed iteration count sums to O(εj^{-1/2}) rather than the quoted O(εj^{-1}). These are inequality-direction or accounting gaps, not circularity, so they do not affect the circularity score.

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

No free parameters are fitted to data in the central claims; the equivalence thresholds and complexity bounds involve problem-dependent constants (L0_f, n, k) treated as inputs. Algorithmic constants (ρ, c, s, β1, ω0, a, Lmin, Lmax) are user choices, not fitted to data. The main caveat is the flawed proof step in Lemma 3.6, which is fixable.

assumptions (5)

domain assumption Assumption 1.1: f smooth, Lipschitz and descent inequality; h,g convex Lipschitz with computable prox/subgradient; level set compact.
Underlies the majorization Lemma 4.1 and the convergence analysis.
domain assumption Assumption 2.1: retraction satisfies global bounds (2.3).
Needed for the uniform bound L in (4.4) and the backtracking termination; not guaranteed for arbitrary retractions on general submanifolds.
standard math Riemannian subdifferential identity ∂_R h(x) = Proj_{T_xM}(∂h(x)) from [65, Theorem 5.1].
Used in optimality conditions and Lemma 2.2.
standard math For Theorem 3.7, cited results [42, Lemmas 5 & 9] and [19, Proposition 9.1.2] on exact penalty equivalence and error bounds.
Imported from prior literature; not proven in the text.
ad hoc to paper The false intermediate claim in Lemma 3.6 that |¯x_i|² ≥ 1/j* for 1≤i≤k; the needed bound ∑_{i=1}^k |x_i|² ≥ k/j* is true by an averaging argument.
The written proof uses an unjustified assertion; the lemma may still be true with a corrected argument.

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Pith. "Pith review of An Inexact Proximal Framework for Nonsmooth Riemannian Difference-of-Convex Optimization." pith.science (2026). https://pith.science/paper/L5CPSHVR

@misc{pith2026250908561,
  author       = {Pith},
  title        = {Pith review of: An Inexact Proximal Framework for Nonsmooth Riemannian Difference-of-Convex Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5CPSHVR}},
  note         = {Machine review of arXiv:2509.08561}
}

abstract

Nonsmooth Riemannian optimization has attracted increasing attention, especially in problems with sparse structures. While existing formulations typically involve convex nonsmooth terms, incorporating nonsmooth difference-of-convex (DC) penalties can enhance recovery accuracy. In this paper, we study a class of nonsmooth Riemannian optimization problems whose objective is the sum of a smooth function and a nonsmooth DC term. We establish, for the first time in the manifold setting, the equivalence between such DC formulations (with suitably chosen nonsmooth DC terms) and their $\ell_0$-regularized or $\ell_0$-constrained counterparts. To solve these problems, we propose an inexact Riemannian proximal DC (iRPDC) algorithmic framework, which returns an $\epsilon$-Riemannian critical point within $\mathcal{O}(\epsilon^{-2})$ outer iterations. Within this framework, we develop several practical algorithms based on different subproblem solvers. Among them, one achieves an overall iteration complexity of $\mathcal{O}(\epsilon^{-3})$, which matches the best-known bound in the literature. In contrast, existing algorithms either lack provable overall complexity or require $\mathcal{O}(\epsilon^{-3})$ iterations in both outer and overall complexity. A notable feature of the iRPDC algorithmic framework is a novel inexactness criterion that not only enables efficient subproblem solutions via first-order methods but also facilitates a linesearch procedure that adaptively captures the local curvature. Numerical results on sparse principal component analysis demonstrate the modeling flexibility of the DC formulaton and the competitive performance of the proposed algorithmic framework.

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