{"id":"3d353a09-2acb-492e-848c-7c6d608fa02a","arxiv_id":"2509.08561","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An inexact Riemannian proximal DC framework solves nonsmooth DC problems on manifolds with O(ε^-3) overall complexity and, on the sphere, DC relaxations exactly match ℓ0-regularized and ℓ0-constrained models for large penalties.","lead":"New optimization framework for nonsmooth difference-of-convex problems on curved spaces proves O(ε^-2) outer iterations and exact equivalence of certain DC sparse penalties to ℓ0 models on the sphere. The methods are tested on sparse PCA, where they beat an ADMM baseline in solution quality and often in speed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Complexity proofs for iRPDC-NFG/AR reverse the tolerance bound; the O(ε^{-3}) overall complexity is not established by the displayed algebra.","rationale":"After checking the main convergence proof, Theorem 4.4 is algebraically consistent as written. Lemma 3.6 does contain a false intermediate assertion (“|¯x_i|² ≥ 1/j*”), but the needed tail bound ∑_{i=k+1}^{j*}|¯x_i|² ≤ 1−k/j* follows from a different argument about the sum of the k largest squares, so the equivalence theorem is repairable rather than collapsed. The genuinely load-bearing gap is in the complexity section: the tolerance bound is used in the wrong direction, and the AR inner-loop count does not match the quoted O(ε_j^{-1}) rate. A reader following the displayed sums cannot derive the headline O(ǫ^{-3}) total as written. I do not share the reader's emphasis on Assumption 2.1 as the weakest point: for standard retractions on compact embedded manifolds, the quadratic term in (2.3) dominates for large tangent vectors, so global retraction constants are plausible; and the finite-equivalence thresholds' dependence on L0_f is a practical estimation issue, not a proof gap. The reader's own list did flag an inconsistent iRPDC-AR complexity statement, and I agree with that part; my concern sharpens it by pinning the inconsistency to specific equations, namely the reversed ε_j bound in Theorem 5.5 and the T_i sum in Algorithm 4. The verdict remains CONDITIONAL: the framework is likely salvageable, but the complexity claims in Theorems 5.5 and 5.7 need a corrected derivation before the O(ǫ^{-3}) headline can be accepted.","tokens_in":28943,"tokens_out":41161,"duration_ms":614602,"concrete_test":"Sum the T_i displayed in Algorithm 4 with δ_{j,0}=ε_j/(8L0_h): compute T_j^{AR}=Σ_{i=0}^{⌈log4(2L0_h ℓ_j^{-1} ε_j^{-1})⌉} ⌈16(ℓ_j/(4^i δ_{j,0})+1)^{1/2}⌉ and check whether T_j^{AR}=O(ε_j^{-1/2}) or O(ε_j^{-1}). Then substitute ε_j^{-1}=O(ǫ^{-2}) from (4.12) and (5.24), together with the outer bound J=O(ǫ^{-2}), and recompute the total in Theorem 5.7. Separately, re-derive Theorem 5.5's inner count after replacing the asserted “ε_j=O(ǫ^{-2})” by “ε_j^{-1}=O(ǫ^{-2})” and verify that Lemma 5.4's O(ε_j^{-1/2} log ε_j^{-1}) bound yields exactly O(ǫ^{-3} log ǫ^{-1}), not a different exponent. If the loop sum is O(ε_j^{-1/2}), the O(ǫ^{-3}) claim is recoverable with an explicit correction; if not, the practical complexity claims fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is in the inner-complexity accounting for the practical algorithms. From (4.12), the outer accuracy satisfies ǫ_j = O(ǫ), and from the first argument of the min in (5.24) the dual tolerance satisfies ε_j ≥ 2cβ1ℓjǫ_j^2/(4L0_h) = Ω(ǫ²) for fixed problem data and sufficiently small ǫ, so ε_j^{-1}=O(ǫ^{-2}). Theorem 5.5's proof instead asserts “ε_j=O(ǫ^{-2})” and then applies Lemma 5.4; that comparison is backwards. With the correct direction, Lemma 5.4 does give O(ǫ^{-1} log ǫ^{-1}) inner iterations per outer, so the O(ǫ^{-3} log ǫ^{-1}) total for iRPDC-NFG is salvageable, but only by rewriting the displayed argument. The iRPDC-AR statement is internally inconsistent in a sharper way. Algorithm 4 runs i=0,...,⌈log4(2L0_h ℓ_j^{-1} ε_j^{-1})⌉ stages, each with Ti=⌈16(ℓj/δj,i+1)^{1/2}⌉ and δj,i=4^iε_j/(8L0_h). Summing the displayed loop gives O(ε_j^{-1/2}) iterations per outer subproblem, not O(ε_j^{-1}) as claimed after (5.32). Since ε_j=Ω(ǫ²), the loop's own total is O(ǫ^{-1}) per outer and O(ǫ^{-3}) overall; if the quoted O(ε_j^{-1}) rate were used instead, the per-outer cost would be O(ǫ^{-2}) and the overall total O(ǫ^{-4}). The paper never reconciles these counts, so the headline O(ǫ^{-3}) complexity rests on an unverified rate. This is a correctness gap in Theorems 5.5 and 5.7, not merely a typo, because the stated proof either uses the wrong direction of the tolerance bound or invokes an O(ε_j^{-1}) rate that the displayed AR loop does not deliver.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29466,"tokens_out":10446,"duration_ms":86653,"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":[{"comment":"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.","section":"§3, Lemma 3.6"},{"comment":"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.","section":"§5.2, Theorem 5.5 proof"},{"comment":"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.","section":"§5.2, Theorem 5.7 and Algorithm 4"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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).","section":"§6.2.2"},{"comment":"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.","section":"Assumption 2.1 and §6.2.1"},{"comment":"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‖.","section":"§5.1, after (5.19)"},{"comment":"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.","section":"Theorem 3.7 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is worth pursuing. The main framework proof (Theorem 4.4) is coherent, and the equivalence results are interesting. The most serious issue is the iRPDC-AR complexity claim: the displayed loop actually supports O(ǫ⁻³) via an O(ε_j⁻¹/²) per-outer rate, but the text claims O(ε_j⁻¹), which with the correct Ω(ǫ²) lower bound on ε_j would give O(ǫ⁻⁴). This is a genuine proof gap, though likely repairable within the manuscript's scope. The Lemma 3.6 false assertion is also local and fixable by an averaging argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nOne 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.\n\nBottom 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.","headline":"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.","tokens_in":869,"tokens_out":1187,"would_cite":true,"duration_ms":28420,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q25","68R10","68U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["difference-of-convex optimization","Riemannian optimization","nonsmooth optimization","inexact proximal gradient","sparse principal component analysis","ℓ0 regularization","sphere manifold","iteration complexity"],"falsifier":"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 $C$.","tokens_in":28747,"feed_emoji":"🧮","tokens_out":10573,"duration_ms":83054,"temperature":0.7,"pith_summary":"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.","feed_headline":"New algorithm family solves sparse manifold problems in O(ε^-3)","feed_subtitle":"On the sphere, DC penalties match ℓ0 sparsity exactly, and the framework returns an ε-critical point in O(ε^-2) outer loops.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the capped-ℓ1 DC relaxation of the ℓ0 norm and the asymptotic equivalence technique that Theorem 3.1 adapts to manifolds.","marker":"[35]"},{"why":"Provides the critical-point lower-bound lemma (no small nonzero entries) that Lemma 3.2 carries over to the sphere.","marker":"[10]"},{"why":"Introduced the DC reformulation of cardinality constraints using $\\|x\\|_1 - |||x|||_k$, used in model (1.6).","marker":"[26]"},{"why":"Defines the ManPG proximal-gradient method over the Stiefel manifold whose subproblem (4.9) and retraction updates iRPDC extends.","marker":"[15]"},{"why":"Establishes global rates of retraction-based methods with the majorization inequality that underlies Lemma 4.1 and the uniform bound (4.4).","marker":"[12]"},{"why":"Provides an inexact Riemannian proximal gradient method whose stopping criteria the new inexactness condition (4.13) generalizes and strengthens.","marker":"[30]"},{"why":"Contributes the accumulative regularization scheme used by iRPDC-AR to reach $O(\\epsilon^{-3})$ overall complexity.","marker":"[34]"},{"why":"Gives the OADMM baseline for nonsmooth Riemannian DC problems, used both in the complexity comparison and the SPCA experiments.","marker":"[67]"}],"fun_headline_variants":["Manifold DC models match ℓ0 sparsity exactly","Inexact Riemannian DC: O(ε^-3) total work","First proof: nonsmooth DC = ℓ0 on sphere","Sparse manifolds solved by proximal DC framework","O(ε^-2) outer, O(ε^-3) overall for DC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Manifold DC models match ℓ0 sparsity exactly","Inexact Riemannian DC: O(ε^-3) total work","First proof: nonsmooth DC = ℓ0 on sphere","Sparse manifolds solved by proximal DC framework","O(ε^-2) outer, O(ε^-3) overall for DC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1560,"prompt_tokens":1098,"completion_tokens":462,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":714,"completion_tokens_details":{"reasoning_tokens":371}},"tokens_in":714,"tokens_out":462,"duration_ms":4771,"temperature":1.0,"reasoning_tokens":371,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:02:34.179959+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $C$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the capped-ℓ1 DC relaxation of the ℓ0 norm and the asymptotic equivalence technique that Theorem 3.1 adapts to manifolds."},{"cited_title":"Bian and X","cited_arxiv_id":null,"evidence_quote":"Provides the critical-point lower-bound lemma (no small nonzero entries) that Lemma 3.2 carries over to the sphere."},{"cited_title":"Gotoh, A","cited_arxiv_id":null,"evidence_quote":"Introduced the DC reformulation of cardinality constraints using $\\|x\\|_1 - |||x|||_k$, used in model (1.6)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the ManPG proximal-gradient method over the Stiefel manifold whose subproblem (4.9) and retraction updates iRPDC extends."},{"cited_title":"Huang and K","cited_arxiv_id":null,"evidence_quote":"Provides an inexact Riemannian proximal gradient method whose stopping criteria the new inexactness condition (4.13) generalizes and strengthens."},{"cited_title":"ADMM for Nonsmooth Composite Optimization under Orthogonality Constraints","cited_arxiv_id":"2405.15129","evidence_quote":"Gives the OADMM baseline for nonsmooth Riemannian DC problems, used both in the complexity comparison and the SPCA experiments."}],"review_version":2}