{"id":"a0c1ee57-d1dc-44ba-b4e4-a6991edb5c33","arxiv_id":"2412.05689","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The landing algorithm converges linearly near local minima for smooth non-convex optimization on the Stiefel manifold under a local Riemannian PŁ condition.","lead":"This paper proves that the landing algorithm, an infeasible retraction-free method for optimizing functions with orthogonality constraints, converges linearly near a local minimum when the function satisfies a local Riemannian Polyak-Lojasiewicz condition. This gives a theoretical reason to trust a cheap method that avoids expensive projections onto the Stiefel manifold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.6's linear-rate proof hinges on Lemma 3.4, an imported off-manifold PŁ-type domination result that is not proved here and is not shown to follow from the stated manifold-only assumptions; the theorem is conditional on this missing bridge.","rationale":"The paper's proof of Theorem 3.6 is otherwise coherent: given Proposition 3.3 and Lemma 3.4, the descent calculation yields the claimed rate, and the distance-to-S maintenance step follows from Lipschitz continuity of Lambda and the chosen step-size bound. The load-bearing weakness is precisely the unproved bridge, Lemma 3.4, which transfers a manifold-only PŁ condition to the off-manifold safety region. The reader identified this same fragility, and I agree with that assessment. The additional observation about alpha_safe requiring a uniform gradient bound G is a symptom of the same under-specification rather than an independent failure. The appropriate disposition is therefore unchanged from the reader's verdict: conditional, with the condition being a self-contained proof or precise restatement of Lemma 3.4 including all smallness and boundedness assumptions needed for its validity. No evidence of a false theorem or circular fitting was found.","tokens_in":10315,"tokens_out":7651,"duration_ms":74178,"concrete_test":"Take [28]'s proof of Lemma 3.4 and re-derive inequality (9) from Assumptions 3.1 and 3.2 alone, recording every additional condition used; if any condition beyond the stated hypotheses appears, such as bounded gradient on St(d,r)_epsilon ∩ D(S,delta) or a delta/epsilon smallness link, then Theorem 3.6's hypothesis block needs that condition. As an independent probe, on the quadratic PCA objective f(x) = -tr(x^T A A^T x D) with d=2, r=1, which is known to satisfy Assumption 3.2, sweep x in St(d,r)_epsilon ∩ D(S,delta) for several values of delta/epsilon and lambda and check whether L(x) <= (1/mu')||Lambda(x)||^2 holds with the paper's mu' expression; a violation would be a concrete counterexample to the lemma as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.6 is an induction whose only quantitative step beyond Proposition 3.3 is Lemma 3.4: for x in St(d,r)_epsilon ∩ D(S,delta), L(x) <= (1/mu')||Lambda(x)||^2. Lemma 3.4 is imported from the authors' prior preprint [28], and its statement here is the sole channel through which Assumption 3.2, a Riemannian PŁ inequality posed only on the Stiefel manifold, controls the merit function at infeasible points. The manuscript neither proves this lemma nor gives the exact hypotheses under which it was derived in [28], and no relation is stated among delta, epsilon, lambda, gamma, mu, hat L, and L' that would make the transfer valid. If the proof of Lemma 3.4 in [28] uses a global gradient bound, a smallness condition such as delta much smaller than lambda(1-epsilon)^2 / hat L, or an off-manifold variant of PŁ, then Theorem 3.6 as stated is missing an assumption. A visible symptom of the same under-specification is that alpha_safe in Proposition 2.2 depends on a uniform bound G on ||grad f||, but G never appears in the hypotheses of Theorem 3.6. None of this shows the theorem is false; it shows the central claim is not self-contained and cannot be verified from the present text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the landing algorithm, a retraction-free infeasible method for optimization over the Stiefel manifold, and proves a local linear convergence rate for smooth non-convex objectives satisfying a local Riemannian Polyak-Łojasiewicz condition. The main result, Theorem 3.6, states that if the merit function L(x) = f(x) + h(x) + γp(x) is initially sufficiently small in a neighborhood of the local-minimum set S, then with a suitable step size the merit function converges linearly: L(x_k) ≤ (1 - αρμ'/2)^k L(x_0). The proof combines a descent estimate for the merit function, a pseudo-gradient domination lemma, and a quadratic-growth property of L. Numerical experiments on PCA and on CNN training with orthogonality-constrained layers illustrate the computational efficiency of the landing algorithm relative to retraction-based and other retraction-free methods.","tokens_in":10570,"tokens_out":3939,"duration_ms":37499,"significance":"If Theorem 3.6 is correct, it gives a clean explanation of the fast empirical convergence observed for the landing algorithm and extends the global finite-time results of Ablin et al. [2] to a local linear rate under a condition weaker than geodesic strong convexity. The paper is honest about the local nature of the result and does not fit parameters; the rate is a theorem and the PŁ constant is an assumption. The PCA experiment code is publicly available, and the numerical comparisons cover both iteration count and wall-clock time. However, the central proof depends on Lemma 3.4, which is imported from the authors' prior work [28] and is not proved here, and the safety-step argument uses a uniform gradient bound G that is not stated among the theorem's hypotheses. These issues make the main result conditional on an unverified bridge between manifold-only PŁ and off-manifold merit domination.","major_comments":[{"comment":"Lemma 3.4 is the only channel through which Assumption 3.2, a Riemannian PŁ inequality stated only on St(d,r) ∩ D(S,2δ), controls the merit function at infeasible points x ∈ St(d,r)_ε ∩ D(S,δ). The lemma is cited from [28] but neither proved nor accompanied by the exact hypotheses under which it was derived there. In particular, the manuscript does not state the smallness relations among δ, ε, λ, γ, μ, L, L', and Ĝ that would make the transfer from the manifold to the off-manifold region valid, and it does not show that Assumptions 3.1 and 3.2 alone imply inequality (9). Since the proof of Theorem 3.6 invokes Lemma 3.4 in the crucial step L(x_{k+1}) ≤ (1 - μ'(αρ - α²L'/2))L(x_k), the main theorem is not verifiable from the present text. This is a load-bearing gap: if Lemma 3.4 requires an additional condition or a different proof, Theorem 3.6 as stated is missing an assumption.","section":"Section III.C, Lemma 3.4 and Theorem 3.6"},{"comment":"The theorem chooses α ≤ min{ρ/L', α_safe}, but α_safe in Proposition 2.2 depends on a uniform bound G on ||grad f(x)||, and G is not defined in the hypotheses of Theorem 3.6. Since L-smoothness on all of R^{d×r} does not imply a global bound on the gradient, the step-size condition is not well-posed as written. A fix would be to define G as a supremum over the compact set St(d,r)_ε ∩ D(S,δ) and state it as part of the theorem, but this is absent. The same gap affects the induction step, which relies on Proposition 2.2 to keep all iterates in the safety region.","section":"Proposition 2.2 and Theorem 3.6"},{"comment":"The induction step uses the inequality ||Λ(x_k)||_F ≤ L' dist(S, x_k) to control dist(S, x_{k+1}). This requires that Λ be Lipschitz on a neighborhood containing S and that Λ vanish on S. The manuscript defines L' as a maximum of Ĝ, L_Λ, and L_L, but it does not explicitly state the domain on which the Lipschitz constant L_Λ applies or verify that Λ(y)=0 for y∈S. The argument is likely repairable, but as written the constants and domains are not fully specified.","section":"Section III.C, proof of Theorem 3.6"}],"minor_comments":[{"comment":"The phrase \"principle component analysis\" should read \"principal component analysis.\"","section":"Abstract"},{"comment":"The line \"⟨gradf(x), ∇p(x)⟩ = ⟨skew(∇f(x)x^T), x(x^T x - I)x^T⟩ = 0\" appears to have a typo in the right-hand side; the second argument should likely be x(x^T x - I), not x(x^T x - I)x^T. Please check the expression.","section":"Section II.C"},{"comment":"The informal statement \"if we set λ = O(L') ... μ' ≈ μ\" is not justified and could be misleading; please either remove it or give a precise quantitative statement.","section":"Section III.B"},{"comment":"The proof of Lemma 3.5 invokes Proposition 2.2 of [24] to pass from a local Euclidean PŁ condition on L to quadratic growth. Since [24] may have additional regularity assumptions, it would be helpful to state explicitly that L is C² and to verify the hypotheses of that proposition.","section":"Section III.C"}],"recommendation":"major_revision","confidential_remarks":"The heavy reuse of Lemma 3.4 from the authors' own prior work [28] is the main risk to the novelty and self-containedness of the paper. If the lemma can be proved within the paper or stated with a complete proof, the main theorem would be a solid contribution. The current manuscript is not ready for publication without this missing bridge."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"New here: a local linear rate for the landing algorithm under local Riemannian PŁ. That is a genuine addition to the prior global finite-time result, and it explains fast empirical behavior. The induction in Theorem 3.6 is clean, the constants are consistent, and the CPU-time experiments show real benefit. The structure is clear.\n\nSoft spots: Theorem 3.6 relies on Lemma 3.4, imported from the authors' prior work [28], which dominates the merit function by the landing field in the off-manifold safety region. This manuscript does not prove it or restate its exact hypotheses. Assumption 3.2 is Riemannian PŁ only on the Stiefel manifold, but Lemma 3.4 must control off-manifold points. The transfer is not shown, and no relation among δ, ε, λ, γ, μ, and L' is stated. If Lemma 3.4 fails for some allowed f, Theorem 3.6 collapses. No contradiction is visible, but the central claim is not self-contained.\n\nRelated symptom: α_safe depends on a uniform bound G on ||grad f||, but G never appears in Theorem 3.6's hypotheses. That is an under-specified assumption, not fatal, but needs stating.\n\nNumerics: the PCA sanity check is fine but single-run, without seeds or error bars. The CNN experiments use objectives that do not satisfy smoothness or PŁ; the paper caveats this, so it is minor.\n\nBottom line: this is a worthwhile rate result for infeasible retraction-free Riemannian optimization. The assembled theorem is new, and the proof architecture is sound if the imported lemma is correct. But the paper cannot be verified from the present text. I would send it to a serious referee with an explicit request to prove or precisely restate Lemma 3.4 and to state the G bound. Engaging is justified; citing immediately is risky until the bridge is closed.","headline":"The paper proves a new local linear rate for the landing algorithm under local Riemannian PŁ, but the main theorem depends on an unproved imported lemma that transfers PŁ from the manifold to off-manifold points, so the result is conditional on that missing bridge.","tokens_in":713,"tokens_out":1115,"would_cite":false,"duration_ms":26480,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C30","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the landing algorithm, an infeasible retraction-free update for orthogonality-constrained problems, converges linearly near a local minimum under a local Riemannian PŁ condition.","keywords":["orthogonality constraints","Stiefel manifold","landing algorithm","retraction-free optimization","infeasible methods","Riemannian Polyak-Łojasiewicz condition","linear convergence","nonconvex optimization"],"falsifier":"Choose an $L$-smooth objective that satisfies the local Riemannian PŁ condition on the Stiefel manifold and simulate the landing algorithm from an initial point with $L(x_0)\\le\\mu'\\rho^2\\delta^2/16$; if any iterate inside the safety region violates $L(x_{k+1})\\le(1-\\alpha\\rho\\mu'/2)L(x_k)$, or if an off-manifold point in $\\operatorname{St}(d,r)_\\epsilon\\cap D(S,\\delta)$ satisfies $L(x)>(1/\\mu')\\|\\Lambda(x)\\|_F^2$, then Lemma 3.4 and Theorem 3.6 fail. A single such example would settle the question.","tokens_in":10069,"feed_emoji":"📉","tokens_out":12756,"duration_ms":108682,"temperature":0.7,"pith_summary":"Optimization with orthogonality constraints is often solved by retraction-based Riemannian methods, but each retraction costs a matrix exponential, inversion, or square root. This paper studies the landing algorithm, an infeasible retraction-free alternative whose iterates are allowed to drift slightly off the Stiefel manifold (the set of matrices with $x^\\top x=I_r$) and then settle back onto it. The paper proves a local linear convergence rate: under Euclidean smoothness and a local Riemannian Polyak–Łojasiewicz condition near the set of local minima, the merit function contracts by a fixed factor every iteration once the iterates enter a specified basin. The rate is $L(x_k)\\le(1-\\alpha\\rho\\mu'/2)^kL(x_0)$, and the distance to the optimum set and the landing-field norm shrink at the same exponential pace. This matters because the result explains the fast empirical convergence reported for the landing algorithm and turns a cheap per-iteration update into a provably fast solver for PCA, low-rank approximation, and orthogonality-constrained neural networks.","feed_headline":"Landing algorithm provably converges linearly near Stiefel minima","feed_subtitle":"Near a Stiefel minimum, the retraction-free landing method shrinks the merit function at an exponential rate.","key_machinery":"The central object is the landing field $\\Lambda(x)=\\operatorname{grad}f(x)+\\lambda\\nabla p(x)$, where $p(x)=\\frac14\\|x^\\top x-I_r\\|_F^2$ penalizes constraint violation; on the manifold it reduces to the Riemannian gradient, and its two terms are mutually orthogonal. The proof rides on the merit function $L(x)=f(x)+h(x)+\\gamma p(x)$ with $h(x)=-\\frac12\\langle\\operatorname{sym}(x^\\top\\nabla f(x)),\\,x^\\top x-I_r\\rangle$, which is designed so that the landing field is an ascent direction for $L$. Three inequalities carry the argument: Proposition 3.3 gives $\\langle\\Lambda,\\nabla L\\rangle\\ge\\rho\\|\\Lambda\\|_F^2$, Lemma 3.4 (pseudo-gradient domination) gives $L(x)\\le(1/\\mu')\\|\\Lambda(x)\\|_F^2$ in the safety region, and Lemma 3.5 converts that into the quadratic-growth bound $L(x)\\ge(\\mu'\\rho^2/4)\\operatorname{dist}(S,x)^2$. Together with a Lipschitz bound on $\\Lambda$ and the safety step-size condition $\\alpha\\le\\min\\{\\rho/L',\\alpha_{\\mathrm{safe}}\\}$, these make one landing step a contraction on $L$ that also stays inside the basin.","core_discovery":"The paper claims that the landing algorithm, which replaces the retraction step with the Euclidean update $x_{k+1}=x_k-\\alpha(\\operatorname{grad}f(x_k)+\\lambda x_k(x_k^\\top x_k-I_r))$, converges locally at a linear rate whenever the objective is twice continuously differentiable and $L$-smooth in the whole space and satisfies the Riemannian PŁ inequality $f(x)-f_S^*\\le(1/2\\mu)\\|\\operatorname{grad}f(x)\\|_F^2$ in a $2\\delta$-neighborhood of the local-minimum set on the Stiefel manifold. The proof tracks the merit function $L(x)=f(x)+h(x)+\\gamma p(x)$ rather than $f$ itself, and shows that with a safe step size and an initial merit value below $\\mu'\\rho^2\\delta^2/16$, the merit function decays as $(1-\\alpha\\rho\\mu'/2)^k$, while the iterates never leave the safety region $\\operatorname{St}(d,r)_\\epsilon\\cap D(S,\\delta)$. The same mechanism yields linear rates for the distance to the minimum set and for the landing-field norm.","pith_inferences":["Editorial extension: the same proof template should transfer to retraction-free updates on other embedded constraint sets, such as Grassmann or hyperbolic manifolds, whenever a matching off-manifold pseudo-gradient domination inequality can be established.","Editorial extension: for a quadratic objective the constants $\\mu'$, $\\rho$, and $L'$ can be computed in closed form, so the theorem predicts an explicit contraction factor; measuring the empirical ratio $L(x_{k+1})/L(x_k)$ in the PCA experiment would test that prediction directly.","Editorial extension: the theorem is stated for the merit function $L$, and the paper only partially spells out the transfer to the original objective $f$; wherever the correction term $h$ is small near the optimum set, the same exponential statement applies to $f$ itself.","Editorial extension: in stochastic/minibatch settings, a local PŁ condition together with bounded gradient noise would plausibly give linear convergence down to a noise floor, which would match the multi-stage learning-rate-decay behavior seen in the CNN experiments, though that extension requires separate analysis."],"forward_implications":["If Theorem 3.6 is correct, every iterate started in the basin $D(S,\\delta)\\cap\\operatorname{St}(d,r)_\\epsilon$ with $L(x_0)\\le\\mu'\\rho^2\\delta^2/16$ satisfies $L(x_k)\\le(1-\\alpha\\rho\\mu'/2)^kL(x_0)$, an exponential contraction in the merit function.","Since each landing step is a Euclidean update, the per-iteration cost is about one gradient evaluation plus the low-rank penalty term, avoiding the $O(dr^2)$ or heavier retraction cost, so the runtime advantage over retraction-based methods grows with the number of columns $r$.","The distance to the local-minimum set and the landing-field norm inherit linear rates from the merit function via quadratic growth and the Lipschitz property of $\\Lambda$.","The assumptions cover standard quadratic-type Stiefel problems such as principal component analysis, so the linear rate applies to a common machine-learning task, not only to constructed examples.","The safety step size keeps all iterates within the prescribed constraint-violation tolerance, so the infeasible path stays controlled while the objective declines exponentially."],"supporting_citations":[{"why":"introduces the landing algorithm and its retraction-free update rule, the object whose local rate this paper proves.","marker":"[1]"},{"why":"supplies the safety region, the merit function, the safety step-size bound, and the empirical observation of fast convergence that motivates the result.","marker":"[2]"},{"why":"provides the pseudo-gradient domination lemma that transfers the PŁ condition to the off-manifold safety region.","marker":"[28]"},{"why":"establishes that quadratic-type Stiefel problems satisfy the local Riemannian PŁ condition, grounding Assumption 3.2 in standard examples.","marker":"[22]"},{"why":"gives the equivalence result used to derive the quadratic growth of the merit function from the local PŁ inequality.","marker":"[24]"},{"why":"supplies the canonical metric and the closed form of the Riemannian gradient on the Stiefel manifold used to define the landing field.","marker":"[13]"},{"why":"supports the premise that the iterates can first reach a neighborhood of a stationary point, making the local-rate analysis a meaningful refinement.","marker":"[27]"}],"fun_headline_variants":["Retraction-free landing algorithm achieves local linear rate","Infeasible method proves local linear convergence on Stiefel","Landing algorithm: local linear rate without retractions","Orthogonal constraints? Landing converges locally linear","No retraction needed: landing method has local linear rate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The linear-rate proof requires that for every point in the off-manifold safety region near a local minimum the merit function is bounded by a constant times the squared landing-field norm, a transfer of the on-manifold PŁ condition to nearby infeasible points that the paper imports from its companion work.","fun_headline_variants_meta":{"raw":{"variants":["Retraction-free landing algorithm achieves local linear rate","Infeasible method proves local linear convergence on Stiefel","Landing algorithm: local linear rate without retractions","Orthogonal constraints? Landing converges locally linear","No retraction needed: landing method has local linear rate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000536,"raw_usage":{"total_tokens":2555,"prompt_tokens":908,"completion_tokens":1647,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":1570}},"tokens_in":524,"tokens_out":1647,"duration_ms":12139,"temperature":1.0,"reasoning_tokens":1570,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:27:55.922416+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose an $L$-smooth objective that satisfies the local Riemannian PŁ condition on the Stiefel manifold and simulate the landing algorithm from an initial point with $L(x_0)\\le\\mu'\\rho^2\\delta^2/16$; if any iterate inside the safety region violates $L(x_{k+1})\\le(1-\\alpha\\rho\\mu'/2)L(x_k)$, or if an off-manifold point in $\\operatorname{St}(d,r)_\\epsilon\\cap D(S,\\delta)$ satisfies $L(x)>(1/\\mu')\\|\\Lambda(x)\\|_F^2$, then Lemma 3.4 and Theorem 3.6 fail. A single such example would settle the question.","supporting_citations":[{"cited_title":"Fast and accurate optimization on the orthogonal manifold without retraction","cited_arxiv_id":null,"evidence_quote":"introduces the landing algorithm and its retraction-free update rule, the object whose local rate this paper proves."},{"cited_title":"Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints","cited_arxiv_id":"2303.16510","evidence_quote":"supplies the safety region, the merit function, the safety step-size bound, and the empirical observation of fast convergence that motivates the result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that quadratic-type Stiefel problems satisfy the local Riemannian PŁ condition, grounding Assumption 3.2 in standard examples."},{"cited_title":"Fast convergence to non-isolated minima: four equivalent conditions for c 2 functions","cited_arxiv_id":null,"evidence_quote":"gives the equivalence result used to derive the quadratic growth of the merit function from the local PŁ inequality."},{"cited_title":"The geometry of algorithms with orthogonality constraints","cited_arxiv_id":null,"evidence_quote":"supplies the canonical metric and the closed form of the Riemannian gradient on the Stiefel manifold used to define the landing field."},{"cited_title":"Convergence results for projected line-search methods on varieties of low-rank matrices via łojasiewicz inequality","cited_arxiv_id":null,"evidence_quote":"supports the premise that the iterates can first reach a neighborhood of a stationary point, making the local-rate analysis a meaningful refinement."}],"review_version":1}