{"id":"a0b34bde-8b28-4500-9a22-bde6b48c7130","arxiv_id":"1908.00639","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalized Rayleigh quotient iteration for vector Lagrangian systems with equality constraints, with a cubic-convergence criterion involving a constrained Chebyshev term, unifies and improves several eigenpair and optimization algorithms.","lead":"This paper builds a single mathematical framework, based on vector Lagrangians, for Rayleigh quotient iteration, Riemannian Newton methods, and tensor eigenpair solvers, and gives conditions for quadratic and cubic convergence. It also introduces faster tensor eigenpair algorithms and a practical search that finds most or all complex tensor eigenpairs.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.2's proof applies the tangent-space lower bound (6.7) to the non-tangent vector x_{i+1}-v, leaving the normal component uncontrolled; both the quadratic and cubic rate proofs are incomplete as written.","rationale":"The reader's weakest-assumption analysis identifies the same load-bearing problem: inequality (6.7) is used on a vector that is not shown to be tangent. This is not a cosmetic issue. For an embedded manifold, the difference of two nearby points on M has a normal component of order squared distance, and the projected Hessian is only assumed invertible on the tangent space; indeed, on the sphere the normal direction is annihilated by the projection. The proof of Theorem 6.2 contains no argument controlling this normal component, so the displayed lower bound on ||x_{i+1}-v|| does not follow. The paper has real independent support: the Schur-form algebra is coherent, the recovery of classical RQI, O-NCM, and two-sided RQI is plausible, and the code is a concrete artifact. I do not see a separate internal contradiction beyond this proof gap. The quadratic portion is likely repairable by a tangent/normal decomposition, and the cubic portion additionally requires the retraction's second-order term to cancel, which is a standard but nontrivial step. Because the central theorem is unproven as written but not obviously false, the existing CONDITIONAL verdict is the right one.","tokens_in":25345,"tokens_out":25058,"duration_ms":260063,"concrete_test":"Re-derive the final estimate of Theorem 6.2 in local coordinates: decompose x_{i+1}-v = w_T + w_N at xi, with w_T in Null(JC(xi)), and bound ||w_N|| <= K||x_{i+1}-v||^2 using C^3 regularity of M. Then verify whether the RQI/Chebyshev step satisfies ||tau - exp^{-1}_{xi}(v)|| = O(||xi-v||^2); if so, a second-order retraction yields x_{i+1}-v = tangent vector + O(||xi-v||^3), closing both the quadratic and cubic arguments. If this derivation requires an additional hypothesis on the retraction, Theorem 6.2 must be revised accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in the proof of Theorem 6.2, immediately after (6.13). Inequality (6.7) is a lower bound on ||Pi_{Llambda}Lx(x,R(x))psi|| for psi in Null(JC(x)), i.e., for tangent directions at x. But the proof then applies this bound to the ambient difference x_{i+1}-v, which is not in Null(JC(xi)): both x_{i+1} and v lie on M, so their difference generally has a normal component of order O(||x_{i+1}-v||^2). In concrete cases this normal direction is exactly where the projected Hessian can be singular; for the unit sphere, Pi Lx x = 0 while x is normal. Thus the displayed estimate ||x_{i+1}-v|| <= (1/C)||Pi Lx(x)(x_{i+1}-v)|| is not justified by (6.7). The cubic part has a second, related issue: the proof writes r(xi,tau)=xi+tau+O(||tau||^3), but a second-order retraction only guarantees xi+tau+(1/2)II_{xi}(tau,tau)+O(||tau||^3). The O(||tau||^2) normal term must be shown to cancel with the normal component of xi-v, for example by showing tau approximates the inverse exponential map to second order. No such normal-component or local-coordinate estimate is supplied. The rates may still be true and repairable, but the theorem as written rests on an unproven identification of an ambient displacement with a tangent displacement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a general framework for Rayleigh quotient iteration (RQI) applied to systems L(x, λ) = 0, C(x) = 0, where L is called a vector Lagrangian and C is a constraint. The authors define a generalized Rayleigh quotient R(x) and a generalized projected Hessian Π Lλ Lx, and state a convergence theorem (Theorem 6.2) asserting quadratic convergence of a generalized RQI, cubic convergence of a Rayleigh-Chebyshev variant, and cubic convergence of the plain RQI under a zero condition on a tensor G. They also provide a Schur-form realization of the iteration, apply the framework to tensor eigenpairs, constrained optimization, nonlinear eigenvalue problems, two-sided RQI, and Grassmann/Stiefel manifolds, and report numerical experiments including an algorithm for enumerating complex tensor eigenpairs.","tokens_in":25661,"tokens_out":3136,"duration_ms":30823,"significance":"If the convergence theorem is correct, the paper would provide a unified theory covering classical RQI, Riemannian Newton, and several recent tensor eigenvalue algorithms, while also producing a new Rayleigh-Chebyshev iteration. The numerical results, especially the complex tensor eigenpair enumeration, are potentially valuable and the framework is elegant. The paper ships with open-source code and the derivations are self-contained. However, the central proof has gaps that affect the claimed rates, and the complex enumeration claim relies on an unproven heuristic about random starting points. The framework is significant but the theoretical core needs repair before the claims can be accepted.","major_comments":[{"comment":"The proof applies inequality (6.7) to the ambient difference x_{i+1} - v, but (6.7) is stated only for tangent vectors ψ in Null(JC(x)). Since both x_i and v lie on M, the vector x_{i+1} - v is generally not in Null(JC(x_i)): its normal component is O(||x_{i+1}-v||^2) but the estimate (1/C)||Π Lλ Lx(x)(x_{i+1}-v)|| bounds only the projected, tangent part. Without a separate bound for the normal component, the displayed conclusion ||x_{i+1}-v|| ≤ (1/C)||Π Lλ Lx(x)(x_{i+1}-v)|| does not follow. This gap affects the quadratic convergence claim in Theorem 6.2 and also feeds into the cubic-rate part.","section":"§6, proof of Theorem 6.2, after Eq. (6.13)"},{"comment":"The proof uses the expansion r(x_i, τ) = x_i + τ + O(||τ||^3) for a second-order retraction. A second-order retraction only guarantees r(x_i, τ) = x_i + τ + (1/2) II_{x_i}(τ,τ) + O(||τ||^3), where II is the second fundamental form and the quadratic term is normal. The substitution x_i - v = (x_{i+1}-v) - τ - O(||τ||^3) ignores this O(||τ||^2) normal term, which enters the Taylor expansion of L at second order. The paper provides no argument that this term cancels with the normal component of x_i - v or is absorbed into the tensor G. Without such an estimate, the cubic convergence proof is incomplete.","section":"§6, proof of Theorem 6.2, Chebyshev part"},{"comment":"Even after correcting the retraction expansion, the claim that the term A is dominated by Π Lλ Lx( x̂_i)(x_{i+1}-v) requires the operator Π Lλ Lx to be invertible uniformly on the relevant directions; the proof only states a crude bound ||A|| ≤ D||x_{i+1}-v|| ||x_i-v||. This bound also needs the normal component of x_{i+1}-v to be controlled, which is not established. The argument as written does not rigorously justify the transition from (6.14)-(6.15) to the final rate.","section":"§6, Eq. (6.15) and the A-term argument"}],"minor_comments":[{"comment":"The paper states that the algorithm enumerates all complex eigenpairs using random starting points and a deduplication table, but also acknowledges that 'a detailed global convergence analysis is still needed.' This heuristic is not a theorem, and the claim that all pairs are found rests on numerical experience; the presentation should make this limitation more prominent.","section":"§7.2, complex tensor eigenpairs"},{"comment":"There are numerous typographical and OCR artifacts, such as 'T aylor', 'iterati ons', 'coincident' for 'coincidence', and broken table formatting in §7.2. These should be cleaned up.","section":"Throughout"},{"comment":"The comparison table for SO-NCM vs O-NCM reports only average times and improvement percentages without standard deviations or hardware details, making the claimed 16% improvement hard to assess; a brief description of the experimental setup would help.","section":"§7.1, Table"},{"comment":"The reference to 'the notebook TwoLeftInverses.ipynb' is informal; since the numerical experiments are part of the evidence, the manuscript should cite the specific code version or include the relevant results in the paper.","section":"Remark 6.3"}],"recommendation":"major_revision","confidential_remarks":"The paper's central convergence proof is not rigorous as written, but the gaps appear repairable: the quadratic part needs a tangent/normal decomposition, and the cubic part needs a correct second-order retraction expansion with explicit control of the normal term. The framework and numerical results are interesting, and the paper is self-contained with no circularity. I recommend major revision rather than rejection, provided the authors can supply the missing estimates or adjust the claims accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The paper has a real idea: it puts RQI, Riemannian Newton on embedded manifolds, two-sided RQI, nonlinear eigenvalue solvers, and tensor eigenpair methods into one vector-Lagrangian framework, with a left inverse playing the role of the projection and a Rayleigh quotient giving λ. That is not in the existing literature, and the citations to Gabay, Absil and coauthors, and Jaffe-Weiss-Nadler are appropriate — the paper builds on them rather than ignoring them. Second, the main convergence theorem (Theorem 6.2) is not proved as written.\n\nWhat is genuinely good: the Schur-form derivation is clean; the special cases are checked; the SO-NCM reformulation of O-NCM is simple and plausibly faster; the complex tensor eigenpair search is a real contribution — computing thousands of complex eigenpairs in minutes is something I haven't seen elsewhere; and the code is linked. The paper also explicitly says global convergence for the complex search is not analyzed, which is honest.\n\nThe soft spot is load-bearing. Inequality (6.7) is a lower bound on the projected Hessian for tangent directions ψ ∈ Null(JC(x)). The proof applies it to x_{i+1} − v, which is not tangent: both points lie on the manifold, so their difference has a normal component of order O(||x_{i+1} − v||²). Without controlling that normal component, the estimate ||x_{i+1} − v|| ≤ (1/C)||Π Lx (x)(x_{i+1} − v)|| does not follow. The cubic part has a related problem: the proof writes r(x,τ) = x+τ+O(||τ||³), but a second-order retraction only guarantees x+τ+½ II(τ,τ)+O(||τ||³). That quadratic normal term has to be cancelled, and no such cancellation is shown. The rates may well be true, but the theorem needs repair. Minor: the timing table in 7.2 has no units; presumably seconds, but it should say.\n\nWho this is for: anyone working on tensor eigenpairs, Riemannian optimization, or nonlinear eigenvalue problems. The framework is useful, and the algorithmic sections stand on their own. My recommendation: send it to peer review. A competent referee can require a fixed proof of 6.2 and a cleaned-up numerical description. I wouldn't desk-reject it.","headline":"A genuinely unifying framework for Rayleigh quotient iteration with a real proof gap in the main theorem; worth refereeing, not desk-rejecting.","tokens_in":26190,"tokens_out":3595,"would_cite":true,"duration_ms":32871,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65K10","65F10","65F15","15A69"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single theorem now governs quadratic and cubic convergence for Rayleigh quotient iterations in eigenproblems, constrained optimization, and tensor eigenpairs.","keywords":["Rayleigh quotient iteration","vector Lagrangian","cubic convergence","Riemannian Newton","tensor eigenpairs","nonlinear eigenvalue problem","constrained optimization","second covariant derivative"],"falsifier":"Take a smooth Lagrangian and a curved constraint satisfying the stated hypotheses, for example $C(x)=x_1^2+x_2^2-1$ with $L(x,\\lambda)=F(x)-x\\lambda$ and a Rayleigh quotient $R$ with nonzero differential at the solution. Run iteration (6.8) in high-precision arithmetic from a nearby initial point and compute $Q_i=\\log\\|x_{i+1}-v\\|/\\log\\|x_i-v\\|$. If $Q_i$ does not approach $2$ for some such problem while all stated hypotheses hold, the missing normal-component control breaks the claimed quadratic rate; if $Q_i$ always approaches $2$, the control is automatic in these cases.","tokens_in":25107,"feed_emoji":"📐","tokens_out":16393,"duration_ms":136379,"temperature":0.7,"pith_summary":"The paper claims that a single iteration scheme, generalized Rayleigh quotient iteration, solves any system $L(x,\\lambda)=0$ with equality constraints $C(x)=0$, provided $\\lambda$ is chosen by a consistent Rayleigh quotient $R(x)$. The system covers eigenvector and invariant-subspace problems, constrained optimization, nonlinear eigenproblems, and tensor eigenpairs. Theorem 6.2 states that under $C^2$ smoothness of the Lagrangian and uniform invertibility of the projected Hessian on the tangent space, the generalized RQI converges quadratically; under $C^3$ smoothness and a second-order retraction, a Rayleigh–Chebyshev variant with a correction term converges cubically. The same theorem supplies a criterion, vanishing of a tensor $G$ built from second derivatives of $L$ and the differential of $R$, for plain RQI already to be cubic. This reduces many separate convergence proofs to one argument and yields new algorithms, including a unitary tensor RQI that computes most complex eigenpairs of moderate tensors quickly.","feed_headline":"One theorem unifies quadratic and cubic Rayleigh quotient convergence","feed_subtitle":"Eigenproblems, tensor eigenpairs, and constrained optimization reduce to one iteration scheme with a cubic variant.","key_machinery":"The central object is the vector Lagrangian $L(x,\\lambda)$ together with its constraint $C(x)$, and the generalized Rayleigh quotient $R(x)$ that selects $\\lambda$ at each iterate. The machinery that carries the argument is the projected generalized Hessian $\\Pi_{L_\\lambda} L_x$, obtained from a left inverse $L_\\lambda^{-}$ of $L_\\lambda$, restricted to the tangent space $JC(x)\\eta=0$; uniform invertibility of this map is inequality (6.7). For the cubic statements the load-bearing object is the tensor $G(x)[\\eta^{[2]}] = -\\tfrac{1}{2} L_{xx}(x)[\\eta^{[2]}] - L_{x\\lambda}(x)[\\eta, JR[\\eta]] - \\tfrac{1}{2} L_{\\lambda\\lambda}(x)[(JR[\\eta])^{[2]}]$, which acts as the second covariant derivative of the Lagrangian along the iteration direction. The Schur form — solving $\\zeta=-L_x^{-1}L_\\lambda$ and $\\xi=L_x^{-1}L$, then projecting with $(JC\\,\\zeta)^{-1}JC\\,\\xi$ — is the computational realization that links the abstract theorem to classical resolvent equations and to tensor and eigenvector algorithms.","core_discovery":"On the paper's own terms, the discovery is that one construction governs all of these iterations. Let $L(x,\\lambda)$ be a vector Lagrangian with constraint $C(x)=0$, and let $R$ be any $C^1$ Rayleigh quotient with $R(v)=\\mu$ at a solution. Choose a left inverse $L_\\lambda^{-}$ of $L_\\lambda$, set $\\Pi_{L_\\lambda}=I - L_\\lambda L_\\lambda^{-}$, and solve the projected Hessian equation $\\Pi_{L_\\lambda} L_x(x,R(x))\\,\\eta = -\\Pi_{L_\\lambda} L(x,R(x))$ on the tangent space $JC(x)\\eta=0$, updating by a retraction $x_{i+1}=r(x_i,\\eta)$. This iteration converges quadratically to $(v,\\mu)$ whenever the projected Hessian is uniformly invertible on the tangent space. If $L$ is $C^3$ and the retraction is second-order, the Rayleigh–Chebyshev step that appends the correction $T$ defined by $\\Pi_{L_\\lambda} L_x T = \\Pi_{L_\\lambda} G$ converges cubically, and when the projection of $G(v)$ vanishes the unmodified RQI is already cubic. The paper also gives the Schur-form realization of these equations when $L_x$ is invertible, which in the eigenvector case reproduces the classical resolvent formula and in constrained optimization reproduces the Riemannian Newton update equations.","pith_inferences":["The theorem's statement can be read as implicitly requiring a second-order normal estimate for the displacement $x_{i+1}-v$, since the proof applies the tangent-space lower bound (6.7) to it; if that estimate is not automatic on curved constraints, adding it as an explicit hypothesis is the cleanest repair.","The freedom to choose any consistent $R$ suggests a design principle: choose the Rayleigh quotient so that the projected $G(v)$ vanishes, obtaining cubic convergence from plain RQI rather than paying for the Chebyshev correction; testing different left inverses on tensor eigenpairs would show whether such choices are practical.","The paper's own numerical remark that a detailed global convergence analysis for the all-complex-eigenpair search is still needed, and that the last ten percent of pairs dominate runtime, points to the natural next problem of proving basin-of-attraction results or using RQI as a local accelerator inside a global polynomial solve.","Because the Schur form becomes unstable when $L_x$ is nearly singular, the tangent form is the natural fallback near convergence; a hybrid that switches forms based on the conditioning of $L_x$ would extend the framework to harder nonnormal problems."],"forward_implications":["For the eigenvector problem with the standard Rayleigh quotient, the generalized RQI is exactly classical RQI, and the condition $\\Pi_{L_\\lambda}(v)G(v)=0$ reproduces the known cubic convergence for normal matrices.","For constrained optimization with $H=J_C^T$, the projected-Hessian equation is Riemannian Newton on the embedded manifold, giving a unified quadratic-convergence proof for that method and for feasible projected SQP.","For real tensor eigenpairs, the Schur-form RQI is equivalent to the existing Newton-correction iteration on the sphere but avoids forming the projected Hessian; the paper reports about 16% faster runs, and about 34% faster with one redundant tensor evaluation removed.","For nonnormal matrices, the two-sided RQI is recovered by writing left and right eigenvectors as a constrained Lagrangian, and its cubic convergence follows from the same $G$ criterion.","A unitary version of the tensor RQI computes most complex eigenpairs of moderate tensors in minutes, and identifies the real pairs as a by-product without a homotopy run."],"supporting_citations":[{"why":"Supplies the classical Rayleigh quotient iteration for invariant subspaces that the paper generalizes.","marker":"[1]"},{"why":"Supplies the definitions and properties of first- and second-order retractions used in the convergence theorem.","marker":"[2]"},{"why":"Establishes the equivalence between feasible projected SQP and Riemannian Newton for constrained optimization, which the paper extends to general Lagrangians.","marker":"[3]"},{"why":"Supplies the Riemannian Hessian and projection formulas used to identify the projected Hessian with $\\Pi_H L_x$ in constrained optimization.","marker":"[4]"},{"why":"Introduces the left-inverse choice of the Lagrange multiplier estimate that serves as the generalized Rayleigh quotient.","marker":"[12]"},{"why":"Provides the reduced quasi-Newton method with feasibility improvement whose multiplier estimate is the same Rayleigh quotient formula.","marker":"[13]"},{"why":"Supplies the nonlinear eigenvalue problem setting and the Newton iteration baseline that the general RQI recovers.","marker":"[15]"},{"why":"Supplies the tensor eigenpair Newton-correction algorithms against which the Schur-form RQI is benchmarked.","marker":"[16]"},{"why":"Supplies the two-sided Rayleigh quotient iteration whose cubic convergence is rederived from the paper's $G$ criterion.","marker":"[24]"},{"why":"Supplies the nonlinear Rayleigh functional method used to define $R$ for nonlinear eigenproblems in Section 7.6.","marker":"[27]"}],"fun_headline_variants":["Unified RQI: quadratic and cubic convergence from one construction","Cubic convergence in RQI linked to second covariant derivative and Chebyshev term","One iteration scheme for eigenproblems, tensors, and optimization with cubic convergence","New RQI constructs unify quadratic and cubic convergence via retraction geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is uniform invertibility of the projected Hessian on the tangent space, inequality (6.7), together with the unstated premise that the displacement from the current point to the solution has a normal component controlled at second order, so that the tangent-space lower bound can honestly be applied to it.","fun_headline_variants_meta":{"raw":{"variants":["Unified RQI: quadratic and cubic convergence from one construction","Cubic convergence in RQI linked to second covariant derivative and Chebyshev term","One iteration scheme for eigenproblems, tensors, and optimization with cubic convergence","New RQI constructs unify quadratic and cubic convergence via retraction geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001034,"raw_usage":{"total_tokens":4396,"prompt_tokens":1031,"completion_tokens":3365,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":3296}},"tokens_in":647,"tokens_out":3365,"duration_ms":21685,"temperature":1.0,"reasoning_tokens":3296,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:41:51.986228+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth Lagrangian and a curved constraint satisfying the stated hypotheses, for example $C(x)=x_1^2+x_2^2-1$ with $L(x,\\lambda)=F(x)-x\\lambda$ and a Rayleigh quotient $R$ with nonzero differential at the solution. Run iteration (6.8) in high-precision arithmetic from a nearby initial point and compute $Q_i=\\log\\|x_{i+1}-v\\|/\\log\\|x_i-v\\|$. If $Q_i$ does not approach $2$ for some such problem while all stated hypotheses hold, the missing normal-component control breaks the claimed quadratic rate; if $Q_i$ always approaches $2$, the control is automatic in these cases.","supporting_citations":[{"cited_title":"Absil, R","cited_arxiv_id":null,"evidence_quote":"Supplies the classical Rayleigh quotient iteration for invariant subspaces that the paper generalizes."},{"cited_title":"Absil, J","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between feasible projected SQP and Riemannian Newton for constrained optimization, which the paper extends to general Lagrangians."},{"cited_title":"Absil, R","cited_arxiv_id":null,"evidence_quote":"Supplies the Riemannian Hessian and projection formulas used to identify the projected Hessian with $\\Pi_H L_x$ in constrained optimization."},{"cited_title":"Gabay , Minimizing a differentiable function over a differential man ifold, Journal of Optimization Theory and Applications, 37 (1982), pp","cited_arxiv_id":null,"evidence_quote":"Introduces the left-inverse choice of the Lagrange multiplier estimate that serves as the generalized Rayleigh quotient."},{"cited_title":"Gabay , Reduced quasi-Newton methods with feasibility improvemen t for nonlinearly constrained optimization , Springer Berlin Heidelberg, Berlin, Heidelberg, 1982, pp","cited_arxiv_id":null,"evidence_quote":"Provides the reduced quasi-Newton method with feasibility improvement whose multiplier estimate is the same Rayleigh quotient formula."},{"cited_title":"Güttel and F","cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear eigenvalue problem setting and the Newton iteration baseline that the general RQI recovers."},{"cited_title":"Jaffe, R","cited_arxiv_id":null,"evidence_quote":"Supplies the tensor eigenpair Newton-correction algorithms against which the Schur-form RQI is benchmarked."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-sided Rayleigh quotient iteration whose cubic convergence is rederived from the paper's $G$ criterion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear Rayleigh functional method used to define $R$ for nonlinear eigenproblems in Section 7.6."}],"review_version":1}