{"id":"bd96ef45-7fc6-4356-a09e-ffc16d97a8d4","arxiv_id":"2412.09746","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"QMSR trains a quadratic manifold with a greedy algorithm and reconstructs data from sparse samples via a linear encoder on the sampled components.","lead":"This paper introduces QMSR, a method that reconstructs high-dimensional data from a few measured components using curved, quadratic approximation surfaces instead of flat linear spaces. It reports accuracy gains of orders of magnitude over standard linear methods on wave-like and transport-dominated data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central QMSR claim depends on the conditioning of P V, but QDEIM points are chosen from the first m singular vectors while the greedy V can contain high-order singular vectors; no condition numbers or norm-gap estimates connect Prop.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the sparse encoder is (PV)^+Ps, and all theoretical guarantees are stated in the norm induced by (PV)^+, not in the Euclidean norm used in every experiment. This concern is central because the headline claim is that QMSR with m=2r sparse samples matches full-data quadratic-manifold reconstruction accuracy. If (PV)^+ is ill-conditioned, the encoder can be unstable precisely when V contains high-order singular vectors, which the greedy method explicitly does. The paper gives no bound on kappa(PV), no report of its value, and no equivalence argument between the weighted norm of Proposition 2 and the measured relative error. This is a missing link between theory and empirical claim, and it is not merely a matter of consensus: it is a correctness risk in transferring the method to new transport-dominated problems. The proof flaw in Eq. (14) is real and should be corrected, but it is less load-bearing because the key identity (PV)^+P W=0 does hold for the actual minimizer of Eq. (12), so the theoretical conclusions survive with a corrected formula. The numerical evidence is substantial and the released code is a concrete asset; the experimental comparisons are favorable but consistently show that oversampling from m=r to m=2r dramatically improves QMSR accuracy, which is consistent with a conditioning effect and further motivates the requested diagnostic. On balance, the paper deserves conditional acceptance, with the conditioning analysis and proof correction as required revisions.","tokens_in":16719,"tokens_out":10948,"duration_ms":125228,"concrete_test":"Using the released code, instrument Algorithm 1 to record kappa(PV)=sigma_max/sigma_min for every (r,m) grid point in Figures 3, 7, and 10, and for the same test points compute the ratio rho = ||(PV)^+P(g(f(Ps))-s)||_2 / ||g(f(Ps))-s||_2. If kappa(PV) is modest (say <=1e3) and rho is near 1 in all m=2r regimes where QMSR matches the full-data QM curve, the conditioning objection is empirically discharged for these benchmarks. If kappa(PV) is large (>=1e8) or rho >> 1, then the Euclidean accuracy is not controlled by Proposition 2 and the paper needs a conditioning analysis or a different sampling construction before the central claim can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the sparse encoder f_{V,P}(Ps)=(PV)^+Ps in Eq. (9). Proposition 2 proves optimality only in the norm ||(PV)^+P(hat-s-s)||_2, whereas the reported errors in Eq. (17) and all figures are Euclidean. The sampling operator P is constructed by QDEIM from Phi_m, the first m left singular vectors (Section 3.4), but the greedy construction of V (Section 3.2) deliberately selects singular vectors with indices much larger than m, e.g., phi(140) and phi(182) in Section 4.1.2. Nothing in the paper guarantees that rows chosen for low-order modes keep P V well-conditioned for such high-order columns, and no condition number or norm-gap estimate is reported for any of the three experiments. If P V is ill-conditioned, (PV)^+ can amplify the sampling residual by a factor of kappa(PV), so the observed agreement with full-data quadratic-manifold reconstructions may be a property of these particular benchmarks rather than of the method. A separate proof defect: Eq. (14) in Proposition 1 is dimensionally inconsistent, since h(es)^T is 1-by-p while gamma I is p-by-p. The identity (PV)^+P W=0 can be recovered from the correct ridge minimizer W=(I-V(PV)^+P)S A^T(A A^T+gamma I)^{-1}, so the proof is repairable, but the printed argument is invalid as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"QMSR extends quadratic manifold approximations to the setting where only m sparse components of a high-dimensional vector are observed. Given training data, the method constructs a sampling operator P by applying QDEIM to the first m left singular vectors, greedily selects r left singular vectors V using an objective built from the sparse linear encoder f_{V,P}(Ps)=(PV)^+Ps, and then fits the quadratic decoder weight W by ridge regression. The final approximation is g_{V,W}(f_{V,P}(Ps)). The paper proves idempotence of this map, an optimality property with respect to the weighted objective ||(PV)^+P(\\hat{s}-s)||_2, and exact recovery of points that lie on the quadratic manifold. Numerical experiments on three transport-dominated test sets report the relative error (17) and show that QMSR with m=2r sparse samples performs comparably to full-data quadratic manifold reconstruction and significantly better than linear DEIM.","tokens_in":17032,"tokens_out":9817,"duration_ms":104252,"significance":"If the claims are sustained, QMSR is a practically valuable nonlinear alternative to empirical interpolation for sparse data reconstruction, with natural applications in model reduction, sensor placement, and field reconstruction from sparse measurements. The paper has several concrete strengths: it provides reproducible code, the numerical experiments are large-scale and include a comparison against the nonlinear encoder from (8), and the exact-recovery property for points on the manifold is a clean and checkable statement. The main weaknesses are in the theoretical support: the proof of Proposition 1 is invalid as printed, and the optimality result in Proposition 2 does not connect the weighted objective to the Euclidean errors actually measured in the experiments, with no conditioning information on PV reported. These issues are repairable, but they are load-bearing for the paper's theoretical claims.","major_comments":[{"comment":"The expression for W in Eq. (14) is not the minimizer of (12) and is dimensionally inconsistent: h(\\tilde{s})^T is a 1×p row vector, so h(\\tilde{s})^T + \\gamma I is not defined, and a matrix K that depends on a single test point s cannot describe the weight matrix W that is fitted to the whole training set. The idempotence conclusion is nevertheless repairable: with A = h(f_{V,P}(PS)), the minimizer of (12) is W = (I - V(PV)^+P) S A^T (A A^T + \\gamma I)^{-1}, and (PV)^+ P W = 0 follows from (PV)^+ P V = I. The printed proof should be replaced by this corrected argument.","section":"§3.3, Proposition 1, Eq. (14)"},{"comment":"Proposition 2 proves optimality only with respect to the weighted norm ||(PV)^+P(\\cdot)||_2. Because (PV)^+PV=I and (PV)^+PW=0, the objective in (16) reduces to ||\\tilde{s} - (PV)^+Ps||_2 and is independent of the nonlinear term W h(\\tilde{s}); it therefore does not measure or bound the Euclidean reconstruction error (1) or the relative error (17) reported in all figures. No condition number of PV or norm-gap estimate is provided anywhere in Section 4. The greedy bases in Sections 4.1.2 and 4.2.2 contain singular vectors with indices far beyond m (e.g., φ(140) and φ(182) in 4.1.2), while P is constructed from the first m singular vectors, so κ(PV) could be large and the sparse encoder could amplify the sampling residual. I ask the authors to report κ(PV) for each experiment or to give an explicit bound connecting the weighted objective to the Euclidean error; without this, the observed agreement with full-data reconstructions is not explained by the theory.","section":"§3.3, Proposition 2 and Eq. (16)"}],"minor_comments":[{"comment":"The update should read V^i = [φ(j_1), ..., φ(j_i)] rather than V^i = [φ(1), ..., φ(j_i)].","section":"Algorithm 1, line 7"},{"comment":"The method acronym is misspelled as \"QSMR\" in the opening sentence of Section 4 and in several figure captions; it should be \"QMSR\".","section":"Section 4 opening and figure captions"},{"comment":"The justification that (11) and (13) have the same solutions is imprecise as stated: the norm identity ||AZ||_F = ||A||_F for a matrix Z with orthonormal columns requires that the map is an isometry, which is not automatic in the claimed generality with p ≥ m. The equivalence should be argued directly through the left-multiplication by the orthonormal matrix Φ, rather than by citing the identity in that form.","section":"Remark 1"},{"comment":"The sentence \"we aim to select P such that the data S can be well approximated in the space spanned by the first m left-singular vectors\" is vague, since P acts by sampling rows of S, not by projecting S; a more precise statement about the role of QDEIM in making the sampled rows of Φ_m well conditioned would help the reader.","section":"Section 3.4"},{"comment":"The claim that QMSR with m=2r samples achieves \"comparable accuracy\" to full-data reconstruction is stated without a quantitative tolerance; specifying the accuracy gap, for example in terms of the plotted error curves, would make the claim more precise.","section":"Section 4.1.2"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the central empirical contribution appears sound, but the proof of Proposition 1 is invalid as printed and the conditioning/norm-gap issue in Proposition 2 is a real gap between the theory and the reported Euclidean errors. Both are fixable within the manuscript's scope, so I recommend major revision rather than rejection. I would also encourage the editor to ensure the revised version includes the corrected derivation of W and at least a reported condition number for the sampling basis in each experiment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: the core of this paper is solid and worth engaging, but it needs a revision. The new idea is a sparse encoder f(P s) = (PV)^+ P s on quadratic manifolds, plus a greedy training procedure that selects singular vectors with that encoder in the objective. That is genuinely new relative to the prior quadratic-manifold and empirical-interpolation literature. The numerical evidence is strong: on three transport-dominated problems, QMSR with m=2r samples matches full-data quadratic-manifold reconstruction and beats QDEIM by orders of magnitude. Code is available, the train/test split is honest, and the paper does not overclaim—it says comparable, not better, than full-data reconstruction.\n\nThe soft spots are two, and they are in different weight classes. First, Eq. (14) in the proof of Proposition 1 is dimensionally inconsistent: h(es)^T is 1 x p, gamma I is p x p, and the expression for K cannot be right. The result is repairable—the ridge solution is W = (I - V(PV)^+P) S A^T (A A^T + gamma I)^{-1} with A = h(f(P S))—and with that, (PV)^+P W = 0 follows immediately. So the theorem likely holds, but the printed proof is wrong and should be fixed.\n\nSecond, and more substantive: the optimality in Proposition 2 is measured in the (PV)^+ -induced norm, while the experiments report Euclidean error. QDEIM constructs P from the first m left singular vectors, but the greedy V can contain singular vectors with indices much larger than m (the examples list phi(140), phi(182)). Nothing guarantees P V is well-conditioned for those high-order columns, and no condition numbers or norm-gap estimates are reported. The greedy objective likely steers away from badly conditioned choices in practice, but the paper does not say that or analyze it. This is a moderate gap, not a fatal one: the method's success on three benchmarks is genuine, but the theory does not yet explain why it should generalize.\n\nI also note the Gauss-Newton comparison sweeps a wide set of damping coefficients and reports the best; that is generous to the nonlinear encoder, so the conclusion that the linear sparse encoder is sufficient is not inflated.\n\nWho should read this: anyone working on gappy POD, DEIM, sensor placement, or transport-dominated model reduction. It deserves a serious referee—fix the proof, add conditioning discussion, and it should be acceptable.\n\nRecommendation: send to peer review with a request for revision, not a desk reject.","headline":"Solid new method with strong numerics; a repairable proof flaw and missing conditioning analysis need fixing before acceptance.","tokens_in":17556,"tokens_out":4067,"would_cite":true,"duration_ms":38074,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F55","62H25","65F30","68T09","65F20","65M22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Quadratic manifold sparse regression reconstructs high-dimensional data vectors from a small number of samples and matches full-data accuracy on transport-dominated problems.","keywords":["quadratic manifolds","sparse regression","empirical interpolation","model reduction","dimensionality reduction","QDEIM sampling","transport-dominated dynamics","Kolmogorov barrier"],"falsifier":"Compute the ratio of the weighted residual $\\|(PV)^+ P (\\hat{s} - s)\\|_2$ to the Euclidean error $\\|\\hat{s} - s\\|_2$ over a test set of transport-dominated data: if this ratio grows large while the manifold fit itself stays good, the optimality result no longer controls the error QMSR reports. Directly, find a dataset where $PV$ is ill-conditioned under QDEIM sampling; QMSR errors far above full-data quadratic manifold reconstruction would falsify the claim that $m = 2r$ samples generally suffice.","tokens_in":16521,"feed_emoji":"🎯","tokens_out":8373,"duration_ms":76873,"temperature":0.7,"pith_summary":"Quadratic manifold sparse regression (QMSR) reconstructs a high-dimensional data vector from a small number of its measured components, without assuming the data lie near a linear subspace. The method trains a quadratic decoder and a sparse linear encoder together via a greedy selection of singular vectors, with the sample locations chosen by the QDEIM rule. The paper's central claim is that on transport-dominated and wave-like data, QMSR with $m = 2r$ samples matches the accuracy of reconstructing from all components, and outperforms empirical interpolation and gappy POD by orders of magnitude. It also proves that any data point lying exactly on the trained quadratic manifold is recovered exactly from the sparse samples when the sampled reduced basis has full column rank.","feed_headline":"Quadratic manifolds beat linear sparse recovery by orders of magnitude","feed_subtitle":"Using just m = 2r samples matches full-data quadratic manifold accuracy on transport problems.","key_machinery":"The central object is the quadratic manifold decoder $g_{V,W}(c) = Vc + Wh(c)$ with $h(c) = [c_1 c_1, c_1 c_2, \\ldots, c_r c_r]^\\top$, paired with the sparse linear encoder $f_{V,P}(Ps) = (PV)^+ Ps$. The load-bearing identity is $(PV)^+ P W = 0$ for $W$ fitted by the regularized least-squares step; this identity makes the reconstruction idempotent, collapses the nonlinear projection onto the manifold to a linear least-squares problem in the norm induced by $PV$, and yields the exact-recovery corollary. Sampling locations are chosen by QDEIM on the first $m$ singular vectors, and the greedy training objective selects singular vectors specifically for this sparse encoder rather than for full-data reconstruction.","core_discovery":"The discovery is that nonlinear manifold expressivity and sparse observations can be combined through one specific encoder-decoder pair: encode the samples $Ps$ as $f_{V,P}(Ps) = (PV)^+ Ps$, then decode as $g_{V,W}(c) = Vc + Wh(c)$, where $h$ collects the quadratic products of the reduced coordinates. If the weight matrix $W$ is fitted by the proposed sparse greedy least-squares objective, the composed map is idempotent, the reconstruction is the unique minimizer of the weighted residual $\\|(PV)^+ P(\\hat{s} - s)\\|_2$ over the manifold, and every manifold point is recovered exactly whenever $PV$ has full column rank. The numerical experiments on Vlasov density fields, acoustic wave pulses, and rotating detonation data show relative errors close to full-data quadratic manifold reconstruction with only $m = 2r$ sparse samples, while linear empirical interpolation is orders of magnitude less accurate.","pith_inferences":["The annihilation identity $(PV)^+ P W = 0$ suggests the same sparse encoder could be paired with other trained decoders, polynomial or neural, as long as the fitted weight matrix satisfies an analogous annihilation condition.","The sample placement and conditioning of $PV$, not just the manifold's fit, is the practical bottleneck; a sensor-placement criterion that minimizes the condition number of $PV$ could be a natural extension of the QDEIM choice.","The $m = 2r$ heuristic is established only on the three test problems; datasets with different Kolmogorov n-width decay would test whether the required oversampling factor depends on the conditioning of $PV$.","A head-to-head comparison against nonlinear sparse reconstruction methods based on neural networks would clarify whether the improvement over linear methods comes mainly from the quadratic manifold or from the sparse regression formulation itself."],"forward_implications":["Transport-dominated fields that defeat linear approximation can be reconstructed from a small number of sensor samples using quadratic manifolds, with errors matching full-data reconstruction.","The exact-recovery property means QMSR inherits the key interpolation-style guarantee of empirical interpolation: points on the learned nonlinear manifold are reproduced exactly from $m \\geq r$ sparse samples when $PV$ has full column rank.","The sparse linear encoder is numerically sufficient in the tested examples: it performs like solving the nonlinear sparse least-squares problem with Gauss-Newton iterations, so the nonlinear projection can be avoided in practice.","Oversampling beyond $m = r$ avoids the instability seen in empirical interpolation at $m = r$; the paper uses $m \\in \\{r, 2r, 3r, 4r\\}$ and reports comparable accuracy already at $m = 2r$.","Because the decoder is quadratic, the method remains a component-wise polynomial of the reduced coordinates, so it can be used within projection-based model reduction workflows that require a differentiable decoder."],"supporting_citations":[{"why":"Supplies the QDEIM sampling operator used to select sparse sample locations from the singular vectors.","marker":"[17]"},{"why":"Provides the full-data greedy quadratic manifold construction that QMSR adapts to the sparse encoder objective.","marker":"[47]"},{"why":"Defines empirical interpolation and DEIM, the linear baselines that QMSR is compared against and whose sampling ideas it extends.","marker":"[8, 15]"},{"why":"Introduces gappy POD, the linear sparse reconstruction baseline the paper positions QMSR against.","marker":"[20]"},{"why":"Establishes quadratic manifolds as nonlinear reduced spaces, the geometric foundation QMSR builds on.","marker":"[29, 43]"},{"why":"Documents the instability of DEIM and gappy POD with certain sampling points, which motivates oversampling and explains the $m = r$ behavior in the experiments.","marker":"[40]"},{"why":"Provides the rotating detonation wave dataset used in the third numerical experiment.","marker":"[12]"}],"fun_headline_variants":["Sparse m=2r samples beat linear methods on quadratic manifolds","Quadratic manifold recovery: order-of-magnitude gains from sparse data","Nonlinear projection: sparse samples match full-data quadratic manifold","m=2r sparse samples: quadratic manifolds beat linear by orders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The accuracy guarantee depends on the sampled reduced basis being well conditioned; the paper proves optimality in the norm induced by the pseudo-inverse of $PV$, not directly in the Euclidean error that the experiments report, so a poorly conditioned sample set could break the link between theory and measured accuracy.","fun_headline_variants_meta":{"raw":{"variants":["Sparse m=2r samples beat linear methods on quadratic manifolds","Quadratic manifold recovery: order-of-magnitude gains from sparse data","Nonlinear projection: sparse samples match full-data quadratic manifold","m=2r sparse samples: quadratic manifolds beat linear by orders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001435,"raw_usage":{"total_tokens":5725,"prompt_tokens":823,"completion_tokens":4902,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":4826}},"tokens_in":439,"tokens_out":4902,"duration_ms":33994,"temperature":1.0,"reasoning_tokens":4826,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:46:05.072226+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the ratio of the weighted residual $\\|(PV)^+ P (\\hat{s} - s)\\|_2$ to the Euclidean error $\\|\\hat{s} - s\\|_2$ over a test set of transport-dominated data: if this ratio grows large while the manifold fit itself stays good, the optimality result no longer controls the error QMSR reports. Directly, find a dataset where $PV$ is ill-conditioned under QDEIM sampling; QMSR errors far above full-data quadratic manifold reconstruction would falsify the claim that $m = 2r$ samples generally suffice.","supporting_citations":[{"cited_title":"Drmaˇc and S","cited_arxiv_id":null,"evidence_quote":"Supplies the QDEIM sampling operator used to select sparse sample locations from the singular vectors."},{"cited_title":"Everson and L","cited_arxiv_id":null,"evidence_quote":"Introduces gappy POD, the linear sparse reconstruction baseline the paper positions QMSR against."},{"cited_title":"Peherstorfer, Z","cited_arxiv_id":null,"evidence_quote":"Documents the instability of DEIM and gappy POD with certain sampling points, which motivates oversampling and explains the $m = r$ behavior in the experiments."},{"cited_title":"Investigations on Projection-Based Reduced Order Model Development for Rotating Detonation Engine","cited_arxiv_id":"2404.10323","evidence_quote":"Provides the rotating detonation wave dataset used in the third numerical experiment."}],"review_version":1}