{"id":"c87793b9-2885-47f1-80f2-69b76e43e917","arxiv_id":"1908.02249","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A rank-constrained linear representation with separate input and output subspaces yields reduced-order LQR controllers that stabilize the Ginzburg-Landau system at lower rank and suppress vortex shedding more effectively than DMD or OMD in the tested cases.","lead":"This paper introduces low-rank Dynamic Mode Decomposition (lrDMD), a data-driven model reduction method that fits a rank-constrained linear map between separate input and output subspaces, and uses it to build reduced-order LQR feedback controllers for two unsteady fluid systems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Flat-plate control superiority is supported by single deterministic runs with no quantified LQR cost; the abstract's 'cost lower' claim is not yet established.","rationale":"I read the paper as a computational proposal: a rank-constrained linear model with distinct left/right subspaces, solved by a non-convex optimization, is applied to LQR control. The strongest_claim is comparative superiority. In good faith, the formulation is sound: Eq. (6) is a valid relaxation of Eq. (2), Eq. (7) gives the optimal D for fixed subspaces, and the GL rank-5 stabilization result is a genuine demonstration that a single shared subspace (DMD/OMD) can fail while separate row/column subspaces succeed. The adjoint projection errors provide a plausible mechanism. However, the flat-plate evidence for the headline 'cost lower' claim is thin: only single deterministic runs, a binary table, and no quantitative cost. The reader's weakest_assumption (solver local minima) is reasonable but secondary here; the lrDMD solver is initialized from a DMD-feasible point and the subspace projection is designed to be monotone, so local-minimum risk is at least bounded for reconstruction. The central gap is the lack of any robustness or repetition evidence for the nonlinear-flow control comparison. Because the paper's method is sound and promising but the comparative claim is not yet established with the reported evidence, I agree with the CONDITIONAL verdict and do not propose changing it.","tokens_in":15850,"tokens_out":16491,"duration_ms":196764,"concrete_test":"For the flat-plate configuration at t0 = 210 with default snapshot spacing, run each controller from at least 20 independent small random perturbations of the initial vorticity field (e.g., 1% amplitude perturbations), and also repeat with snapshot windows n = 150 and n = 250. For every closed-loop run compute the realized LQR cost J = sum_{k=t0}^{T} (q_k^T Q q_k + S u_k^2) using the same Q, S as the Riccati design. If lrDMD's mean cost is not lower than OMD and DMD with non-overlapping confidence intervals, the abstract's 'cost lower' claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that lrDMD produces better reduced-order LQR controllers than DMD or OMD. The GL section gives a clean rank-5 stabilization demonstration and a plausible adjoint-capture mechanism, so the formulation itself is not the weak point. The weak point is the only nonlinear-flow evidence for the comparative claim: Section 5.2.3 reports one deterministic closed-loop trajectory per activation time (t0 = 170, 190, 210) and a binary pass/fail table (Table 5) for two snapshot spacings. No integrated control cost J is computed, no repeated trials or perturbation of initial conditions are reported, and the figures show raw CD and actuation strength only. Since the abstract asserts lrDMD suppresses vortex shedding 'at a cost lower than either DMD or OMD,' the absence of a quantified cost and uncertainty estimate is directly load-bearing: if another snapshot window, initial perturbation, or rank shifted the ordering, the headline claim would not generalize. This is distinct from and more immediate than the solver-local-minimum concern: even granting Algorithm 1 reliably reaches a good local minimum, its DMD-feasible initialization makes it unlikely to be worse than DMD in reconstruction error, and the control comparison still rests on single runs.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes low-rank Dynamic Mode Decomposition (lrDMD), a reduced-order modeling method that solves the rank-constrained least-squares problem min_{L,D,R} ||Y - LDR^T X||_F^2 with separate input and output subspaces, and compares it with DMD and OMD. Two numerical solvers are presented: a heuristic subspace-projection method and a Riemannian trust-region gradient method. The reduced-order models are used to design full-state LQR feedback controllers, which are tested on the linearized complex Ginzburg-Landau equation in the globally unstable regime and on two-dimensional incompressible flow past an inclined flat plate at Re=100. The paper claims that lrDMD achieves lower reconstruction error, stabilizes the GL system at lower rank, provides improved estimates of adjoint modes, and suppresses flat-plate vortex shedding at lower control cost than DMD or OMD.","tokens_in":16066,"tokens_out":5560,"duration_ms":57426,"significance":"The core idea of using separate input and output subspaces for data-driven reduced-order models is a meaningful extension of DMD and OMD, particularly for non-normal flow systems where adjoint information matters for control. The GL experiments provide a clean, full-order-validated demonstration: lrDMD stabilizes the system at rank 5 where DMD and OMD fail, and the adjoint-mode projection error is reduced. The authors are also honest about the non-convexity of the optimization problem in Remark 2. However, the strongest comparative claims—lower reconstruction error in general and lower control cost on the flat plate—are not yet supported by the evidence as presented. The flat-plate comparison rests on single deterministic runs with no quantified cost, and the reconstruction-error comparison is in-sample. If these gaps are fixed, the paper could make a solid contribution to data-driven flow control.","major_comments":[{"comment":"The abstract claims that lrDMD suppresses vortex shedding from the inclined flat plate \"at a cost lower than either DMD or OMD,\" but no quantitative control cost is reported anywhere in Section 5.2. The evidence consists of CD vs. time plots and actuation-strength plots for three activation times (t0 = 170, 190, 210) and a binary pass/fail table (Table 5) for two snapshot spacings. This is insufficient to establish a cost ordering. The authors should compute and report an integrated cost, such as the actual LQR objective J = sum (q^T Q q + u^T S u) evaluated on the closed-loop nonlinear trajectories, or an equivalent metric (e.g., time-integrated deviation of CD from the steady-state value plus control effort), for each method, activation time, and snapshot spacing. Without such a quantitative comparison, the headline \"cost lower\" claim is not load-bearing evidence and may not generalize.","section":"Section 5.2.3, Figs. 7-8, Table 5; abstract"},{"comment":"The reconstruction errors reported in Table 2 are computed as epsilon = ||Y - A_hat X||_F on the same data matrices X, Y that were used to construct each reduced-order model. This is an in-sample evaluation of a training objective, and lrDMD has more degrees of freedom than DMD or OMD in optimizing that objective, so its lower error is partly expected. This does not demonstrate better predictive accuracy or a generally superior model. The paper should include an out-of-sample test, for example by withholding a portion of the time series or snapshots from a different initial condition, and reporting the prediction error on that held-out data. Without such evidence, the \"lower reconstruction error\" claim in the abstract is overstated.","section":"Section 3.1, Table 2"},{"comment":"The formula for D in Algorithm 1, line 10, includes an extra factor R^T L: D = (L^T Y X^T R)(R^T X X^T R)^{-1}(R^T L). This is inconsistent with Eq. (7), which gives D*(L,R) = (L^T Y X^T R)(R^T X X^T R)^{-1}. Since L and R are generally different orthogonal matrices, R^T L is not the identity, and as printed the algorithm does not compute the least-squares optimal D for the given L and R. Please correct the formula or explicitly clarify whether R^T L is an intentional part of a different definition. If the implementation actually uses Eq. (7), then this is a typo that should be fixed; otherwise the algorithm and the numerical results may be based on a different model than the one analyzed in Section 2.3.","section":"Appendix A.1, Algorithm 1, line 10"},{"comment":"All claimed advantages of lrDMD over DMD and OMD depend on the non-convex optimization problem (6) being solved to a sufficiently good local minimum. Remark 2 states that the gradient method has no global convergence guarantee, and Algorithm 1 is a heuristic initialized from the DMD subspace. The paper does not report any sensitivity analysis with respect to initialization or convergence diagnostics. For example, one could initialize from the OMD solution, from random orthogonal matrices, or from several perturbed DMD bases and quantify the spread in reconstruction error and closed-loop control performance. Without this, the single-run comparisons in Sections 3.1 and 5 cannot rule out the possibility that the observed lrDMD advantages are artifacts of the particular local minimum found by the proposed initialization. This is a load-bearing concern for the generality of the central claim.","section":"Remark 2 and Section 3"}],"minor_comments":[{"comment":"The heading \"Mathematical F ormulation\" contains an erroneous space; it should read \"Mathematical Formulation.\"","section":"Section 2 heading"},{"comment":"In the paragraph after Eq. (14), the sentence \"the eigenvector of X that corresponds to the largest eigenvalue\" should refer to the eigenvector of F, not X.","section":"Section 5.1.3"},{"comment":"The notation Y = U^T Y and X = U^T X reuses symbols X and Y for the projected data matrices; this is confusing and should be clarified, for example with new symbols X_p, Y_p.","section":"Section 3.1"},{"comment":"The check and cross symbols in Table 5 do not render correctly in the text version (they appear as box glyphs); ensure that the caption and symbols display properly in the final PDF.","section":"Table 5"},{"comment":"The paper should clarify that T in Eq. (15) is the number of timesteps advanced by the nonlinear solver between control updates, and that Table 5 varies this parameter for the snapshot data matrices; this distinction is implicit but not stated clearly.","section":"Section 5.2.3"},{"comment":"Reference [36] is listed as \"forthcoming\"; if the associated CDC paper has appeared, the complete publication details should be provided.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript builds on the authors' own prior work [36] and on Heas and Herzet [37], and the novel contribution is mainly the separate input/output subspace formulation and its control applications. The reviewer is not concerned about self-citation per se, but the paper should carefully delineate which parts are new relative to [36] and [37]. The GL control results are the strongest part of the paper, while the flat-plate comparative claim needs quantitative support. If the authors can add an out-of-sample reconstruction test, a quantified flat-plate control cost, and solver sensitivity checks, the paper could be acceptable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe useful thing here is not the algorithm—lrDMD is a repackaging of the rank-constrained least-squares problem from Heas and Herzet (2016) with the solver from the authors' own CDC paper—but the demonstration that letting the input and output subspaces differ pays off in reduced-order LQR control. The Ginzburg-Landau section is the strongest part: at rank 5, lrDMD stabilizes a system that DMD and OMD cannot, and it recovers the adjoint mode more accurately at every rank tested. Those are clean, interpretable results, and they give the paper a reason to exist.\n\nThe flat-plate part is where I part company with the abstract. The claim that lrDMD suppresses vortex shedding 'at a cost lower than either DMD or OMD' is not supported by anything the paper actually computes. There is no integrated LQR cost, no repeated trials, no perturbation of initial conditions. Table 5 is pass/fail for two snapshot spacings, and Figures 7–8 are single deterministic runs. It may well be true—the GL results make it plausible—but the evidence is exactly the kind of single-run comparison that has a habit of reversing when you move to a new snapshot window or add noise. The stress-test note is right, and it is more immediate than the solver-convergence worry.\n\nAlso worth flagging: the reconstruction-error improvement in Table 2 is in-sample, so it cannot carry the weight of a generalizability claim. lrDMD is initialized from the DMD subspace, so it is essentially guaranteed to match DMD on training data; showing lower error on the same data is not news. The authors are honest about the non-convex global-optimization issue in Remark 2, which I appreciate.\n\nOn balance the paper is sound and worth engaging. The math is careful, the GL validation is genuinely out-of-sample, and the idea deserves a serious referee. But the comparative claims for the nonlinear flow need quantitative support before they appear in print. I'd send it out with a request for repeated runs, integrated control costs, and an out-of-sample reconstruction test, not desk-reject it.\n\nFor your reading group: maybe. It's useful for anyone working on data-driven flow control, and the gap between the abstract and the flat-plate evidence is a good discussion point.","headline":"lrDMD is a solid application of a known rank-constrained idea; the GL results are clean, but the flat-plate 'cost lower' claim is not backed by any computed cost.","tokens_in":16608,"tokens_out":4312,"would_cite":true,"duration_ms":39734,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes lrDMD, a rank-constrained linear model with distinct input and output subspaces, and shows it builds full-state LQR controllers that stabilize the Ginzburg-Landau system at lower rank and suppress flat-plate vortex…","keywords":["flow control","Dynamic Mode Decomposition","low-rank approximation","model reduction","LQR feedback control","adjoint modes","Ginzburg-Landau equation","vortex shedding"],"falsifier":"On a small linear system where the global rank-$r$ optimum can be found by exhaustive search, run the lrDMD solvers from many random initial subspaces instead of the DMD subspace and compare reconstruction error and the smallest rank of a stabilizing LQR controller. If a substantial fraction of random starts land in local minima with errors no better than DMD or OMD, or fail to stabilize at the rank the paper reports, the central improvement is initialization-dependent rather than a property of the rank-constrained formulation.","tokens_in":15580,"feed_emoji":"🌊","tokens_out":12210,"duration_ms":110314,"temperature":0.7,"pith_summary":"The paper argues that a reduced-order model for unsteady fluid flows should not be forced to use the same low-dimensional subspace for the input and the output of its linear map. It proposes low-rank Dynamic Mode Decomposition (lrDMD), which solves $\\min_{L,D,R}\\|Y-LDR^{T}X\\|_F^2$ over separate orthonormal left and right subspaces $L$ and $R$. On the linearized Ginzburg-Landau equation in the globally unstable regime, lrDMD stabilizes the system with a rank-5 full-state LQR controller, whereas DMD and OMD leave an unstable eigenvalue at that rank. On the unsteady flow over a flat plate inclined at 35 degrees, lrDMD suppresses vortex shedding with lower actuation cost than DMD and matches or exceeds OMD, especially when the controller is switched on after nonlinearities have grown. The paper's point is that the separate input and output subspaces, not merely the rank constraint, are what make the reduced-order controllers better.","feed_headline":"Rank-constrained DMD stabilizes flows at lower rank","feed_subtitle":"Letting the input and output subspaces differ makes full-state LQR control of unstable flows work at lower order.","key_machinery":"The load-bearing object is the rank-constrained least-squares surrogate with the closed-form optimal $D^{*}$ and the observation that the objective depends only on the subspaces spanned by $L$ and $R$. The paper solves it with two methods: a fast subspace-projection iteration that alternates between an optimal $L$ for a fixed projected basis and an Orthogonal Procrustes update for $R$, initialized from the DMD/POD subspace, and a Riemannian trust-region gradient method on the product Grassmann manifold initialized by the subspace projection. In the controller construction, the LQR Riccati equation is solved by Galerkin projection onto the row-space basis of the rank-$r$ model, which for lrDMD is the right subspace $R$; the same projection choice is made for DMD and OMD, so the comparison isolates the learned model rather than the controller projection.","core_discovery":"The paper's central claim is that the best rank-$r$ linear model of a dynamical system from snapshot data is not generally obtained by projecting onto one subspace, as DMD and OMD do, but by solving the rank-constrained least-squares problem $\\min_{L,D,R}\\|Y-LDR^{T}X\\|_F^2$ over orthonormal $L,R\\in\\mathbb{R}^{m\\times r}$. For fixed $L$ and $R$ the optimal middle factor is $D^{*}=(L^{T}YX^{T}R)(R^{T}XX^{T}R)^{-1}$, and the remaining problem depends only on the subspaces spanned by $L$ and $R$, so it lives on a product of Grassmann manifolds. The freedom to choose the input subspace $R$ and output subspace $L$ separately is what the paper identifies as the source of improvement: it fits the data more closely, represents the adjoint mode of the unstable Ginzburg-Landau eigenmode more faithfully, and yields full-state LQR controllers that stabilize that system at rank 5, suppress vortex shedding behind a 35-degree flat plate from later nonlinear initial conditions, and recover the true optimal actuator location at lower rank than DMD or OMD.","pith_inferences":["If the separate input and output subspaces are the active ingredient, lrDMD-type models should also improve other data-driven tasks in which the row space matters, such as non-intrusive receptivity and sensitivity analysis of nonnormal flows.","The comparison inherits whatever local minima the non-convex solvers find; a natural stress test is to initialize the Grassmannian optimization from many random subspaces and measure how often reconstruction error and stabilizing controller rank stay better than OMD.","The rank-constrained model is not tied to LQR, so the same lrDMD representation could be plugged into other controller designs, such as output-feedback or $\\mathcal{H}_\\infty$ synthesis, for problems where full-state measurements are unavailable."],"forward_implications":["At a fixed rank, lrDMD achieves lower snapshot reconstruction error than DMD and OMD on the flat-plate data, and in several cases the fast subspace-projection solver is both more accurate and cheaper than the gradient solver.","lrDMD stabilizes the Ginzburg-Landau system with a rank-5 LQR controller, while DMD and OMD leave an unstable eigenvalue at rank 5; at rank 9 all three approaches approach the full-order optimal controlled spectrum.","lrDMD recovers the true optimal actuator location for the Ginzburg-Landau system at rank 5, where DMD selects the wrong location until rank 9.","For flow over the inclined flat plate, lrDMD returns the flow to the steady state from later, more nonlinear initial conditions and does so with lower actuation cost than DMD, while also giving a lower projection error for the unstable adjoint mode at a given rank."],"supporting_citations":[{"why":"Defines the DMD approximation and supplies the baseline reduced-order model that lrDMD is compared against.","marker":"[25, 26]"},{"why":"Defines Optimal Mode Decomposition and the same-subspace optimization that lrDMD relaxes by allowing separate input and output subspaces.","marker":"[27, 38]"},{"why":"Establishes the closed-form rank-constrained dynamic-mode-decomposition solution that motivates the lrDMD optimization problem.","marker":"[37]"},{"why":"Provides the gradient-based solver on the Grassmann manifold used for the lrDMD gradient method.","marker":"[36]"},{"why":"Supplies the Ginzburg-Landau parameters, LQR weighting, actuator model, and full-order optimal actuator location used as ground truth.","marker":"[28]"},{"why":"Supplies the flat-plate unstable steady state, actuator model, and feedback-control scenario used for the vortex-shedding demonstration.","marker":"[21]"},{"why":"Provides the fast immersed-boundary flow solver used to generate the flat-plate snapshot data.","marker":"[50]"},{"why":"Supplies the manifold-optimization toolbox used for the trust-region gradient solves of lrDMD and OMD.","marker":"[41]"},{"why":"Supplies the Riemannian trust-region algorithm used by the gradient-based solver.","marker":"[42]"},{"why":"Provides the closed-form solution to the Orthogonal Procrustes problem used in the subspace-projection initialization.","marker":"[52]"}],"fun_headline_variants":["Rank-constrained DMD: better control at lower order","Low-rank DMD cuts controller order for unstable flows","Separate subspaces make DMD control flows at lower rank","lrDMD stabilizes flows with lower-order controllers","Distinct input-output subspaces improve DMD control"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the numerical solvers for the non-convex problem find a good enough local minimum; the paper states that the gradient method has no global convergence guarantee and that the subspace-projection method is a heuristic initialized from the DMD subspace. If those solvers often land in poor local minima on other systems, the claimed advantages over DMD and OMD would not generalize.","fun_headline_variants_meta":{"raw":{"variants":["Rank-constrained DMD: better control at lower order","Low-rank DMD cuts controller order for unstable flows","Separate subspaces make DMD control flows at lower rank","lrDMD stabilizes flows with lower-order controllers","Distinct input-output subspaces improve DMD control"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1577,"prompt_tokens":1020,"completion_tokens":557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":478}},"tokens_in":636,"tokens_out":557,"duration_ms":6140,"temperature":1.0,"reasoning_tokens":478,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:49:52.043316+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a small linear system where the global rank-$r$ optimum can be found by exhaustive search, run the lrDMD solvers from many random initial subspaces instead of the DMD subspace and compare reconstruction error and the smallest rank of a stabilizing LQR controller. If a substantial fraction of random starts land in local minima with errors no better than DMD or OMD, or fail to stabilize at the rank the paper reports, the central improvement is initialization-dependent rather than a property of the rank-constrained formulation.","supporting_citations":[{"cited_title":"Low-Rank Dynamic Mode Decomposition: An Exact and Tractable Solution","cited_arxiv_id":"1610.02962","evidence_quote":"Establishes the closed-form rank-constrained dynamic-mode-decomposition solution that motivates the lrDMD optimization problem."},{"cited_title":"Low-rank dynamic mode decomposition using Rie- mannian manifold optimization","cited_arxiv_id":null,"evidence_quote":"Provides the gradient-based solver on the Grassmann manifold used for the lrDMD gradient method."},{"cited_title":"H2 optimal actuator and sensor placement in the linearised complex ginzburg–landau system","cited_arxiv_id":null,"evidence_quote":"Supplies the Ginzburg-Landau parameters, LQR weighting, actuator model, and full-order optimal actuator location used as ground truth."},{"cited_title":"Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators","cited_arxiv_id":null,"evidence_quote":"Supplies the flat-plate unstable steady state, actuator model, and feedback-control scenario used for the vortex-shedding demonstration."},{"cited_title":"A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions","cited_arxiv_id":null,"evidence_quote":"Provides the fast immersed-boundary flow solver used to generate the flat-plate snapshot data."},{"cited_title":"Manopt, a matlab toolbox for optimization on manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies the manifold-optimization toolbox used for the trust-region gradient solves of lrDMD and OMD."},{"cited_title":"Trust-region methods on rieman- nian manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies the Riemannian trust-region algorithm used by the gradient-based solver."},{"cited_title":"A generalized solution of the orthogonal procrustes problem","cited_arxiv_id":null,"evidence_quote":"Provides the closed-form solution to the Orthogonal Procrustes problem used in the subspace-projection initialization."}],"review_version":1}