{"id":"b7979e07-2d92-45c1-9bf0-1741785958f3","arxiv_id":"1908.01047","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Incremental SVD-based online DMD and DMD-with-control algorithms for time-varying systems, demonstrated on EEG error-related potentials.","lead":"This paper develops incremental versions of Dynamic Mode Decomposition that track systems whose behavior changes over time, using incremental singular value decomposition instead of matrix inversion. It tests them on EEG brain signals and shows they can predict error-related potentials when standard online DMD struggles with low-rank data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reduced-order update (Eqs. 33, 38) replaces A_k by its rank-r projection without an error bound, so the central 'appropriately reduced model' claim is not established; proofs of the exact update also contain false orthogonality identities.","rationale":"I read the paper as claiming incremental SVD gives exact full-rank RLS updates plus a practical reduced-order approximation. The exact algebra checks out: the false row-orthogonality line in Appendix C is a proof defect, not a counterexample to the final formula, which can be recovered by noting X_{k+1}V_{k+1} = rho U_k Sigma_k V_sk,1 + x_{k+1} v_sk,2. The reduced-order equations are the weakest load-bearing component because they are what distinguishes the paper from online DMD [35]. The paper provides no error analysis for rank truncation, and the experiments compare only against a heuristic baseline in the singular regime, so the improvement cannot be attributed to correct reduced-order updates. This matches the reader's weakest assumption at Eqs. (33) and (38). The conditional verdict remains appropriate: the full-rank algorithm is plausible and supported by the provided code, and the truncation issue is addressable with added analysis or experiments rather than requiring rejection.","tokens_in":25003,"tokens_out":29073,"duration_ms":292089,"concrete_test":"Generate full-rank random time-varying data with known A_k (e.g., n=20, window w=40, rho=0.9, r=5), run Eq. (33), and at each step compute the rank-r truncated least-squares operator Abar^{LS}_k = Y_k Vbar_k Sigmabar_k^{-1} Ubar_k^* from the current window. Plot the Frobenius-norm distance between Abar^{trunc}_k and Abar^{LS}_k over time. If the distance grows with k or is not bounded by a small multiple of the discarded singular values, the recursive truncation is not tracking the reduced-order least-squares solution and the paper needs an explicit error bound or a different recursive reduced update. As a control, run the same experiment with r = rank(X_k) to verify that the exact full-rank update matches batch least squares, isolating the proof issue from the truncation issue.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central algorithmic claim is that a reduced-order DMD operator can be maintained online. Equations (33) and (38) are not derived from a rank-r minimization; they are the exact rank-n recursion (32)/(37) with A_k replaced by Abar_k and with vbar, Sigmabar, Ubar substituted for the full SVD factors. This recursion is exact only if the previous operator already equals the rank-r least-squares operator. Once a truncation is made, the error from earlier steps is carried forward, and no bound in Theorem 1 or 2 links sigma_thr to the difference between Abar_{k+1}^{trunc} and the rank-r truncated least-squares solution, nor to prediction error. The sentence 'If X_{k+1} is well-approximated by its projection' asserts, without proof, that exactness survives the substitution. The EEG comparison is against online DMD with a heuristic singular-data initialization, so it does not isolate this error. Separately, the proofs of the exact theorem are not clean: in Appendix C, equation (57) uses 'rows in V_sk are orthogonal,' which would imply V_sk,1 V_sk,1^* = I, false for an (n+1)x n matrix with orthonormal columns (scalar n=1 is a counterexample). The same issue appears in Appendix D. The exact formulas appear to be recoverable by a corrected argument, so the load-bearing unresolved piece is the truncated reduced-order update.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops incremental, SVD-based algorithms for online Dynamic Mode Decomposition (DMD) and DMD with control (DMDc) for time-varying systems, in both a weighted (discounted-data) and a sliding-window formulation. The authors derive recursive updates for the least-squares DMD operator from incremental SVD updates, state reduced-order variants obtained by truncating to the leading r singular values, and argue that the approach gives access to time-varying singular values, handles singular data matrices, and avoids storing the data matrix. The autonomous-system algorithms are evaluated on EEG error-related-potential data and compared with the online DMD of [35]; the non-autonomous algorithms are illustrated on randomly generated linear time-varying systems.","tokens_in":25363,"tokens_out":11547,"duration_ms":111491,"significance":"If the full-rank recursion is correct and the reduced-order extension can be rigorously justified, this is a potentially useful streaming DMD method: it avoids storing the data matrix, provides singular values for model-order selection, and can operate when the empirical covariance is singular. The out-of-sample EEG prediction protocol, the comparison with online DMD, and the public MATLAB code are concrete strengths, as is the fact that the full-rank weighted update reduces to a standard recursive least-squares-like correction. The significance is tempered, however, by the absence of any error analysis for the truncated updates and by flaws in the written proofs of the exact updates; both bear directly on the central claim that an 'appropriately reduced model' can be maintained online.","major_comments":[{"comment":"The reduced-order update is asserted but not derived from a rank-r least-squares problem. Equations (33) and (38) are obtained by replacing the full SVD factors in the exact recursion (32)/(37) with their leading-r counterparts and replacing A_k by \\bar A_k, but this substitution is not exact once a previous truncation has been made: the error from earlier steps is carried forward, and no bound in Theorem 1 or 2 links the threshold \\sigma_thr (or the rank r) to either the difference between \\bar A_{k+1}^{trunc} and the truncated least-squares operator or to future prediction error. The sentence 'If X_{k+1} is well-approximated by its projection' is an assumption, not a proof of exactness. The same gap appears in the DMDc analogues in Theorems 3 and 4. This is load-bearing for the abstract's claim that the algorithms 'allow us to determine an appropriately reduced model at each time.' I ask the authors either to provide a rigorous error analysis of the truncated recursion or to reframe the reduced-order claims as a heuristic and add experiments that directly quantify the truncation error.","section":"§4, Theorem 1 (Eq. 33) and Theorem 2 (Eq. 38)"},{"comment":"The proofs of the exact updates contain false orthogonality identities. In Appendix C, V_sk = [V_sk,1; v_sk,2] is an (n+1) x n matrix with orthonormal columns, so it satisfies V_sk^* V_sk = I; the step at Eq. (57) uses V_sk,1 V_sk,1^* = I and V_sk,1 v_sk,2^* = 0, which are not implied and are in fact false (for n = 1, V_sk is a unit 2-vector, and V_sk,1 V_sk,1^* = |a|^2 is not 1 unless |a|=1). The same problem appears in Appendix D, where V\\'sk V\\'sk^* = I and V_hatsk,1 V_hatsk,1^* = I are used even though these matrices are generally projections, not identities, when the number of columns is smaller than the row dimension. The final identities, e.g. Eq. (58), may well be recoverable by a corrected argument starting from S_k V_sk = U_sk \\Sigma_sk in the full-rank weighted case, but the written proofs are not valid as they stand and must be rewritten.","section":"Appendix C (Eq. 57) and Appendix D (Eq. 64)"},{"comment":"The complexity claim 'O(nr + r^3)' and the conclusion that the incremental approach is better when 'r < n^{2/3}' are not justified by the equations in the paper. The update in (32) computes A_k x_{k+1} and adds a rank-one correction to an n x n matrix, which is O(n^2) for a dense A_k; even the projected update (34) requires \\bar U_{k+1}^* \\bar A_k \\bar U_{k+1}, which is not simply the stored reduced operator \\tilde A_k unless additional structure is specified. The authors should state exactly which matrices are stored and updated, and give an operation count for those objects, before claiming a complexity advantage over streaming DMD or online DMD.","section":"§4, paragraph after Eq. (39)"},{"comment":"The EEG comparison with online DMD is partly confounded by the baseline's initialization. In the singular-data case, the authors initialize online DMD with the zero matrix and P_init = alpha I, as suggested in [35], so the poor performance of online DMD in Figures 3–5 may reflect this heuristic initialization rather than an intrinsic advantage of the incremental SVD update. The experiment on well-conditioned raw EEG data (Figure 6), where the two methods perform similarly, supports the authors' own explanation that the difference is due to the ill-conditioned dataset, but the text should state this limitation more prominently and avoid presenting the singular-case comparison as a direct algorithmic superiority result.","section":"§5, Figures 3–6 and accompanying text"}],"minor_comments":[{"comment":"In the windowed cost function (8), the weight of the oldest sample x_{k-w+1} should be rho^{w-1}, not rho^k as written in the displayed definition of X_k; this appears to be a typo that should be fixed for consistency with the subsequent weighted case.","section":"§2.4, Eq. (14)"},{"comment":"The reduced-order quantities \\bar v_{sk,2} and \\bar v_{\\hat sk,2} are not explicitly defined. The authors should specify whether they are the bottom block rows of the right singular vectors of the truncated incremental SVD update, and state the dimensions of all reduced-order factors.","section":"§4, Theorem 1 and Theorem 2"},{"comment":"The symbol y_k is overloaded: it denotes both the recorded EEG value and the arguments of max and min in the denominator. Using different symbols for the bounds would remove ambiguity.","section":"§5, Eq. (40)"},{"comment":"When w < n, the definition q = w is used, but the decremented matrix \\'chi_k has w - 1 columns, so its rank can drop below q. The treatment of zero singular values and the dimensions of U_{\\'chi_k} in this case is not discussed and should be clarified.","section":"Appendix B, Proposition 2"},{"comment":"In the display following the definition of the reduced DMDc model, the formula for \\tilde B_c uses unbolded V_c(\\Sigma_c)^{-1} U_{cb}^* without bars, while the neighboring formulas use barred quantities; this notation should be made consistent.","section":"§2.2, reduced DMDc formulas"}],"recommendation":"major_revision","confidential_remarks":"The paper is worth pursuing: the exact full-rank recursion appears algebraically recoverable from a corrected argument, the code is provided, and the EEG study is thoughtful. The main risk is the unproved truncated update, which is central to the claimed contribution of maintaining an appropriately reduced model online; that gap, together with the invalid orthogonality steps in the appendices, requires substantial revision before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is useful and modest: replace the Sherman-Morrison update in online DMD with incremental SVD updates. That gives access to singular values at each step and handles singular data matrices, which is a real advantage over [35] in exactly the regime where EEG data after filtering is rank-deficient. The full-rank recursive formulas in Theorems 1-4 are algebraically plausible; the scalar case matches recursive least squares, and the code is on GitHub. The EEG predictions are out-of-sample, which is more than many papers in this area do.\n\nThe soft spots are not fatal, but they are real. First, the proofs of the exact updates contain a false identity: Appendix C (and similarly D) claims that rows of V_sk being orthogonal implies V_sk,1 V_sk,1^* = I, which is not true for an (n+1)x n matrix with orthonormal columns. The stress-test note is correct about that. The formulas look recoverable by a corrected argument, but as written the derivation doesn't go through. Second, the reduced-order updates in (33) and (38) are asserted, not derived. The paper says \"if X is well-approximated by its projection\" and then substitutes truncated SVD factors into the exact recursion. That is a heuristic, and there is no error bound linking sigma_thr to the DMD operator error or to prediction error. This matters because \"determine an appropriately reduced model\" is a central claim. It may hold in practice, but the paper does not establish it. Third, the computational complexity comparison is stated loosely (\"when r < n^{2/3}\") without a detailed operation count, so I would read that as rough guidance rather than a rigorous result.\n\nThe EEG experiments are a reasonable demonstration, but the comparison against online DMD is a little lopsided: online DMD without singular-value access is handicapped on exactly this data. The raw-data experiment in Figure 6 partly addresses that, and the conclusion that the benefit shows up mainly in the ill-conditioned case is honest.\n\nWho is this for? Someone working on streaming DMD or BCI applications who needs a low-memory, singular-tolerant update. It is an incremental contribution, not a breakthrough. But it deserves a serious referee: the algorithmic idea is sound, the code is there, and the gaps are addressable in revision. I would accept it for peer review and ask for a corrected proof and some numerical evidence on the truncation error.\n\nRecommendation: send it out, with a clear request to fix the proof and quantify the reduced-order approximation.","headline":"Incremental SVD for online DMD is a sensible incremental step, but the proof has a real gap and the reduced-order claim is unquantified.","tokens_in":25849,"tokens_out":3155,"would_cite":true,"duration_ms":29541,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F15","93B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Incremental SVD updates let dynamic mode decomposition track time-varying systems online, even when the data matrix is singular.","keywords":["dynamic mode decomposition","incremental SVD","time-varying systems","online DMD","DMD with control","EEG error-related potentials","reduced-order modeling","streaming data"],"falsifier":"Construct a time-varying linear system with known $A_k$, stream its data through the weighted incremental DMD, and at each step compare the updated operator with the batch least-squares solution recomputed from all stored data; if the truncated update's error grows without bound as the singular-value threshold increases while the batch solution stays accurate, the unquantified truncation step is the cause. A sharper check is numerical: verify whether $U_{xk+1}\\Sigma_{xk+1}-\\rho U_{xk}\\Sigma_{xk}V_{sk,1}=x_{k+1}v_{sk,2}$ still holds when singular values below $\\sigma_{\\mathrm{thr}}$ are discarded.","tokens_in":24830,"feed_emoji":"🧠","tokens_out":6255,"duration_ms":56226,"temperature":0.7,"pith_summary":"This paper develops streaming algorithms for dynamic mode decomposition (DMD) that maintain a time-varying low-dimensional linear model of high-dimensional data as new measurements arrive. The central idea is to replace the Sherman-Morrison matrix-inverse update used by earlier online DMD with incremental SVD updates, so the singular values of the data matrix are available at every step. That availability lets the user truncate the model to a reduced order at each time and keeps the algorithm well-defined even when the data matrix is singular. The authors give one weighted (discounting old data) and one windowed (sliding window) version, extend both to systems with control inputs (DMDc), and test the autonomous versions on EEG recordings of error-related potentials.","feed_headline":"Streaming DMD now tracks time-varying systems on singular data","feed_subtitle":"Incremental SVD updates keep a reduced-order linear model current at every step, even when the data matrix is singular.","key_machinery":"The carrying mechanism is the incremental singular value decomposition of a data matrix with a new column appended. For a weighted update, the new SVD factors are obtained from the SVD of the small broken-arrow matrix $S_k=[\\Sigma_{xk}\\ \\rho^{-1}U_{xk}^*x_{k+1}]$, which gives the rank-one correction term $v_{sk,2}\\Sigma_{xk+1}^{-1}U_{xk+1}^*$; the windowed version first removes the oldest column with a similar low-rank modification and then appends the newest. The key identity used in both proofs is $U_{xk+1}\\Sigma_{xk+1}-\\rho U_{xk}\\Sigma_{xk}V_{sk,1}=x_{k+1}v_{sk,2}$, which converts the difference between old and new data matrices into the outer product that drives the operator update. This machinery makes the DMD update cost $O(nr+r^3)$ per step for a reduced order $r$, independent of the total data length.","core_discovery":"The paper claims that the time-varying DMD operator can be updated in rank-one form: $A_{k+1}=A_k+(y_{k+1}-A_k x_{k+1})v_{sk,2}\\Sigma_{xk+1}^{-1}U_{xk+1}^*$, where the correction term comes from the incremental SVD of the augmented data matrix. Because the update is built on the SVD rather than on the inverse covariance, it continues to work when $X_kX_k^*$ is singular, a case where the Sherman-Morrison-based online DMD requires a heuristic initialization. The same construction yields reduced-order projected operators by keeping only singular values above a threshold, and analogous updates for the pair $(A_k,B_k)$ in DMD with control. On preprocessed EEG data, the SVD-based algorithms reconstruct and predict error-related potentials with lower normalized RMS error than the online DMD baseline, and the dominant DMD modes align with frontal-lobe activity during erroneous events.","pith_inferences":["A testable extension the authors leave implicit: feed the same incremental SVD machinery through a nonlinear observable map (extended DMD style) and compare tracking on the same time-varying systems; the singular-value access would carry over, but the truncation-error question would become more acute.","The complexity comparison suggests incremental DMD is preferable to Sherman-Morrison online DMD when the reduced order $r$ satisfies $r