An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling

Mustaffa Alfatlawi; Vaibhav Srivastava

REVIEW 4 major objections 5 minor 1 cited by

An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Incremental SVD updates let dynamic mode decomposition track time-varying systems online, even when the data matrix is singular.

desk verdict Incremental SVD for online DMD is a sensible incremental step, but the proof has a real gap and the reduced-order claim is unquantified. read the letter →

arxiv 1908.01047 v3 pith:OYUESVQC submitted 2019-08-02 eess.SP math.OC

Mustaffa Alfatlawi , Vaibhav Srivastava This is my paper · ORCID

classification eess.SPmath.OC MSC 65F1593B30

keywords dynamicmodedecompositionincrementalSVDtime-varyingsystemsonlineDMDwithcontrolEEGerror-relatedpotentialsreduced-ordermodelingstreamingdata

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

The load-bearing premise is that, after truncating the SVD to the leading $r$ singular values, the exact algebraic identity used to derive the rank-one update still holds closely enough that the resulting reduced-order DMD operator is accurate; the paper provides no error bound linking the truncation threshold to operator error (equations (33) and (38)).

Editorial extensions

If this is right

A streaming implementation can maintain a time-varying DMD model without storing the full data matrix, using only the current operator and the incremental SVD factors.
The algorithm keeps working when the data covariance is singular, a regime where Sherman-Morrison updates need a heuristic pseudo-inverse initialization.
Users can select or adapt the reduced model order online by thresholding singular values, and the projected operator gives low-dimensional prediction at each time step.
For systems with known exogenous inputs, the DMDc extension tracks both the dynamics matrix and the input matrix online, with better prediction than ignoring the control term in the paper's numerical example.
On EEG data, the resulting models capture the time-varying error-related potential and produce dominant spatial modes consistent with frontal-lobe activity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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 $r<n^{2/3}$; for larger $r$, the $O(nr+r^3)$ update may lose that advantage.
The lack of a bound linking the singular-value threshold to operator error means practitioners must tune $\sigma_{\mathrm{thr}}$ empirically; a natural next step would be a perturbation bound for the truncated update, which would turn the heuristic threshold into a provable accuracy knob.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Who 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.

Recommendation: send it out, with a clear request to fix the proof and quantify the reduced-order approximation.

Referee Report

4 major / 5 minor

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.

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 (4)

[§4, Theorem 1 (Eq. 33) and Theorem 2 (Eq. 38)] 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.
[Appendix C (Eq. 57) and Appendix D (Eq. 64)] 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.
[§4, paragraph after Eq. (39)] 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.
[§5, Figures 3–6 and accompanying text] 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.

minor comments (5)

[§2.4, Eq. (14)] 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.
[§4, Theorem 1 and Theorem 2] 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.
[§5, Eq. (40)] 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.
[Appendix B, Proposition 2] 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.
[§2.2, reduced DMDc formulas] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the incremental DMD updates are algebraic consequences of the least-squares DMD definition, and the EEG predictions are evaluated on future samples not used to fit the model.

full rationale

The paper's core derivations (Theorems 1–4) are self-contained algebraic manipulations of the least-squares DMD / DMDc definitions (Eqs. 15, 18) combined with external incremental SVD updates from Refs. [1,2,26]. The update formulas (32), (37), (43), and (46) are obtained by substituting the incremental SVD factors into the definition of the next DMD operator and simplifying; they do not assume the target result. No fitted parameter is renamed as a prediction: the EEG experiments stream data, update the operator from past and current measurements, and then forecast a 64-sample future window that was not used in the update at that step. The hyperparameters ρ, w, and σthr are selected by the authors, but this is model selection, not a derivation step that reduces to the prediction target. The paper does not rely on self-citation: Ref. [35] (Online DMD) is by different authors, and the incremental SVD machinery comes from independent literature. The main caveats are correctness/approximation concerns rather than circularity: the truncated updates (33) and (38) are asserted by the statement 'If X_{k+1} is well-approximated by its projection' without an error bound linking σthr to DMD operator error, and the proofs of Theorems 1 and 2 contain orthogonality identities (e.g., Vsk,1Vsk,1∗ = I) that are false for the indicated matrix dimensions. These issues undermine the rigor of the reduced-order claim, but they do not make the derivation circular. The derivation is not equivalent to its inputs by construction, and no step reduces a predicted quantity to a fitted parameter.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The algorithms do not postulate new physical entities. The main free choices are discount factor, window length, singular value threshold, and initial window size. The most fragile mathematical assumption is that the truncated SVD update inherits the exact update without a quantified error bound.

free parameters (4)

Discount factor rho = 0.1, 0.2, 0.4, 0.8 in EEG experiments; 0.9 in control experiments
User-chosen forgetting factor in the weighted cost functions (8) and (11); performance varies with rho, so it is a tuning parameter.
Window length w = 512 samples for EEG; 40 for synthetic DMDc
Sliding-window length in the windowed cost function; chosen by hand and affects adaptivity and error.
Singular value threshold sigma_thr = 0.01 and 0.001 for EEG
Threshold determines the reduced order r (26 to 33 modes in the EEG study); chosen by hand, not estimated from the data model.
Initial window length w0 = 512 samples, with 128 in Appendix F
Initial DMD model is fit to a pre-event window; affects initialization and the reported prediction comparisons.

assumptions (4)

domain assumption High-dimensional data are generated by an unknown low-dimensional time-varying linear dynamics.
Stated in Section 2.3.1 and used to justify the DMD approximation in the EEG application.
ad hoc to paper Truncating the SVD to the leading r singular values leaves the update formula (33) valid without an error term.
Equations (33) and (38) are asserted if X_{k+1} is well-approximated, but no error bound or condition relating truncation error to update error is provided.
standard math Minimum-norm least-squares solution properties of the reduced SVD hold for singular data matrices.
Used in Lemma 1(ii) and Lemma 2(ii) to define the minimum-norm DMD and DMDc operators when the data matrix is singular.
domain assumption Preprocessing with CAR and 1-10 Hz band-pass filtering makes the EEG data matrix singular while preserving the ErrP signal.
Section 5; the claimed advantage over online DMD relies on this, and the raw-data experiment shows both methods perform similarly when the data matrix is well-conditioned.

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Pith. "Pith review of An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling." pith.science (2026). https://pith.science/paper/OYUESVQC

@misc{pith2026190801047,
  author       = {Pith},
  title        = {Pith review of: An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYUESVQC}},
  note         = {Machine review of arXiv:1908.01047}
}

read the original abstract

Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying high-dimensional data. For systems in which such underlying low-dimensional dynamics is time-varying, a time-invariant approximation of such dynamics computed through standard DMD techniques may not be appropriate. We focus on DMD techniques for such time-varying systems and develop incremental algorithms for systems without and with exogenous control inputs. We build upon the work in [35] to scenarios in which high dimensional data are governed by low dimensional time-varying dynamics. We consider two classes of algorithms that rely on (i) a discount factor on previous observations, and (ii) a sliding window of observations. Our algorithms leverage existing techniques for incremental singular value decomposition and allow us to determine an appropriately reduced model at each time and are applicable even if data matrix is singular. We apply the developed algorithms for autonomous systems to Electroencephalographic (EEG) data and demonstrate their effectiveness in terms of reconstruction and prediction. Our algorithms for non-autonomous systems are illustrated using randomly generated linear time-varying systems.

Figures

Figures reproduced from arXiv: 1908.01047 by the authors.

**Figure 2.** The average response to an event at t = 0: the mean ERP (confidence level = 95%) at the FCz channel (left panel) and the topographical view for brain activity across all channels (right panel). The top and middle panels show the patterns during the correct event and the erroneous event, respectively. The bottom panel shows that ErrP obtained by subtracting the signal associated with the correct event from that of th… view at source ↗

**Figure 3.** The mean of the normalized RMS prediction error computed over all iterations as well as the associated 95% [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 5.** Figure 5: However, if we choose a smaller initial window to initialize the DMD model, we observe [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 4.** Figure 4: Predicted ERP signal at channel FCz using incremental DMD with [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Normalized RMS error for the predicted ERP signal at channel FCz for correct events (left panel) and [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: The predicted ERP signal at channel FCz based on well conditioned EEG datasets using incremental DMD [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: Topographical views for the real part part of the 4 dominant DMD modes during correct events using threshold [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: Topographical views for the real part of the 4 dominant DMD modes during erroneous events using threshold [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: The left panel show the continuous time DMD eigenvalues for [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Reconstructed ERP signal at channel FCz using incremental DMD with [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: Normalized RMS error for the reconstructed ERP signal at channel FCz for correct events (left panel) and [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]

**Figure 12.** Figure 12: The Frobenius norm of prediction error for a future-window of 10 samples using (a) weighted incremental [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

**Figure 13.** Figure 13: Normalized RMS error for a future-window of 64 samples of EEG states at channel FCz using incremental [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]

**Figure 14.** Figure 14: Normalized RMS error of ERP prediction using weighted incremental DMD with initial window of 128 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]

**Figure 15.** Figure 15: Normalized RMS error of ERP prediction using windowed incremental DMD with different window sizes [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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OPIDMD combines online proximal gradient descent with physics-informed matrix constraints to learn time-varying linear models of dynamical systems, claiming state-of-the-art short-term prediction on noisy benchmarks.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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