{"id":"0be8c86e-8d9a-4d90-8762-2ef892dee643","arxiv_id":"2507.03770","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors propose esDMD, a single-basis streaming DMD variant, but the central theorem is false and the pseudocode has an apparent update error.","lead":"This paper describes a faster variant of streaming dynamic mode decomposition that keeps one basis instead of two, aiming for a constant-factor speedup. The supplied proof that one basis always suffices is incorrect, and the algorithm text appears to mishandle coordinate updates after basis compression.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is false: exact equality span(Q_k)=span(X_k)=span(Y_k) fails at k=1 unless x_2 is parallel to x_1, and fails at any later step where y_{k+1} introduces a new direction. The single-basis claim is unsupported without an explicit shift-invariant subspace assumption.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing defect: the simultaneous span equality is not a theorem but a shift-invariance/low-rank condition. My analysis confirms this and locates the precise internal contradiction: in the induction step, since x_{k+1}=y_k and y_k in span(Qk)=span(Xk), the column space of X cannot grow when a new snapshot is appended. Therefore any genuinely new direction in y_{k+1} makes span(Q_{k+1}) strictly larger than span(X_{k+1}), breaking the equality claimed in Theorem 1. This is an internal inconsistency, not merely a disagreement with external consensus. The concrete test with a diagonal linear system makes the failure visible at k=1 and k=2. The reader's remaining concerns about the Algorithm 1 coordinate-rotation error, missing quantitative error metrics, and weak efficiency evidence are secondary but do not change the verdict: the central claim is not supported as stated. Since the reader already rejected the paper, my stress-test confirms that rejection should stand. No change to the verdict is needed, though a revised version that states the low-rank/shift-invariance assumption explicitly, corrects the proof, and fixes the update step could make the contribution conditionally acceptable.","tokens_in":8907,"tokens_out":7189,"duration_ms":79620,"concrete_test":"Run Algorithm 1 exactly as printed on the linear system n=3, A=diag(1,2,3), x1=[1,1,1]^T, x_i=A x_{i-1}, y_i=x_{i+1}, with r=3 and epsilon=0. At each i compute the residuals r_X(i)=||(I-Q_iQ_i^T)X_i||_F and r_Y(i)=||(I-Q_iQ_i^T)Y_i||_F. Theorem 1 predicts both residuals are zero at every step. For i=1, [x1,y1]=[x1,x2] has rank 2, so Q1 has two columns while X1 is one-dimensional, giving r_X(1)>0. For i=2, x3 is linearly independent of span{x1,x2}, so the algorithm appends a direction and Q2 has three columns while X2 has rank two, giving r_X(2)>0. If the residuals are nonzero, the theorem is falsified; if they are zero, the implementation must be modifying the basis or the data in a way not described in the paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central justification for esDMD is Theorem 1, which asserts that a single orthonormal basis Qk simultaneously spans Xk and Yk for every k. This is not true for generic streaming data. At k=1, X1=[x1] and Y1=[y1]=[x2]; the initialization Q1=QR([x1 y1]) gives span(Q1)=span{x1,x2}, so span(Q1)=span(X1) holds only if x2 lies in span{x1}. The induction step has the same flaw in sharper form: assuming span(Qk)=span(Xk)=span(Yk), we get yk in span(Qk)=span(Xk). Since x_{k+1}=yk, this gives span(X_{k+1})=span(Xk) exactly. If y_{k+1} has a component orthogonal to Qk, the algorithm appends pk+1, making span(Q_{k+1})=span(Qk,pk+1) strictly larger than span(X_{k+1}); the claimed double equality is lost exactly when a new direction appears. Thus Theorem 1 is only valid under an unstated assumption that the entire trajectory lies in a fixed r-dimensional shift-invariant subspace. A generic autonomous trajectory, such as x_i=diag(1,2,3)^{i-1}x1, violates it already at k=1 or k=2. Because the constant-factor efficiency claim and the 'no loss of accuracy' statement rest on this theorem, the central theoretical support for esDMD collapses until the low-rank/shift-invariance hypothesis is made explicit and the algorithm is shown to preserve it under POD compression.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an efficient streaming dynamic mode decomposition (esDMD) that maintains a single orthonormal basis instead of the two bases used in sDMD, claiming a constant-factor reduction in computational cost and memory with no loss of accuracy. The central theoretical result, Theorem 1 in Section III, asserts that one basis Q_k can simultaneously span X_k and Y_k for every time step. The paper also reports numerical comparisons on a nonlinear oscillatory system and a Kuramoto oscillator network, and it provides a code repository.","tokens_in":9240,"tokens_out":10344,"duration_ms":119813,"significance":"If the single-basis claim were correct, the paper would offer a useful simplification of sDMD with concrete constant-factor savings and would be of genuine interest to the streaming-DMD community. The paper is clearly written, identifies a real redundancy in maintaining two Gram matrices for the same stream, and provides reproducible code and experiments on two canonical systems. However, Theorem 1 is false for generic data, and the numerical validation does not quantify accuracy. The central theoretical and empirical support for the 'no loss of accuracy' claim is therefore not established, so the contribution as stated cannot be accepted.","major_comments":[{"comment":"Theorem 1 is false as stated. For k=1, Q1=QR([x1 y1]) spans the space generated by x1 and y1=x2, whereas span(X1)=span({x1}) and span(Y1)=span({x2}); unless x2 is parallel to x1, no single vector can equal both column spaces. The induction step repeats the error: after appending the new direction p_{k+1} of y_{k+1}=x_{k+2}, the proof claims span(Q_{k+1})=span(X_{k+1})=span(Y_{k+1}), but y_{k+1} is not in X_{k+1}, so the two spans differ in general. The theorem therefore holds only under an unstated assumption that the trajectory lies in a fixed shift-invariant subspace. Since Theorem 1 is the paper's justification for replacing two bases with one and for the 'no loss of accuracy' claim, the central theoretical support of esDMD is invalid.","section":"Section III, Theorem 1 and Eq. (8)"},{"comment":"Algorithm 1 has a stale-coordinate bug in the compression branch. When cols(Q'_i)>r, the basis is rotated to Qi=Q'_i W', but the stored tilde y_- is expressed in the pre-rotation basis; the subsequent update tilde xi <- tilde y_- (which is taken because new_direction is false in that branch) therefore uses coordinates inconsistent with Qi, and the updates to GXi and Ci mix rotated and unrotated quantities. Additionally, the branch cols(Q'_i)<=r uses G''_Xi, G''_Yi, and C''_i before they are assigned, so the pseudocode is undefined in that branch. Even under a valid low-rank assumption, these issues make the algorithm as written non-reproducible.","section":"Section III-A and Algorithm 1"},{"comment":"The numerical validation reports only final-time eigenvalue and frequency overlays together with per-iteration execution times. No quantitative accuracy metric (e.g., eigenvalue error, mode amplitude error, or prediction error), no time-resolved accuracy curve, and no repeated-trial statistics for the spectra are provided. The claim that esDMD 'does not compromise accuracy' is therefore not supported even empirically; the two test systems are low-dimensional and may satisfy the shift-invariant subspace condition that Theorem 1 would require.","section":"Section IV, Figs. 1 and 2"}],"minor_comments":[{"comment":"The statement of Theorem 1 uses r both as an unspecified rank in Qk in R^{n x r} and as a bound in the induction hypothesis cols(Qk)<r; the relationship between these two uses is never defined.","section":"Section III, Theorem 1 proof"},{"comment":"The line 'y_parallel_i <- Qi-1 (Q_i^T yi)' appears to be a typo for Q_{i-1} Q_{i-1}^T y_i; as written, the dimensions are inconsistent.","section":"Algorithm 1"},{"comment":"Algorithm 2 calls UPDATE BASIS((xi, yi), i), but Algorithm 1 declares the procedure with the single argument (xi, yi); the calling convention should be made consistent.","section":"Algorithm 2"},{"comment":"The static bytecode-instruction counts (approximately 350 versus 515) are implementation-specific and are not a substitute for a precise per-iteration operation count, especially since the dominant costs depend on r and n.","section":"Section IV-C"}],"recommendation":"reject","confidential_remarks":"For the editor: I do not see a way to repair Theorem 1 within the current scope; the paper would need to be reframed as an approximate streaming method under an explicit low-rank/shift-invariance assumption, with corrected update equations, error bounds, and quantitative experimental validation. With that reframing and the existing code, a future submission could be worth reconsidering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nMain takeaway: the paper's central theorem is false as stated, and the algorithm has a missing assignment in one branch, so the current preprint does not support its accuracy and efficiency claims. The idea behind it is not crazy, but it needs a rewrite.\n\nWhat is actually new: the reformulation of streaming DMD with a single basis, initialized by QR of the first snapshot pair, is a modest but real variant. It avoids maintaining two bases, which is a legitimate constant-factor saving. The paper is clearly written, engages the relevant literature, and the code is available. The two test problems are standard.\n\nWhere it falls short: Theorem 1 claims a single basis Q_k spans both X_k and Y_k for all k. The induction fails exactly when y_{k+1} has a component orthogonal to Q_k. At that step, Q_{k+1} spans a strictly larger space than X_{k+1}, because x_{k+1}=y_k already lies in Q_k. So the equality breaks, and the 'no loss of accuracy' claim is unsupported. The theorem only holds under an unstated shift-invariant subspace assumption, which is restrictive for generic streaming data.\n\nThe pseudocode in Algorithm 1 has a concrete bug: in the branch where the basis grows without compression, it uses G''_Xi, G''_Yi, C''_i that are never assigned. So the algorithm is not reproducible as written. The coordinate update after compression is also under-specified.\n\nThe experiments compare spectra visually and give timing bars, but no quantitative eigenvalue or reconstruction errors, and the timing evidence is bytecode instruction counts. That is not enough to substantiate the claimed efficiency gain.\n\nThat said, the core idea is salvageable. If the authors restate the theorem as an approximation under an explicit low-rank/shift-invariance assumption, fix the pseudocode, and add quantitative error and timing comparisons, this could be a useful contribution to the streaming DMD niche.\n\nRecommendation: send to peer review. The flaws are serious but fixable, and the paper deserves a careful referee rather than a desk reject.\n\nRegards,","headline":"The main theorem is false as stated and the pseudocode has a missing assignment, but the single-basis idea is salvageable with a rewrite.","tokens_in":9791,"tokens_out":6244,"would_cite":false,"duration_ms":66553,"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 argues that the streaming dynamic mode decomposition can run on a single orthonormal basis instead of two, cutting computation and storage by a constant factor without sacrificing accuracy.","keywords":["dynamic mode decomposition","Koopman operator","modal analysis","nonlinear dynamical systems","stream processing","orthonormal basis","Gram matrix","proper orthogonal decomposition"],"falsifier":"Run esDMD on a trajectory where new directions keep entering after the basis has been compressed to r columns, and at each step test whether every column of Xk and Yk lies in span(Qk); if any non-negligible direction falls outside, the claimed equality of the two column spaces with the single basis does not hold.","tokens_in":8652,"feed_emoji":"⚡","tokens_out":9203,"duration_ms":96011,"temperature":0.7,"pith_summary":"The paper argues that the streaming dynamic mode decomposition (sDMD) can be reformulated so that only one orthonormal basis is updated online instead of two. The key observation is that in a streaming trajectory each new snapshot pair can be written as (yi-1, yi), so every new direction in the data enters through the y sequence and a single evolving basis can track both snapshot matrices. The authors prove a theorem stating that a basis Qk exists with span(Qk)=span(Xk)=span(Yk) at every step, and use it to build esDMD, which keeps the same Gram-matrix machinery but halves the per-step basis bookkeeping. Numerical tests on a nonlinear oscillator and a hundred-oscillator Kuramoto network show the same dominant eigenvalues and frequencies as sDMD and batch DMD at a lower measured execution time. If correct, this means online modal analysis can run at a constant-factor lower cost with no loss of fidelity.","feed_headline":"One streaming basis replaces two in dynamic mode decomposition","feed_subtitle":"Reformulated streaming dynamic mode decomposition keeps accuracy while using one orthonormal basis instead of two.","key_machinery":"The load-bearing object is the single shared orthonormal basis Qk, initialized as QR([x1 y1]) and updated by appending the normalized leftover ek+1 = yk+1 - Qk Qk^T yk+1 when its size exceeds tolerance, then rotated by the top r eigenvectors of the padded Gram matrix G'_{Yk}. The identity that carries the argument is the shift relation yi = xi+1 (equivalently xi = yi-1), which makes every new column of Xk an old column of Yk and thereby lets one basis span both data matrices. The low-dimensional operator takes the form \\tilde{A}_k = C_k G_{X_k}^+, with Ck and GXk accumulated from projected snapshot coordinates, so no explicit pseudoinverse of the full data matrix is needed.","core_discovery":"On its own terms, the paper's central discovery is Theorem 1: for an autonomous discrete-time system where streaming snapshot pairs satisfy yi = xi+1, there exists a single orthonormal basis Qk with rank at most r such that span(Qk)=span(Xk)=span(Yk) for every time step. The proof builds Qk inductively by Gram-Schmidt, starting from the QR factorization of the first pair [x1 y1], and appending only the component of the next y snapshot that is not already captured; because the next x snapshot equals the previous y snapshot, the authors conclude no separate x-basis is needed. Consequently the two-basis update of sDMD is redundant, and esDMD computes the same reduced operator \\tilde{A}_k = C_k G_{X_k}^+ from one Gram matrix, compressed by a proper orthogonal decomposition step when the basis exceeds r columns. The paper presents this as a constant-factor reduction in computational complexity and memory with accuracy matching the two-basis method.","pith_inferences":["The equality span(Qk)=span(Xk)=span(Yk) in Theorem 1 is stronger than what the update actually maintains; what the construction guarantees is that span(Qk) continues to contain span(Yk) and, through the shift relation, span(Xk). If only this weaker containment holds, the constant-factor gain survives but the accuracy guarantee becomes approximate and trajectory-dependent.","A natural extension is to quantify accuracy loss when new directions appear after compression; one could instrument esDMD to report the residual norm ||yk+1 - Qk Qk^T yk+1|| and stop compressing when it exceeds a threshold.","The same single-basis trick should apply to online Koopman estimation and other two-matrix streaming algorithms, since any method maintaining separate bases for X and Y on shift-structured data inherits the redundancy.","The measured constant factor depends on the reference implementation; an optimized sDMD could close some of the gap, so future work should benchmark theoretical operation counts rather than static bytecode instructions."],"forward_implications":["Per-step work drops by a constant factor, so the same streaming hardware can track dynamic modes at a higher sampling rate or with more margin for other processing.","Memory footprint for the online basis decreases because one Gram matrix and one basis are stored instead of two.","The reformulation inherits the same POD compression step, so existing sDMD implementations can be switched to the single-basis version with minimal algorithmic change.","Because the reduced operator is identical in form, all downstream uses of sDMD mode estimates, such as prediction, modal analysis, and Koopman-type filters, carry over unchanged."],"supporting_citations":[{"why":"Introduces the two-basis streaming DMD method that this paper reformulates into a single-basis version.","marker":"[9]"},{"why":"Provides the batch DMD algorithm used as the accuracy benchmark in the numerical experiments.","marker":"[6]"},{"why":"Its joint column-space basis computation inspires the QR initialization of the single shared basis.","marker":"[21]"},{"why":"Supplies the least-squares formulation that the reduced streaming operator inherits.","marker":"[11]"}],"fun_headline_variants":["Single-basis streaming DMD cuts cost, keeps accuracy","One orthonormal basis enough for streaming DMD","Streaming DMD: one basis, same accuracy, less work","Efficient streaming DMD drops to one Gram matrix","Reformulated streaming DMD needs single basis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The premise the whole argument stands on is that a single rank-r orthonormal basis can keep representing the column spaces of both the snapshot matrix and its one-step shift throughout the stream, with the discarded directions contributing no accuracy.","fun_headline_variants_meta":{"raw":{"variants":["Single-basis streaming DMD cuts cost, keeps accuracy","One orthonormal basis enough for streaming DMD","Streaming DMD: one basis, same accuracy, less work","Efficient streaming DMD drops to one Gram matrix","Reformulated streaming DMD needs single basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1410,"prompt_tokens":781,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":397,"completion_tokens_details":{"reasoning_tokens":550}},"tokens_in":397,"tokens_out":629,"duration_ms":7581,"temperature":1.0,"reasoning_tokens":550,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:03:28.568629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run esDMD on a trajectory where new directions keep entering after the basis has been compressed to r columns, and at each step test whether every column of Xk and Yk lies in span(Qk); if any non-negligible direction falls outside, the claimed equality of the two column spaces with the single basis does not hold.","supporting_citations":[{"cited_title":"Dynamic mode decomposition for large and streaming datasets,","cited_arxiv_id":null,"evidence_quote":"Introduces the two-basis streaming DMD method that this paper reformulates into a single-basis version."},{"cited_title":"Dynamic mode decomposition of numerical and experi- mental data,","cited_arxiv_id":null,"evidence_quote":"Provides the batch DMD algorithm used as the accuracy benchmark in the numerical experiments."},{"cited_title":"On dynamic mode decomposition: Theory and applications,","cited_arxiv_id":null,"evidence_quote":"Its joint column-space basis computation inspires the QR initialization of the single shared basis."},{"cited_title":"Robust dynamic mode decomposition,","cited_arxiv_id":null,"evidence_quote":"Supplies the least-squares formulation that the reduced streaming operator inherits."}],"review_version":1}