{"id":"7864277b-1b5f-419a-9b23-dfa73e0d42ed","arxiv_id":"2501.07714","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quantized EDMD predictor estimates converge, as data grows, to a regularized least-squares solution of the unquantized problem, with finite-data errors of order ϵ.","lead":"Dither quantization of state and control snapshots changes the linear predictor that Koopman-based system identification learns from data. This paper proves that in the large-data limit the quantized estimate matches a regularized least-squares solution, with the regularization strength set by the quantization step size, and validates the effect on four control benchmarks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 3 in Theorem 1 is violated by the thin-plate-spline dictionary used in the paper's own pendulum and Van der Pol experiments, so the stated large-data regularization result does not cover the numerical validation.","rationale":"I agree with the reader's weakest_assumption: the analyticity condition is the most load-bearing gap because Theorem 1 is the paper's principal contribution and the thin-plate-spline dictionary in the pendulum and Van der Pol experiments is explicitly non-analytic. The proof of Theorem 1 depends on Taylor expansion and interchange of limits, so Assumption 3 is not a removable technicality within the argument as written. The finite-data Theorem 2 proof also contains an algebraic step I would double-check: the expression for G_ε obtained from the Woodbury expansion does not obviously match the direct expansion of (Φ+ + Φ+_ε)(Ψ+Ψ_ε)ᵀ[(Ψ+Ψ_ε)(Ψ+Ψ_ε)ᵀ]⁻¹, and the invertibility of the O(ε) matrix Ξ_ε is asserted without proof. However, the large-data regularization result is the headline, so the analyticity mismatch is the single concern that most directly threatens the central claim. I am not treating the use of non-analytic RBFs as evidence of error in the simulations; rather, the theorem's hypotheses and the experiments' dictionaries are not aligned. A concrete computational check of distances to centers, plus an analytic-dictionary control, would settle whether the mismatch is substantive or merely cosmetic. Since the reader already reached CONDITIONAL on this basis, my stress-test does not change the verdict.","tokens_in":23643,"tokens_out":6116,"duration_ms":68013,"concrete_test":"For the pendulum experiment in Section V-A, compute d_min = min over snapshots t and centers j of ‖x_t − c_j‖ and compare it with ε/2 for each word-length b = 4,...,15 (ε from Eq. 14). If d_min < ε/2 for any b, Assumption 3 is violated. Then re-run the same identification pipeline with a Gaussian RBF dictionary, which is analytic everywhere and satisfies Assumption 3, keeping centers and widths matched. If the relative-error slopes and the regularized-bias direction in Fig. 1(a)–(c) are unchanged, the practical claim survives under weaker assumptions; if they change materially, the reported validation lies outside the theorem's stated scope.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1: as T→∞, the dither-quantized EDMD estimate converges almost surely to the minimizer of the unquantized least-squares objective plus an O(ε²) regularization term. The proof in Appendix B expands the residual r(x_{t+1}+e^x_{t+1}, ξ_t+η_t) in an infinite Taylor series and interchanges the T→∞ limit with the infinite sum, which is justified only under Assumption 3 (Section IV-A): r must have an absolutely convergent Taylor series with radius of convergence larger than ε/2 on the data-collection region. However, in Section V-A the paper selects thin-plate-spline RBFs, ψ(x)=‖x−c‖²log‖x−c‖, which are only C¹ at c; the second derivative has a logarithmic singularity and no Taylor expansion exists at c. Since the 100 centers are drawn uniformly in the unit box and the trajectories explore the same box, the minimum distance from a snapshot to a center shrinks as T grows, so for fixed ε the radius condition will eventually fail; for finite T with thousands of samples it is already likely to fail for some center. Consequently, Theorem 1 is not applicable to two of the four reported case studies, and those experiments cannot be cited as validation of the theorem. This does not disprove the qualitative phenomenon, but it means the stated theorem and its numerical support are mismatched.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the effect of dither quantization on the EDMD-based lifted linear predictor for nonlinear control systems. The state and input snapshots are quantized before lifting, and the least-squares estimate of the predictor matrices [A,B] is compared with the unquantized estimate. The central theoretical claims are: (1) in the large-data limit T→∞, the quantized estimate converges almost surely to the minimizer of the unquantized quadratic objective plus an O(ε²) regularization term of the form tr(Gβ(ε))+tr(GᵀGΓ(ε)) (Theorem 1); (2) for finite T, the difference between the quantized and unquantized estimates is O(ε) (Theorem 2); and (3) when the observables themselves are quantized, the limiting objective is exactly the unquantized objective plus (ε²/12)(‖A‖²+‖B‖²), giving an explicit regularized least-squares recovery rule (Theorem 3). The paper also reports numerical experiments on a negatively damped pendulum, a Van der Pol oscillator, a bilinear DC motor, and a KdV PDE, with a Koopman-based linear MPC tracking problem, showing error scaling and cost degradation with decreasing word length.","tokens_in":23931,"tokens_out":7087,"duration_ms":74508,"significance":"If the main results are correct, the paper makes a useful conceptual contribution: dither quantization in EDMD is not merely unstructured noise but acts as a deterministic, dictionary-dependent implicit regularization in the large-data limit, with an explicit O(ε²) scaling that connects quantization resolution to prediction/control performance. This is potentially valuable for networked and resource-limited Koopman-based control. The paper is self-contained in its derivations and does not rely on black-box external results for the central theorems; it uses standard tools (Schuchman's condition, strong laws of large numbers) and provides an interesting special case in Section VI with an explicit closed-form regularity term. The experimental validation covers four different systems and includes both prediction and MPC, which is commendable. However, the stated Theorem 1 excludes the thin-plate-spline dictionaries used in two of the four experiments, and the proofs of Theorems 1 and 2 have gaps in the interchange of limits, uniform convergence, and control of inverse matrices.","major_comments":[{"comment":"The pendulum and Van der Pol experiments use a dictionary containing 100 thin-plate-spline RBFs ψ(x)=‖x−c‖²log‖x−c‖. This function is not C² at its center c and has no Taylor expansion at c. Since the centers are drawn uniformly in the unit box and the trajectories explore the same box, for any fixed ε the data will, with high probability, contain snapshots within distance ε/2 of a center as T grows; for the reported finite T with 200,000 snapshots and 100 centers this is already likely. Consequently Assumption 3, which requires the residual r to have an absolutely convergent Taylor series with radius of convergence larger than ε/2, is violated in exactly the experiments that are used to validate Theorem 1. The paper should either replace the thin-plate-spline dictionary with an analytic dictionary (e.g., Gaussian RBFs) or weaken Assumption 3 and rework Theorem 1 to cover non-smooth observables; as it stands, two of the four case studies cannot be cited as numerical support for the large-data theorem.","section":"§V-A, §V-B and §IV-A, Assumption 3"},{"comment":"The proof of Theorem 1 expands the residual in an infinite Taylor series and then interchanges the limit T→∞ with the infinite sum. Assumption 3 only states pointwise absolute convergence of the Taylor series for each fixed data point; it does not provide the uniform domination over t, over G, or over the tail k>n that is needed to interchange the order of the T-limit and the n-limit. Furthermore, Eq. (52) passes the argmin through the almost-sure limit of the objective functions without a uniform-convergence argument over the parameter set A×B. These are load-bearing steps for the paper's central result. The argument can likely be repaired with additional boundedness and domination assumptions (or by proving epi-convergence/consistency of the minimizers), but as written the proof of Theorem 1 is incomplete.","section":"Appendix B, Eq. (50)–(52)"},{"comment":"The finite-data proof writes ̅G = G − G(ΨΨᵀΞ_ε⁻¹ + I)⁻¹ + Π_ε(̅Ψ̅Ψᵀ)⁻¹ and then concludes ∥G_ε∥=O(ε) from ∥Ξ_ε∥=O(ε) and ∥Π_ε∥=O(ε). This conclusion requires uniform control of the inverses: the term (ΨΨᵀΞ_ε⁻¹ + I)⁻¹ contains Ξ_ε⁻¹, whose norm may grow like 1/ε, and the full-row-rank assumption on ̅Ψ does not by itself give a lower bound on the smallest singular value of ̅Ψ̅Ψᵀ that is uniform as ε→0. Without explicit bounds of the form ∥(ΨΨᵀΞ_ε⁻¹+I)⁻¹∥≤C and ∥(̅Ψ̅Ψᵀ)⁻¹∥≤C/λ_min, the claimed O(ε) estimate is not established. This is a essential gap in Theorem 2, which is the basis for the finite-data error scaling used throughout the experimental section.","section":"Theorem 2 proof, Eq. (22)–(23)"},{"comment":"Corollary 3 states that E[y_k(y_ℓ)³]=0 follows from pairwise uncorrelatedness of {y_t}; this implication is false in general, and the proof needs to use the specific independence structure of products of disjoint and overlapping quantization errors. Additionally, Corollary 4 invokes Kolmogorov's strong law (Lemma 5) for sequences a_t e_t; for the law to apply, the e_t must be independent of the coefficients a_t, not merely mean-zero and uncorrelated. The paper should explicitly assume that the dither noises are independent of the underlying trajectory and control sequence and then state the independence of the product sequence. As written, the applications of the strong law in Appendix B are not fully justified, although they are likely repairable.","section":"Appendix A, Corollaries 3–4"}],"minor_comments":[{"comment":"In Lemma 1, the phrase \"characteristic equation\" should be \"characteristic function\".","section":"§II-C"},{"comment":"In the fourth-moment computation, the text says \"the second inequality follows\" but the step is an equality; the wording should be corrected.","section":"Appendix A, Corollary 3 proof"},{"comment":"The definition of ̅U in the proof of Theorem 3 is inconsistent with the definition in Section III: it lists ũ1,...,ũ_{T−1} and omits the term ũ_T that appears in Φ+. The indexing should be harmonized.","section":"Appendix C, definitions of ̃Ψ and ̅U"},{"comment":"The paper would benefit from error bars or confidence intervals in the experimental figures; with 50 Monte Carlo realizations, the slopes and non-monotonicities discussed in Section V-B could be assessed more reliably.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The core idea is sound and the paper is likely to become a useful contribution after revision, but the main theorem's assumptions exclude the non-smooth dictionaries used in the paper's own experiments, and the proofs of Theorems 1 and 2 have genuine gaps in limit interchange and inverse-matrix control. I do not recommend rejection because the qualitative claims are plausible and the numerical trends match the predicted scaling; however, the authors should either prove the theorems under conditions that cover their experiments or modify the experiments to match the analyticity assumption, and they must fill the proof gaps. The reliance on the authors' own preprints [18], [19] is for background and does not create a circularity problem in the main derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is a legitimate incremental extension of the authors' own earlier work on dither quantization for DMD/EDMD, now covering control-affine systems with MPC. The headline result — dither quantization acts as a deterministic O(ε²) ridge regularization in the large-data limit and as an O(ε) perturbation in the finite-data regime — is exactly the kind of practical insight practitioners want. The experiments on four systems show the predicted logarithmic error scaling versus word-length, and the observation that coarser quantization sometimes improves MPC cost (Van der Pol, b=8 and b=4) is reported honestly and attributed to regularization. I buy the qualitative phenomenon.\n\nThe proof gaps are real and not cosmetic. Theorem 1's proof expands the residual in a Taylor series and interchanges limits; Assumption 3 requires absolute convergence with radius greater than ε/2, but the paper's own pendulum and Van der Pol experiments use thin-plate-spline RBFs, ψ(r)=r² log r, which are not analytic at the centers. Since centers are drawn uniformly in the same box the trajectories explore, the assumption eventually fails. So the two flagship experiments are not actually a validation of Theorem 1 as stated. That is a genuine mismatch and needs to be fixed, either by switching the experiments to analytic dictionaries (Gaussians, polynomials) or by proving a weakened version of the theorem under C² or C^k assumptions.\n\nThe other soft spots are secondary but real: the SLLN step in the proof of Theorem 1 (and Corollary 4) applies Kolmogorov's theorem to sequences a_t e_t that are correlated over time; this can likely be repaired with a martingale SLLN argument, but as written it is a gap. Theorem 2's finite-data bound uses inverses of O(ε) perturbed matrices without showing the unperturbed gramian has uniformly bounded inverse; the conclusion O(ε) is plausible but the proof is under-specified. Theorem 3, the quantized-observable special case, is clean and new. No code or error bars, which hurts reproducibility but is not a fatal flaw for this venue.\n\nOverall: the paper deserves a serious referee and heavy revision. I would send it back for major revision, not desk-reject. The core insight is useful and likely correct, but the mismatch between Theorem 1 and its numerical support must be addressed. Readers in data-driven control and networked control who care about quantization would get value from it; it could become a solid contribution after revision.","headline":"Solid incremental extension of the authors' own quantization–EDMD work; the large-data regularization theorem is plausible but not actually validated by the paper's own non-analytic dictionaries.","tokens_in":24445,"tokens_out":3781,"would_cite":true,"duration_ms":38370,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B30","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Dither-quantized training data turn Koopman linear-predictor identification into a regularized least-squares problem, with a penalty of order the square of the quantization step.","keywords":["Koopman operator","Extended Dynamic Mode Decomposition","dither quantization","system identification","model predictive control","lifted linear predictor","regularized least squares","quantization resolution"],"falsifier":"Run modified EDMD on a controlled system whose dictionary contains a thin-plate spline centered on a training trajectory, with very large $T$ and fixed $\\epsilon$; compute the almost-sure limit of $[\\tilde{A},\\tilde{B}]$ and compare it with the minimizer of the unquantized objective plus $\\operatorname{tr}(G\\beta(\\epsilon))+\\operatorname{tr}(G^{\\top}G\\Gamma(\\epsilon))$ evaluated by finite-difference second-derivative matrices. If the limits do not match, Theorem 1 fails exactly where Assumption 3 is violated.","tokens_in":23434,"feed_emoji":"📉","tokens_out":15284,"duration_ms":129323,"temperature":0.7,"pith_summary":"Sending state and control snapshots through a dither quantizer before Koopman-based identification is not just adding noise: this paper claims it is equivalent, in the large-data limit, to solving a regularized least-squares problem. The regularization penalty grows as the square of the quantization step $\\epsilon$, through two matrices $\\beta(\\epsilon)$ and $\\Gamma(\\epsilon)$ built from power series coefficients of the squared prediction residual, so the estimate from quantized data converges almost surely to a shrunk version of the unquantized estimate. In the finite-data regime the estimate differs from the unquantized one by $O(\\epsilon)$, which predicts the linear log-error versus word-length slope seen in the experiments. If true, this gives practitioners a rule for how many bits are needed to keep a prescribed identification error, and it explains why moderate quantization can even improve downstream model predictive control performance by acting as a regularizer.","feed_headline":"Coarse data quietly regularize Koopman predictors","feed_subtitle":"Dither-quantized snapshots give a regularized EDMD estimate; error shrinks like the quantization step squared.","key_machinery":"The load-bearing object is the one-step squared residual $r(x_{t+1},x_t,u_t)=\\|\\varphi(x_{t+1})-A\\varphi(x_t)-Bu_t\\|^2$, viewed as a function of the dither-quantized snapshots. Dither quantization makes each component of the error $\\tilde{x}-x$ and $\\tilde{u}-u$ i.i.d. uniform on $[-\\epsilon/2,\\epsilon/2]$, so in the power series expansion of $r$ around the noisy snapshot, odd-order terms vanish almost surely under the strong law of large numbers and even-order terms converge to deterministic traces involving second derivatives of $\\varphi$. These traces assemble into the matrices $\\beta(\\epsilon)$ and $\\Gamma(\\epsilon)$ that produce the regularization. The finite-$T$ bound uses the mean-value theorem plus the gradient bound $\\|\\nabla\\varphi_i\\|\\le c_\\varphi$ to show $\\|\\Phi_\\epsilon\\|=O(\\epsilon)$ and hence $\\|\\tilde{G}-G\\|=O(\\epsilon)$.","core_discovery":"The central claim is a limit identity. Let $G=[A,B]$ be the lifted linear predictor obtained by modified EDMD from unquantized snapshots, and $[\\tilde{A},\\tilde{B}]$ the predictor from snapshots that have been dithered and quantized with resolution $\\epsilon$. Under assumptions that make the one-step residual $r(x_{t+1},x_t,u_t)=\\|\\varphi(x_{t+1})-A\\varphi(x_t)-Bu_t\\|^2$ analytic in a neighborhood of the quantized data, Theorem 1 states that as $T\\to\\infty$ the quantized estimate converges almost surely to the minimizer of $\\limsup_{T\\to\\infty}\\frac{1}{T}\\|\\Phi_{+}-A\\Phi-BU\\|^2+\\operatorname{tr}(G\\beta(\\epsilon))+\\operatorname{tr}(G^{\\top}G\\Gamma(\\epsilon))$, with $\\beta,\\Gamma=O(\\epsilon^2)$. Theorem 2 states that for finite full-rank data the difference is $G_\\epsilon$ with $\\|G_\\epsilon\\|=O(\\epsilon)$. When the observables themselves are quantized rather than the raw states, the regularizer becomes the explicit ridge penalty $\\frac{\\epsilon^2}{12}(\\|A\\|^2+\\|B\\|^2)$, and under a data-growth condition on the snapshot covariance the normalized identification error can be driven to zero for any fixed $\\epsilon$.","pith_inferences":["If the asymptotic equivalence holds generally, quantization resolution becomes a regularization dial: deliberately dithering and quantizing before EDMD could improve generalization in small-sample regimes, and one could tune word-length as a hyperparameter just like a ridge penalty.","Because $\\beta$ and $\\Gamma$ depend on second derivatives of the dictionary, quantization's effect is dictionary-dependent; under a fixed bit budget, observables with small curvature over the data distribution should be preferred to minimize induced bias.","A concrete testable extension is to estimate $\\beta(\\epsilon)$ and $\\Gamma(\\epsilon)$ by finite differences of the residual second-derivative matrices on the quantized snapshots and then apply (20); success would turn the asymptotic equivalence into a practical debiasing tool.","The paper's own thin-plate-spline experiments sit outside Assumption 3 because $r^2\\log r$ is not analytic at its center; if convergence still holds there, the true sufficient condition is likely a weaker smoothness or a decay of high-order terms, which would be worth isolating."],"forward_implications":["Designers on bandwidth-constrained hardware can budget bits by the rule that asymptotic bias scales as $\\epsilon^2 = \\left((x_{\\max}-x_{\\min})/2^b\\right)^2$: every extra bit quarters the regularization-induced bias.","Finite-data identification error is $O(\\epsilon)$, so log-relative-error falls with slope about $-\\log 2$ per bit; the reported experiments show this slope.","Quantizing observables directly rather than states removes all dictionary dependence from the regularizer, yielding $(\\epsilon^2/12)(\\|A\\|^2+\\|B\\|^2)$, and Lemma 2 says enough data can wash out quantization entirely in that setting.","The unquantized predictor is recoverable in principle by solving the debiased problem (20), so the theory points toward a correction algorithm once $\\beta$ and $\\Gamma$ are estimated.","MPC with quantized-data predictors approaches unquantized performance as word-length grows; non-monotone cases such as one oscillator at $b=8$ are consistent with beneficial regularization rather than pure error."],"supporting_citations":[{"why":"Defines EDMD, the estimation algorithm whose quantized-data variant this paper analyzes.","marker":"[7]"},{"why":"Introduces the modified EDMD for control systems and the lifted linear predictor [A,B] that this paper studies under quantization.","marker":"[8]"},{"why":"Provides the dither quantization model and the statistical independence properties of dithered quantization errors.","marker":"[17]"},{"why":"Supplies the characteristic-function condition (Lemma 1) guaranteeing independence of the quantization error from the signal.","marker":"[25]"},{"why":"Establishes large-data convergence of DMD/EDMD estimates, the regime Theorem 1 extends to quantized snapshots.","marker":"[9]"},{"why":"Is the strong law of large numbers used repeatedly to pass T to infinity and kill odd-order error terms.","marker":"[39]"},{"why":"Provides convex-process minimizer asymptotics used in the proof of Lemma 2 on data-driven mitigation of quantization.","marker":"[40]"}],"fun_headline_variants":["Quantized data regularize Koopman predictors","Coarse snapshots yield smoother Koopman models","Dithering adds implicit ridge penalty to EDMD","Koopman meets limited bandwidth: quantization helps","Quantization-induced regularization for linear predictors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The large-data theorem requires that the squared prediction residual, as a function of the measured snapshot, can be expanded in a convergent power series around every point the quantizer can produce, with radius bigger than half the quantization step; this is violated by the thin-plate-spline radial basis functions used in the paper's experiments, whose $r^2\\log r$ form is not analytic at its center.","fun_headline_variants_meta":{"raw":{"variants":["Quantized data regularize Koopman predictors","Coarse snapshots yield smoother Koopman models","Dithering adds implicit ridge penalty to EDMD","Koopman meets limited bandwidth: quantization helps","Quantization-induced regularization for linear predictors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1610,"prompt_tokens":1104,"completion_tokens":506,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":435}},"tokens_in":720,"tokens_out":506,"duration_ms":5006,"temperature":1.0,"reasoning_tokens":435,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:37:25.217520+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run modified EDMD on a controlled system whose dictionary contains a thin-plate spline centered on a training trajectory, with very large $T$ and fixed $\\epsilon$; compute the almost-sure limit of $[\\tilde{A},\\tilde{B}]$ and compare it with the minimizer of the unquantized objective plus $\\operatorname{tr}(G\\beta(\\epsilon))+\\operatorname{tr}(G^{\\top}G\\Gamma(\\epsilon))$ evaluated by finite-difference second-derivative matrices. If the limits do not match, Theorem 1 fails exactly where Assumption 3 is violated.","supporting_citations":[{"cited_title":"A data–driven approximation of the Koopman operator: extending Dynamic Mode Decomposition,","cited_arxiv_id":null,"evidence_quote":"Defines EDMD, the estimation algorithm whose quantized-data variant this paper analyzes."},{"cited_title":"Dithered quantizers,","cited_arxiv_id":null,"evidence_quote":"Provides the dither quantization model and the statistical independence properties of dithered quantization errors."},{"cited_title":"Dither signals and their e ffect on quantization noise,","cited_arxiv_id":null,"evidence_quote":"Supplies the characteristic-function condition (Lemma 1) guaranteeing independence of the quantization error from the signal."},{"cited_title":"Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator,","cited_arxiv_id":null,"evidence_quote":"Establishes large-data convergence of DMD/EDMD estimates, the regime Theorem 1 extends to quantized snapshots."},{"cited_title":"Sen and J","cited_arxiv_id":null,"evidence_quote":"Is the strong law of large numbers used repeatedly to pass T to infinity and kill odd-order error terms."}],"review_version":1}