{"id":"9e4e227d-b93f-497a-98a6-8311d984b8e3","arxiv_id":"2501.13301","paper_version":4,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"SDMD estimates the Koopman semigroup as I + Δt times the gEDMD generator matrix; the paper's convergence proofs concern the matrix-exponential semigroup, not the operator the method actually uses.","lead":"This paper presents SDMD, an estimator of the stochastic Koopman semigroup built as an Euler step applied to the gEDMD generator approximation. The authors claim convergence guarantees and a neural-network variant, but the guarantees cover a different semigroup than the one actually computed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.14 proves convergence for exp(t·A_N), but SDMD's estimator (Eq. 10) is I + Δt·G^{-1}H; Remark 4.16 hand-waves the missing Euler-step convergence, so the claimed guarantees do not cover the computed operator.","rationale":"I read the paper in good faith as an attempt to build a practical stochastic Koopman estimator by combining gEDMD with an Euler discretization. The underlying algebra of Eq. (10) is plausible as a first-order approximation for small Δt, and the large-data concentration bound of Theorem 4.3 appears to follow the cited framework. However, the advertised 'rigorous convergence guarantees' concern the matrix-exponential semigroup e^{tA_N}, not the operator the method actually returns and whose spectrum is analyzed in the experiments. The paper itself flags the gap only in Remark 4.16, with a bare appeal to Trotter–Kato and no verification of its hypotheses for the estimated operators. This is the reader's weakest assumption as well, and I agree it is the decisive issue. The experimental section compounds the problem: the neural-mass comparison selects the model epoch by maximizing correlation with the ground-truth latent input, and Algorithm 2 computes Ψ'_X despite the text claiming that SDMD-DL avoids it—but the proof gap is more central because it undercuts the main theoretical claim. The underlying estimator may be repairable with a proper convergence theorem for the iterated Euler operator, and the paper itself acknowledges the missing proof implicitly, but as submitted the central claim is not established for the method's actual output. The reader's REJECT therefore stands, and no change to the verdict is needed.","tokens_in":23864,"tokens_out":4982,"duration_ms":48217,"concrete_test":"Recompute the reported experiments (e.g., OU Table 2, triple-well Table 3) with the same data and dictionary matrices, but with the Koopman matrix replaced by expm(Δt G^{-1}H) and, separately, by (I + ΔtG^{-1}H)^n for n = t/Δt; compare the resulting eigenvalues and eigenfunctions against the analytic or reference semigroup spectra. If the exponential version matches theory substantially better than the Euler version at the reported Δt = 0.1, then the numerical validation is testing the object of Theorem 4.14, not the SDMD estimator of Eq. (10). Alternatively or additionally, attempt a rigorous proof or a counterexample that under Assumptions 4.1 and 4.13, sup_{0≤t≤T} ||(I + (t/n)A_N)^n - e^{tA}|| → 0 in the joint limit; the paper currently provides neither.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim (abstract and Section 7) is that SDMD provides rigorous convergence guarantees for approximating the stochastic Koopman semigroup. The theoretical core is Theorem 4.14, which shows the matrix-exponential semigroup e^{tA_N} converges strongly to e^{tA} as N → ∞ under Assumptions 4.10 and 4.13. However, the estimator actually implemented, Eq. (10), is K̂_{N,Δt,m} = I + Δt G^{-1}H, i.e., one step of explicit Euler for the approximated generator A_N = G^{-1}H, not exp(Δt A_N). The experiments then analyze the spectrum of this one-step Euler matrix (e.g., Section 6.3, Table 3). Remark 4.16 asserts, in one sentence and without proof, that iterating the Euler operator (I + ΔtA_N)^n converges to the continuous semigroup, appealing vaguely to Trotter–Kato. It is not shown that the Trotter–Kato hypotheses are satisfied for the sequence A_N arising from the empirical Gram matrices, nor that the composite limit (n → ∞, Δt → 0, N → ∞, together with the earlier m → ∞ limit) commutes in the required way. Without this proof, the rigorous guarantees apply to a different object than the one whose eigenvalues and eigenfunctions the paper reports. This is a load-bearing gap: it is exactly the semigroup approximation property that the framework is advertised to provide.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Stochastic Dynamic Mode Decomposition (SDMD), an estimator of the stochastic Koopman semigroup given by K̂_{N,Δt,m} = I + Δt Ĝ^{-1} Ĥ, where Ĝ and Ĥ are empirical Gram matrices built from dictionary functions and their generator action. The authors claim three convergence guarantees: large-data concentration (Theorem 4.3), zero-sampling-time convergence of the generator approximation (Theorem 4.8), and large-dictionary convergence of the semigroup (Theorem 4.14). A neural-network extension, SDMD-DL, is proposed to learn dictionaries, and experiments on a Stuart-Landau oscillator, an Ornstein-Uhlenbeck process, a triple-well system, and a neural-mass model are reported. The paper's central message is that SDMD provides rigorous, reliable spectral estimates of stochastic Koopman operators while avoiding matrix exponentials.","tokens_in":24230,"tokens_out":4675,"duration_ms":45879,"significance":"If the convergence claims were valid, SDMD would provide a practical and theoretically grounded alternative to gEDMD for stochastic systems, with a computationally cheap one-step update and a neural-network dictionary-learning variant. The paper also makes a valuable effort to connect stochastic Taylor expansion, Galerkin approximation, and Trotter-Kato theory. However, the advertised guarantees do not apply to the object actually computed, and several supporting claims are internally inconsistent. The experimental section is extensive but does not compensate for the theoretical gap. The manuscript, in its current form, does not deliver the rigorous framework it promises.","major_comments":[{"comment":"Theorem 4.14 proves strong convergence of the matrix-exponential semigroups e^{tA_N} to e^{tA}, but the SDMD estimator in Eq. (10) is the explicit Euler operator I + Δt Ĝ^{-1}Ĥ, not e^{Δt A_N}. The spectral analyses in Section 6 (for example, Figures 2, 3, and 7, and Table 3) report eigenpairs of this one-step Euler matrix, not of e^{Δt A_N}. Remark 4.16 asserts without proof that iterating (I + Δt A_N)^n converges to the continuous semigroup, appealing vaguely to Trotter-Kato, but the required hypotheses—such as uniform exponential boundedness of the discrete powers and strong consistency of the Euler scheme for the sequence A_N—are neither stated nor verified. This is a load-bearing gap: the rigorous convergence guarantees claimed in the abstract and Section 7 do not apply to the method's actual output.","section":"Section 4.3.2 and Remark 4.16"},{"comment":"The text in Section 5.2 states that SDMD-DL avoids the heavy evaluation of Ψ'_X by using Ψ_Y directly, but Algorithm 2, line 5, explicitly requires computing Ψ'_X(θ), and the updating formula in Table 1 constructs bH(θ) = Ψ_X(θ)^* Ψ'_X(θ). The claimed computational-efficiency advantage over gEDMD-DL is therefore contradicted by the paper's own pseudocode and table. This is a substantive inconsistency because the avoidance of Ψ'_X is presented as a key benefit of the neural-network extension.","section":"Section 5.2, Algorithm 2, and Table 1"},{"comment":"The zero-sampling-time convergence result is essentially a restatement of the Taylor remainder definition. From Eq. (16), K_{N,Δt} = I + Δt G^{-1}H + o_N(Δt), so A_{N,Δt} - A_N = o_N(Δt)/Δt by Eq. (17). The theorem then follows directly from the defining property of o_N(Δt). No rate, no quantitative bound, and no condition on the uniformity of the remainder are provided. As a result, Theorem 4.8 does not supply a usable convergence guarantee for fixed Δt, which is the regime used in all numerical experiments.","section":"Section 4.2, Theorem 4.8"},{"comment":"The theorem asserts uniform exponential boundedness ∥K^N_t∥, ∥K_t∥ ≤ D e^{ωt} for all N, but the proof only cites [44, Theorem 2.2], which gives individual exponential bounds for each strongly continuous semigroup. The uniformity of the constants D and ω in N is a nontrivial condition required by the Trotter-Kato theorem and is not established. Without it, the uniform-in-N semigroup convergence on compact intervals does not follow from the cited result.","section":"Theorem 4.14 and Assumption 4.13"},{"comment":"The neural-mass experiment acknowledges that SDMD-DL does not converge during training and that the authors select the epoch maximizing the Pearson correlation between the normalized second eigenfunction and the ground-truth latent input. This selection procedure uses information unavailable in practice and is not accompanied by error bars, multiple-seed statistics, or a systematic comparison. The reported superiority of SDMD-DL over EDMD-DL is therefore not a reliable empirical demonstration of the method's robustness.","section":"Section 6.4"}],"minor_comments":[{"comment":"Equation (5) states the exact relation μ = e^{tλ}, while Remark 3.5 computes generator eigenvalues as λ = (μ - 1)/Δt, which is only a first-order approximation. The paper should clarify that the reported generator eigenvalues are approximate and discuss the resulting bias, especially in Tables 2 and 3.","section":"Eq. (5) and Remark 3.5"},{"comment":"The probability bound contains the quantity ϵ - ∥o_N(Δt)∥_F in the definition of ˜ϵ; for small ϵ the argument of the exponentials may become ill-defined. The assumptions under which the bound is meaningful should be stated.","section":"Theorem 4.3"},{"comment":"The existence of o_N(Δt) = lim_{m→∞} Ψ_X^† o_{m,N}(Δt) is asserted, but no argument is given for why this limit exists independently of the data or why the matrix norm of the limit is o(Δt). This is related to the remainder terms being dropped earlier in Eq. (15) and should be justified.","section":"Section 4.1, Eq. (16)"},{"comment":"The manuscript contains numerous typographical and formatting issues, such as the garbled phrase in Section 6.2 ('aware methods (SDMD and gEDMD) for stochastic systems') and inconsistent notation for Δt (sometimes Δ, sometimes ∆t). A careful editorial pass is needed.","section":"General presentation"}],"recommendation":"reject","confidential_remarks":"The main theoretical theorem does not cover the implemented estimator, the claimed efficiency advantage of SDMD-DL is contradicted by the algorithm and table, and the empirical evaluation includes a ground-truth-based epoch selection in a non-converging training run. These are not isolated presentation issues but affect the central claims of the paper. In my view, the manuscript would require a substantially new theoretical analysis and a redesigned experimental protocol to become publishable, which is beyond a routine revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: SDMD is gEDMD's generator matrix followed by an explicit Euler step, and the headline convergence theorem is for e^{tA_N}, not for the Euler iterates the algorithm actually produces. That mismatch is real and load-bearing. But there is genuine work here: the concentration inequalities for large data are careful, the large-dictionary analysis via Trotter-Kato is standard but cleanly presented, and the experiments cover four canonical systems with mostly honest comparisons to EDMD and gEDMD.\n\nWhat is actually new is modest. Eq. (10), K̂ = I + Δt Ĝ^{-1}Ĥ, is exactly gEDMD's generator estimate followed by one explicit Euler step. Remark 3.2 acknowledges the matrix assembly is consistent with gEDMD, so the novelty rests on the semigroup perspective and the neural-network dictionary learning. Both are reasonable, but incremental.\n\nThe soft spots are what kill the 'rigorous convergence guarantees' framing. Theorem 4.14 proves strong convergence of e^{tA_N} to e^{tA} as N → ∞. The method does not compute e^{tA_N}; it computes I + Δt A_N (or iterates of it). Remark 4.16 asserts, without proof, that the iterated Euler operator converges to the continuous semigroup, appealing vaguely to Trotter-Kato. That is not established: the Trotter-Kato hypotheses are never verified for the empirical A_N, and the N → ∞ and Δt → 0 limits are not handled together. The zero-limit theorem (Thm 4.8) is essentially the definition of the Taylor remainder, so it carries no weight. Second, the claimed advantage of SDMD-DL—avoiding Ψ'_X—is contradicted by Algorithm 2 and Table 1, where Ψ'_X is required to build Ĥ. Third, the neural-mass experiment selects the training epoch using the ground-truth latent input, which makes the validation circular; the text acknowledges this, but it undercuts the claim of unsupervised recovery.\n\nNone of these issues is fatal to the practical recipe: Euler-plus-gEDMD is a valid small-Δt estimator, and the experiments show it works on standard examples. But the paper overclaims, and the main theoretical guarantee does not cover the computed object. A serious referee could push for a proper convergence proof for the iterated operator, or at minimum an honest statement of what is and isn't proven.\n\nWho gets value: people who want a cheap stochastic Koopman estimator and are comfortable with first-order accuracy in Δt. The paper is worth reading, and it should be sent to review rather than desk-rejected, but it needs substantial revision before publication.","headline":"SDMD is gEDMD's generator estimate plus an explicit Euler step, and the paper's main convergence theorem is for e^{tA_N} rather than the Euler operator actually computed; the experimental validation in the neural-mass example also selects on ground truth.","tokens_in":24789,"tokens_out":2032,"would_cite":false,"duration_ms":20001,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","60J60","47D06"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stochastic Dynamic Mode Decomposition estimates the Koopman semigroup of a stochastic system directly from data through the first-order Galerkin step $\\hat{K}_{N,\\Delta t,m}=I+\\Delta t\\,\\hat{G}^{-1}\\hat{H}$, with claimed convergence as…","keywords":["stochastic Koopman operator","Markov semigroup","dynamic mode decomposition","sampling time","dictionary learning","neural network basis selection","spectral analysis","convergence guarantees"],"falsifier":"For an Ornstein–Uhlenbeck process with known spectrum, fix a dictionary and compute the eigenvalues of $\\hat K=I+\\Delta t\\,\\hat G^{-1}\\hat H$ for decreasing $\\Delta t$, comparing them with $e^{\\Delta t\\lambda_n}$; also compare the $n$-fold iterate $(\\hat K)^n$ with $e^{t\\hat A_N}$ over fixed time $t=n\\Delta t$. If the one-step spectrum does not approach the true semigroup spectrum as $\\Delta t\\to0$, or the iterates fail to track $e^{t\\hat A_N}$ for small $\\Delta t$, then the claimed convergence guarantees do not cover the method's actual output.","tokens_in":23652,"feed_emoji":"🎲","tokens_out":10802,"duration_ms":100450,"temperature":0.7,"pith_summary":"This paper introduces Stochastic Dynamic Mode Decomposition (SDMD), an estimator for the Koopman semigroup of a stochastic differential equation. Instead of recovering the generator and then exponentiating it, SDMD builds the one-step operator $\\hat{K}_{N,\\Delta t,m}=I+\\Delta t\\,\\hat{G}^{-1}\\hat{H}$ from dictionary evaluations, explicitly carrying the sampling time $\\Delta t$ in the approximation. The authors claim that this estimate converges in the large-data limit, in the zero-limit of the sampling time, and in the large-dictionary limit, so that its eigenvalues and eigenfunctions approximate those of the true stochastic Koopman semigroup. If correct, the framework gives a numerically stable and computationally cheap route to spectral analysis of noisy dynamics, and its neural-network variant automates basis selection. This matters because stochastic spectra encode decay rates, oscillations, metastable transitions, and hidden slow timescales that deterministic methods miss.","feed_headline":"Adding sampling time stabilizes stochastic Koopman estimates","feed_subtitle":"A first-order step approximates the whole stochastic semigroup, avoiding matrix exponentials and recovering spectra from noisy data.","key_machinery":"The load-bearing object is the one-step stochastic Taylor–Galerkin identity $K_{\\Delta t}\\psi \\approx (I+\\Delta t A)\\psi$, which turns semigroup approximation into a linear least-squares problem whose solution is Eq. (10). The finite-dimensional generator $A_N=P_NAP_N$ together with the empirical Gram matrices $\\hat G$ and $\\hat H$ carry the spectral content, while the Trotter–Kato theorem is the mechanism that upgrades generator convergence to semigroup convergence as $N\\to\\infty$. Concentration inequalities for the empirical Gram matrices provide the large-data error bounds.","core_discovery":"The central claim is that the operator $\\hat{K}_{N,\\Delta t,m}=I+\\Delta t\\,\\hat{G}^{-1}\\hat{H}$, built from empirical Gram matrices $\\hat G=(1/m)\\Psi_X^*\\Psi_X$ and $\\hat H=(1/m)\\Psi_X^*\\Psi_X'$, approximates the stochastic Koopman semigroup $K_{\\Delta t}$, not merely its generator. The derivation truncates the stochastic Taylor expansion $K_{\\Delta t}\\psi_j \\approx \\psi_j+\\Delta t\\,A\\psi_j$ and solves the resulting least-squares problem by Galerkin projection, so the estimator carries the sampling time explicitly and avoids forming $\\exp(\\Delta t A)$. The paper claims convergence in three successive limits: large data $m\\to\\infty$ with explicit concentration bounds, zero sampling time $\\Delta t\\to0$ toward the generator approximation $A_N=G^{-1}H$, and large dictionary $N\\to\\infty$ toward the true semigroup via the Trotter–Kato theorem. Consequently the eigenvalues and eigenfunctions of $\\hat K$ approximate those of the stochastic Koopman semigroup, and generator eigenvalues are recovered from $\\lambda=(\\mu-1)/\\Delta t$.","pith_inferences":["Editorial inference: because $\\hat K=I+\\Delta t\\,\\hat G^{-1}\\hat H$ is an explicit Euler step, its iterates inherit a stability constraint; for stiff stochastic systems a practitioner would need $\\Delta t$ below a threshold tied to the spectral radius of $A_N$ for reliable long-horizon iteration, while the paper's own experiments analyze the one-step spectrum.","Editorial inference: SDMD's accuracy is coupled to how well the drift and diffusion coefficients are known or estimated, and the paper does not quantify how errors in those coefficients propagate into $\\hat G$, $\\hat H$, and the computed spectrum.","Editorial inference: the neural-mass result suggests a testable selection rule—calibrating the dictionary by maximizing one dominant eigenfunction's correlation with a known slow variable may tune the whole spectral approximation; checking this across switching rates and noise levels would turn an observed effect into a criterion.","Editorial inference: because the estimator is constructed from i.i.d. initial points and an explicit $\\Delta t$, it may transfer more readily than snapshot-pair EDMD to non-stationary or time-inhomogeneous sampling designs, but the paper does not test this."],"forward_implications":["For small sampling times, SDMD approximates both the semigroup spectrum and, via $\\lambda=(\\mu-1)/\\Delta t$, the generator spectrum; the paper demonstrates this on the Stuart–Landau oscillator, Ornstein–Uhlenbeck process, triple-well potential, and a neural-mass model.","The method removes the need to compute matrix exponentials, so spectral estimation costs roughly the assembly and inversion of an $N\\times N$ Gram system even when the underlying generator is unbounded.","Wrapping the estimator in a neural-network dictionary (SDMD-DL) automates basis selection, and the experiments indicate it can isolate latent slow modulations and transient dynamics that dictionary-learning EDMD misses.","Iterating the one-step estimate, $\\hat K^n$, yields a discrete-time approximation of the continuous semigroup, so long-horizon predictions can be obtained by repeated application of the same matrix.","The claimed guarantees are ordered: for fixed $N$ and $\\Delta t$, bounds improve with $m$; then the discretized generator approaches $A_N$ as $\\Delta t\\to0$; then the Galerkin semigroup approaches the true semigroup as $N\\to\\infty$."],"supporting_citations":[{"why":"Supplies the EDMD/Galerkin snapshot regression that SDMD extends with a stochastic Taylor term.","marker":"[57]"},{"why":"Supplies the generator Gram-matrix assembly for $\\hat G$ and $\\hat H$ and the Galerkin approximation $A_N=G^{-1}H$ that SDMD builds on.","marker":"[23]"},{"why":"Gives the stochastic Taylor expansion $K_{\\Delta t}\\psi\\approx \\psi+\\Delta t A\\psi$ used in the derivation of Eq. (10).","marker":"[43]"},{"why":"Provides the Trotter–Kato theorem that turns generator convergence into semigroup convergence as $N\\to\\infty$.","marker":"[15]"},{"why":"Supplies the graph-norm dictionary assumptions and Galerkin convergence framework used in Section 4.3.","marker":"[32]"},{"why":"Provides the dictionary-learning alternating minimization scheme that SDMD-DL adapts with the $\\hat K$ update rule.","marker":"[29]"},{"why":"Justifies the existence of the large-data EDMD limit invoked in Eq. (16).","marker":"[26]"},{"why":"Provides the perturbation inequality used to prove the large-data concentration bound for $\\hat G^{-1}\\hat H$.","marker":"[45]"},{"why":"Supplies Hoeffding's inequality used to derive the concentration bounds in Lemma 4.5.","marker":"[55]"}],"fun_headline_variants":["Sampling time stabilizes stochastic Koopman estimates","No matrix exponentials: a faster stochastic Koopman","Neural networks automate basis for stochastic Koopman","SDMD: sampling time for stable stochastic Koopman","Sampling time beats exponentials for stochastic Koopman"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence proofs are written for the ideal semigroup built from the approximated generator, but the algorithm actually uses only a one-step linear update; the assertion that iterating that update gives the same semigroup is left unproved.","fun_headline_variants_meta":{"raw":{"variants":["Sampling time stabilizes stochastic Koopman estimates","No matrix exponentials: a faster stochastic Koopman","Neural networks automate basis for stochastic Koopman","SDMD: sampling time for stable stochastic Koopman","Sampling time beats exponentials for stochastic Koopman"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000583,"raw_usage":{"total_tokens":2736,"prompt_tokens":932,"completion_tokens":1804,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":1725}},"tokens_in":548,"tokens_out":1804,"duration_ms":14224,"temperature":1.0,"reasoning_tokens":1725,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:17:23.287081+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For an Ornstein–Uhlenbeck process with known spectrum, fix a dictionary and compute the eigenvalues of $\\hat K=I+\\Delta t\\,\\hat G^{-1}\\hat H$ for decreasing $\\Delta t$, comparing them with $e^{\\Delta t\\lambda_n}$; also compare the $n$-fold iterate $(\\hat K)^n$ with $e^{t\\hat A_N}$ over fixed time $t=n\\Delta t$. If the one-step spectrum does not approach the true semigroup spectrum as $\\Delta t\\to0$, or the iterates fail to track $e^{t\\hat A_N}$ for small $\\Delta t$, then the claimed convergence guarantees do not cover the method's actual output.","supporting_citations":[{"cited_title":"Williams, Ioannis G","cited_arxiv_id":null,"evidence_quote":"Supplies the EDMD/Galerkin snapshot regression that SDMD extends with a stochastic Taylor term."},{"cited_title":"Data- driven approximation of the Koopman generator: Model reduction, system identification, and control","cited_arxiv_id":null,"evidence_quote":"Supplies the generator Gram-matrix assembly for $\\hat G$ and $\\hat H$ and the Galerkin approximation $A_N=G^{-1}H$ that SDMD builds on."},{"cited_title":"Pavliotis","cited_arxiv_id":null,"evidence_quote":"Gives the stochastic Taylor expansion $K_{\\Delta t}\\psi\\approx \\psi+\\Delta t A\\psi$ used in the derivation of Eq. (10)."},{"cited_title":"Engel, S","cited_arxiv_id":null,"evidence_quote":"Provides the Trotter–Kato theorem that turns generator convergence into semigroup convergence as $N\\to\\infty$."},{"cited_title":"Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator","cited_arxiv_id":null,"evidence_quote":"Provides the dictionary-learning alternating minimization scheme that SDMD-DL adapts with the $\\hat K$ update rule."},{"cited_title":"On convergence of extended dynamic mode decomposition to the Koopman operator","cited_arxiv_id":null,"evidence_quote":"Justifies the existence of the large-data EDMD limit invoked in Eq. (16)."},{"cited_title":"Philipp, Manuel Schaller, Septimus Boshoff, Sebastian Peitz, Feliks Nüske, and Karl Worthmann","cited_arxiv_id":null,"evidence_quote":"Provides the perturbation inequality used to prove the large-data concentration bound for $\\hat G^{-1}\\hat H$."}],"review_version":1}