{"id":"4f606a18-abc0-427a-97d4-701c741f629e","arxiv_id":"2512.24953","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A pseudo-resolvent matrix built from EDMD data is exactly the resolvent of the finite-section Koopman operator, and its spectral convergence follows from classical stable-convergence assumptions.","lead":"This paper builds a data-driven approximation of the Koopman operator's resolvent using the Sherman-Morrison-Woodbury identity and proves spectral convergence under a stability assumption, plus an error bound controlled by a residual. A generalist might read it to see whether resolvent/pseudospectrum diagnostics can clean up spectral pollution in Koopman/DMD analysis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence theorems are conditional on the unverified Assumption 4.2; the abstract promises sufficient conditions that never appear in the full text, so the central claim that resolvent diagnostics are a valid data-driven tool is not actually established.","rationale":"The reader's weakest assumption correctly identifies Assumption 4.2 as the load-bearing premise. My independent reading confirms that the paper proves only conditional statements from this assumption, gives no sufficient conditions despite the abstract's promise, and does not connect finite-M empirical computations to the large-data Galerkin object used in the theory. The conditional theorems appear internally consistent, and the numerical comparisons are suggestive, but the central claim is under-supported. This does not move the verdict beyond CONDITIONAL: the paper could be repaired by adding verified sufficient conditions for Assumption 4.2, explicit finite-M bounds, and a direct numerical check of the stability premise.","tokens_in":18067,"tokens_out":8869,"duration_ms":94883,"concrete_test":"In the linear-oscillator experiment of Section 6.2, where K is known analytically, fix z=0 in rho(K) and compute ||(zI-K_N)^{-1}|| for N=20,50,120,250,500 at a fixed finite M, then repeat with M increased by a factor of 10. If the resolvent norms do not stabilize in N, or if the M-dependence is as large as the N-dependence, Assumption 4.2 is not supported and the finite-M/finite-N distinction is material. As an additional branch, for the pendulum with an orthonormal dictionary, test whether sup_N ||(zI-P_N K P_N)^{-1}|| stays bounded for z=0; this would supply the promised measure-preserving sufficient condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.10 and Theorem 5.4 both rest on Assumption 4.2: for each z in rho(K), the finite-section resolvent R_N(z)=(zI-K_N)^{-1} is eventually defined and uniformly bounded in N. This is precisely the condition that excludes spectral pollution, so the main convergence result is conditional on ruling out the phenomenon the paper claims to solve. The text explicitly notes that strong convergence K_N->K and SLLN for the empirical Gram matrices do not imply resolvent convergence (Section 4, after Definition 4.1), but it never supplies a separate data-dependent criterion verifying Assumption 4.2. The abstract states that 'contractive, measure-preserving, and compact settings provide concrete sufficient conditions,' yet no such theorem or proof appears in the full text. Moreover, the numerical sections use finite M throughout, while the theory is formulated for the large-data limit K_N of the Galerkin section; no bound shows that the finite-M empirical resolvent is close to that idealized object. Thus the convergence and error-bound theorems, while conditionally correct, have no verified input for the reported data-driven experiments.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a data-driven spectral diagnostic for Koopman operators based on the resolvent of EDMD finite sections. It constructs a matrix approximation R_N(z) via the Sherman-Morrison-Woodbury identity, establishes in Eq. (5) that this object is exactly (zI-K_N)^{-1}, and then develops convergence theory for the approximate spectrum. Under Assumption 4.2 (stable convergence) it proves local Hausdorff/Kuratowski convergence near isolated eigenvalues; under Assumption 4.5 it claims preservation of algebraic multiplicities. Theorem 5.4 gives an error bound |λ-\\bar{λ}_N| ≤ C η_N with η_N = ||(K-K_N)P_N^λ||. Numerical experiments on a pendulum, the Lorenz system, and a noisy two-oscillator example illustrate threshold-based spectral detection, pseudospectral contours, and mode separation. The convergence theorems are conditional on stability assumptions and largely follow standard operator-approximation results from Chatelin and Mills; the numerical evidence is suggestive but does not verify the preconditions of the theory.","tokens_in":18309,"tokens_out":10074,"duration_ms":95576,"significance":"If the conditional theorems were fully justified, the paper would provide a clean resolvent-based framework for spectral detection and error certification in EDMD finite sections, complementing residual-based methods such as ResDMD. The SMW construction is computationally standard, and Eq. (5) shows that no genuinely new operator is introduced beyond the resolvent of K_N; the substantive content is the set of convergence and error theorems, which import known results from the spectral approximation literature. The paper is explicit about Assumption 4.2 as an external hypothesis, and the linear-oscillator experiment makes a serious attempt to test the error bound using a known flow map. However, the advertised data-driven guarantees are not established: the key stability assumptions are neither verified nor supplied with concrete sufficient conditions, and the finite-data/finite-dictionary limit is conflated. With substantive revision, the conditional results could become a useful contribution.","major_comments":[{"comment":"The justification of Assumption 4.5 is logically invalid. Corollary 4.9 proves only dim(P_N^λ F) ≥ m_λ, while Assumption 4.5 requires equality. Strong convergence of Riesz projections does not prevent extra eigenvalues of K_N inside Γ_λ: a rank-two projection can converge strongly to a rank-one projection, so the lower bound does not imply dimension equality. The upper-semicontinuity part of Theorem 4.10 only excludes eigenvalues of K_N outside B(λ,ε); it does not exclude additional eigenvalues that remain inside arbitrarily small neighborhoods of λ. Consequently, the multiplicity-preservation conclusion of Theorem 4.10 and the definition of \\bar{λ}_N in Theorem 5.4 are not established. An argument excluding spurious eigenvalues accumulating at λ is required.","section":"§4, Assumption 4.5 and Corollary 4.9"},{"comment":"The entire convergence theory rests on Assumption 4.2, which is precisely a condition that rules out spectral pollution. The abstract promises that 'contractive, measure-preserving, and compact settings provide concrete sufficient conditions,' but no theorem in the full text supplies such conditions. The text itself notes, after Definition 4.1, that strong convergence of K_N to K and SLLN for the Gram matrices do not imply resolvent convergence, yet it never verifies Assumption 4.2 for the EDMD dictionaries used in the paper. Without a data-dependent criterion or a more restricted claim, the main convergence theorem is conditional on an unverified input that essentially assumes away the phenomenon the method claims to solve.","section":"§4, Assumption 4.2 and Abstract"},{"comment":"The theory is formulated for the infinite-data limit K_N = lim_{M→∞} Ψ_X^† Ψ_Y, but all numerical experiments use finite M, and no bound connects the finite-M empirical resolvent to this idealized K_N. The SLLN statement is too vague for deterministic or non-i.i.d. trajectories, especially with unbounded dictionaries, and in §6.2 the numerical study sets ndata = 30N, so M and N grow together. The theorems take a fixed K_N and let N→∞; a joint limit or a finite-M perturbation bound is needed before the numerical evidence can support the convergence claims.","section":"§4 Definition 4.1 and §6 experiments"},{"comment":"The error bound is not a practical a posteriori estimate in a data-driven setting because η_N = ||(K-K_N)P_N^λ|| requires exact knowledge of the true Koopman operator K. The only numerical verification (§6.2) is possible because the linear oscillator flow map is known in closed form; in the nonlinear examples no η_N is computed. The Conclusion's claim that the method offers 'a quantifiable measure of reliability that is often missing' is therefore not supported. Please clarify that Theorem 5.4 is an operator-theoretic convergence-rate statement, not a data-computable certificate, or provide a computable surrogate with rigorous control.","section":"§5, Theorem 5.4"}],"minor_comments":[{"comment":"The arXiv title emphasizes singular-value diagnostics, whereas the full text is titled 'Data-Driven Spectral Analysis through Pseudo-Resolvent Koopman Operator...' and the two abstracts use different terminology and promises. Please harmonize title and abstract with the content actually presented.","section":"Title/Abstract"},{"comment":"The notation R_N(z) is used for both the discrete Koopman resolvent and the generator resolvent. This is a source of confusion; consider distinguishing them, e.g., R_N^K(z) and R_N^A(z).","section":"Remark 3.2"},{"comment":"The remark that resolvent stability 'prevents the formation of spurious eigenvalues in the vicinity of U' is too strong. If spurious eigenvalues lie inside U, then for z near U but outside U the distance condition in Definition 4.14 can fail, making the implication vacuous. Rephrase to describe what the condition actually prevents.","section":"Definition 4.14 and Remark 4.15"},{"comment":"Detection thresholds are reported inconsistently: the text states thresholds 2.5e-14 and 3.5e-14, while Figure 1 captions list 9.2e-14, 1.6e-13, and 1.9e-13. Please align the text and figures.","section":"§6.1 / Figure captions"},{"comment":"The symbol M is used both for the invariant subspace and for the data size M elsewhere in the paper. Use a different letter, e.g., V or E, for the subspace.","section":"Lemma 5.3"},{"comment":"The proof is only a citation to [4, Theorem 6.15]. Since this is the main error-bound result, please state the relevant theorem or include a short proof, and verify that the hypotheses of [4] match Assumptions 4.2 and 4.5 exactly.","section":"Theorem 5.4 proof"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to combine two versions or abstracts, and the title/abstract mismatch should be corrected. The conditional theorems are standard and plausible, but the central data-driven claim is not delivered: Assumption 4.2 and Assumption 4.5 are not verified, the derivation of Assumption 4.5 is logically invalid, and the finite-M versus infinite-M gap is unaddressed. These are load-bearing issues, but they are fixable in a major revision: add concrete sufficient conditions (or explicitly narrow the claims), repair the multiplicity-preservation argument, and provide a finite-data perturbation result or clearly restrict the theory to the ideal infinite-data finite-section operators. I would not reject, because the conditional framework is coherent and the experimental comparisons are useful."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you work on Koopman spectral computation: it is a competent repackaging of known spectral approximation theory with a useful diagnostic wrapper, but the central convergence claim rests on an assumption the paper never verifies, and the abstract promises more than the text delivers.\n\nWhat is actually new is thin. The central object, by the paper's own Eq. (5), is exactly the resolvent of the EDMD matrix K_N; the Sherman-Morrison-Woodbury identity is only a computational reorganization. The convergence theorems are applications of Chatelin and Mills to Galerkin sections. That said, the paper does a real service by turning resolvent norms and singular vectors into a practical spectral pollution diagnostic: contour plots of 1/||R_N(z)||, extraction of pseudomodes, and a comparison against ResDMD on a noisy two-oscillator example. The experiments show the tool can separate nearby spectral bands and pick out Hamiltonian spectra where gEDMD gives spurious damping. That is worth having.\n\nSoft spots, in proportion. The load-bearing issue is Assumption 4.2: stable convergence, i.e., uniform boundedness of the finite-section resolvents for z outside the true spectrum. This is precisely the condition that excludes spectral pollution, so the main theorem is conditional on the phenomenon the paper claims to diagnose. The abstract says contractive, measure-preserving, and compact settings provide concrete sufficient conditions, but no such theorem or proof appears in the full text. The large-data limit M→∞ and dictionary limit N→∞ are also blurred; the theory is for the expected Gram matrices, while all experiments use one finite trajectory, with no bound on the gap.\n\nThere is also a logical slip in the proof of Assumption 4.5. The text says the assumption is valid because Corollary 4.9 shows dim(P_N^λ F) ≥ m_λ. The assumption requires equality. The inequality is useful, but it does not establish the assumption.\n\nNumerics are suggestive but not fully supported: no code or data, no error bars, and the one quantitative verification fits C = 0.04 to the same data points it then claims to confirm. That is a consistency check, not an independent test.\n\nFor a reading group: maybe. For citation in my own work: no. As an editor, I would send it to peer review—not because the math is new, but because the diagnostic is useful and the conditional framework is mostly correct. The referee should require the authors to either verify the stability assumption in their examples, or scale back the claims and correct the Assumption 4.5 discussion.","headline":"Competent but largely derivative repackaging of known spectral approximation results, with a useful diagnostic wrapper and an unverified stability assumption at the load-bearing point.","tokens_in":18900,"tokens_out":3576,"would_cite":false,"duration_ms":442713,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","47A10","65J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A data-driven pseudo-resolvent built directly from time series recovers the Koopman spectrum with proved Hausdorff convergence and residual-based error bounds.","keywords":["Koopman operator","pseudo-resolvent","Sherman-Morrison-Woodbury","spectral pollution","resolvent norm","pseudospectrum","spectral convergence","data-driven spectral analysis"],"falsifier":"Take a system with a known purely continuous spectrum (for example, a chaotic map) and a fixed dictionary, compute $R_N(z)$ on a fine grid of $z$, and compare the inverse-norm sublevel sets with the known spectral support; persistent spurious sublevel sets or failure of the norm to diverge at genuine spectral locations as $N$ grows would indicate that Assumption 4.2 fails for that dictionary. Alternatively, for a linear oscillator with known eigenvalues, compute the ratio $|\\lambda - \\bar{\\lambda}_N|/\\eta_N$ for increasing $N$; if that ratio grows without bound, the asserted $N$-independent error bound is violated.","tokens_in":17840,"feed_emoji":"📊","tokens_out":6164,"duration_ms":51208,"temperature":0.7,"texified_at":"2026-08-05T20:46:17.810472+00:00","pith_summary":"This paper aims to show that the resolvent of a finite-dimensional Koopman approximation can be constructed directly from time-series data using the Sherman-Morrison-Woodbury identity, and that its norm, computable as the reciprocal of the smallest shifted singular value, provides a reliable spectral indicator. The central theoretical result is that, under a stable-convergence assumption, the approximate spectra of these finite sections converge to the true Koopman spectrum in the Hausdorff metric near isolated eigenvalues, with algebraic multiplicities preserved. A companion error bound controls the arithmetic mean of the approximate eigenvalues by a residual quantity that can be estimated in practice. A sympathetic reader would care because this turns resolvent-norm diagnostics on data-driven Koopman models from a heuristic into a certified spectral detection tool.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8712,"prompt_tokens":800,"completion_tokens":7912,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":800,"completion_tokens_details":{"reasoning_tokens":7186}},"feed_headline":"Resolvent-norm diagnostic recovers true Koopman spectrum","feed_subtitle":"A directly computed resolvent turns data-driven spectral search into a certified method with error bounds.","key_machinery":"The central object is the data-driven pseudo-resolvent matrix obtained by applying the Sherman-Morrison-Woodbury identity to $(zI - \\Psi_X^\\dagger \\Psi_Y)^{-1}$: $R_N(z) = (1/z)I + (1/z^2) \\Psi_X^\\dagger (I - (1/z) \\Psi_Y \\Psi_X^\\dagger)^{-1} \\Psi_Y$. This expression is mathematically equal to the resolvent of the finite-section Koopman operator $K_N$ but is computed directly from the data, and its norm's reciprocal is the key spectral diagnostic. The identity carries the argument by connecting the raw data matrices to the resolvent, which then feeds the convergence theory and the error bounds.","core_discovery":"The paper's central claim is that the finite-section resolvent $R_N(z) = (zI - K_N)^{-1}$ of the standard data-driven Koopman approximation can be formed directly from the raw data matrices via the Sherman-Morrison-Woodbury identity, without ever assembling $K_N$. The reciprocal of its norm, $1/\\|R_N(z)\\|$, is the smallest singular value of the shifted finite-section operator and functions as a spectral indicator. The paper proves that if the finite-section resolvents are eventually uniformly bounded on the true resolvent set (Assumption 4.2), then for an isolated eigenvalue the set of approximate eigenvalues inside an isolating contour converges to the true eigenvalue in the Kuratowski/Hausdorff se","pith_inferences":["The stability assumption is stated without data-dependent sufficient conditions; verifying it for a specific dictionary would require numerically checking that ∥R_N(z)∥ stays bounded as N grows on a grid of z in the resolvent set.","The residual-based and resolvent-based diagnostics are reciprocal views of the same pseudospectral object; combining both could yield a method that both certifies individual eigenvalues and separates nearby modes.","The residual-bound theorem suggests a practical dictionary-selection rule: increase N until the ratio of eigenvalue error to the residual quantity stays below a chosen threshold, though the constant C must be estimated.","Extending the construction to stochastic systems via an Itô-calculus version of the cross matrix would provide a data-driven resolvent for stochastic differential equations, where spectral pollution is equally problematic."],"forward_implications":["The reciprocal of the smallest shifted singular value of the finite-section operator identifies spectral candidates: probes where it is small are near the true spectrum.","Under the stable-convergence assumption, the finite-section spectra converge locally to the Koopman spectrum in Hausdorff distance, so spectral pollution disappears and true eigenvalues are not missed.","For an isolated eigenvalue with preserved multiplicity, the arithmetic mean of the approximate eigenvalues obeys a residual-controlled bound, giving an a posteriori error estimate.","Because the generator pseudo-resolvent is built directly from data, one avoids the matrix logarithm, preventing spurious dissipation in conservative systems.","The leading right and left singular vectors provide a minimum-residual pseudomode and the corresponding optimal forcing direction, enabling mode separation."],"fun_headline_variants":["Direct resolvent from data certifies Koopman spectrum","Resolvent norm via data skips Koopman matrix build","Smallest shifted singular value locates true eigenvalues","Certified spectral search from finite-section resolvents","Resolvent diagnostic gives error-bounded Koopman spectrum"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is Assumption 4.2: for every z in the true resolvent set, the finite-section resolvents are eventually defined and uniformly bounded; the paper assumes this rather than deriving it from data or dictionary properties, and it also relies on the empirical Gram matrices converging in the large-data limit before the finite-section operators are meaningful.","fun_headline_variants_meta":{"raw":{"variants":["Direct resolvent from data certifies Koopman spectrum","Resolvent norm via data skips Koopman matrix build","Smallest shifted singular value locates true eigenvalues","Certified spectral search from finite-section resolvents","Resolvent diagnostic gives error-bounded Koopman spectrum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1427,"prompt_tokens":787,"completion_tokens":640,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":571}},"tokens_in":531,"tokens_out":640,"duration_ms":6295,"temperature":1.0,"reasoning_tokens":571,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:27:13.603219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a system with a known purely continuous spectrum (for example, a chaotic map) and a fixed dictionary, compute $R_N(z)$ on a fine grid of $z$, and compare the inverse-norm sublevel sets with the known spectral support; persistent spurious sublevel sets or failure of the norm to diverge at genuine spectral locations as $N$ grows would indicate that Assumption 4.2 fails for that dictionary. Alternatively, for a linear oscillator with known eigenvalues, compute the ratio $|\\lambda - \\bar{\\lambda}_N|/\\eta_N$ for increasing $N$; if that ratio grows without bound, the asserted $N$-independent error bound is violated.","supporting_citations":[],"review_version":1}