{"id":"c024ff95-bf84-4604-93aa-3693d554855b","arxiv_id":"2608.02661","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Filtered EDMD projects the Koopman action onto forward-intersection subspaces that need not be invariant, provably preserving every represented nonzero Koopman eigenpair while suppressing spectral pollution.","lead":"This paper introduces Filtered EDMD, a data-driven method that removes spurious eigenvalues in Koopman spectral analysis by projecting onto dictionary subspaces that need not be invariant. It gives nonlinear dynamics practitioners a way to keep genuine oscillation and decay rates while discarding the artifacts that standard EDMD cannot distinguish.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Filtered EDMD preserves represented nonzero eigenpairs, but Proposition 2 does not imply removal of spectral pollution: admissible non-invariant ranges can create new spurious eigenvalues.","rationale":"The reader's weakest assumption concerned sampling identifiability and numerical-rank tolerance. I find the sampling theory largely sound: Proposition 7 gives eventual exact sample identification of W_{S1,m}, and Assumption 1 is a mild eventual full-rank condition. The more central issue is logical, not statistical: Proposition 2 is a one-way preservation guarantee. It keeps genuine nonzero represented eigenpairs, but it does not prevent A_S from having additional eigenvalues that are not Koopman eigenvalues at all. The minimal counterexample above shows this can happen for an admissible S even with K|_G injective and Vmax = 0. This does not invalidate the paper's formal theorems, which are carefully stated as preservation results, but it does undermine the strongest-claim reading that Filtered EDMD 'removes spectral pollution.' It also reinforces the reader's objection to the word 'certifies' in the abstract. I therefore keep the CONDITIONAL verdict: the theoretical core is sound, but the claims should be narrowed to preservation of represented nonzero eigenpairs and reduction of spurious content, not elimination or certification.","tokens_in":24939,"tokens_out":35307,"duration_ms":456437,"concrete_test":"Run the 3-dimensional example in exact arithmetic: H = span{f, h, h2} with orthonormal basis, K f = 2f + h2, K h = f, K h2 = f + h2, G = span{f, h}, S1 = span{f}, projection P_S1(f)=f, P_S1(h2)=0. Compute the eigenvalues of A_S1 = P_S1 ∘ (K|_G) and of K. If 2 ∈ sigma(A_S1) but 2 ∉ sigma(K), then the 'remove spectral pollution' formulation is false as stated, and the paper should be revised to claim preservation/reduction rather than elimination of spurious eigenvalues.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The inference that Filtered EDMD 'removes spectral pollution while keeping the genuine nonzero spectrum' is not entailed by Proposition 2. Proposition 2 only guarantees that if Kf=lambda f with lambda != 0 and f in G, then A_S f = lambda f. It says nothing about eigenvectors of A_S that are not Koopman eigenfunctions. Since admissible S need not be invariant, an eigenvector f in S of A_S can satisfy Kf = lambda f + w with w in KG \\ S; the projector P_S kills w and produces eigenvalue lambda even though f is not a Koopman eigenfunction. Concretely, at the abstract linear-operator level used in the paper's framework, take H = span{f, h, h2} orthonormal and K f = 2f + h2, K h = f, K h2 = f + h2, with G = span{f, h}. Then KG = span{f, h2}, S1 = G ∩ KG = span{f}, and Vmax = 0, so S1 is admissible. The projection onto S1 along h2 gives A_S f = 2f and A_S h = f, so 2 is in sigma(A_S), whereas sigma(K) = {0, (3 ± sqrt(5))/2}. Thus the filtered compression itself can manufacture a nonzero spurious eigenvalue. The paper's Remark 2 notes projections differ away from the invariant core, but it does not address spurious eigenvalues created there. The central guarantee should therefore be stated as preservation of represented nonzero Koopman eigenpairs; claims of 'removing spectral pollution' or 'certifying' a spectrum require an additional no-spurious theorem or explicit caveat.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Projected Koopman Operator Approximation (PKOA) framework and a Filtered EDMD construction. For a dictionary space G, an admissible subspace S is required to lie between the maximal Koopman-invariant subspace V^G_max and the one-step compatible intersection G ∩ KG, and the filtered operator is A_S = P_S ∘ (K|_G) for a projection P_S onto S. Proposition 2 shows that every Koopman eigenpair represented in G with nonzero eigenvalue is preserved by A_S. The forward-intersection chain S_{j+1} = S_j ∩ K S_j is proposed as a canonical choice of admissible subspaces, stabilizing at the maximal invariant core when K|_G is injective. The paper analyzes two projection geometries, coordinate-orthogonal and L2(μ)-orthogonal, and provides SVD-based algorithms together with an almost-sure convergence theorem under a sampling-identifiability assumption. Numerical experiments on a Kronecker flow, a polynomial system, and the Van der Pol oscillator compare the method with EDMD, T-SSD, RFB-EDMD, and ResDMD.","tokens_in":25227,"tokens_out":4699,"duration_ms":59517,"significance":"The preservation theorem and the forward-intersection construction are clean and potentially useful: the paper correctly separates the question of range selection from projection geometry, and Proposition 3 gives a canonical hierarchy of compatible subspaces. The convergence proof (Theorem 1 and Proposition 7) is a genuine contribution, and the algorithms are explicit and reproducible in structure. The main advertised strength—'removing spectral pollution'—is, however, not established by the stated theorem, because preservation of genuine eigenpairs does not exclude newly created spurious eigenvalues. If the claims are appropriately qualified, the framework is a reasonable contribution to the EDMD literature; as written, the central claim needs correction or additional theory.","major_comments":[{"comment":"The central claim that Filtered EDMD 'removes spectral pollution' is not entailed by Proposition 2. Proposition 2 only guarantees that if Kf = λf with λ≠0 and f∈G, then A_S f = λf. Since admissible S need not be invariant, an eigenvector f∈S of A_S can satisfy Kf = λf + w with w ∈ KG\\S; the projection P_S kills w and creates λ as an eigenvalue even though f is not a Koopman eigenfunction. For a concrete counterexample at the level of the paper's abstract linear framework, take H = span{f,h,h2}, Kf = 2f + h2, Kh = f, Kh2 = f + h2, G = span{f,h}. Then KG = span{f,h2} and S1 = G ∩ KG = span{f}, so S1 is admissible. The projector onto S1 along h2 gives A_S f = 2f and A_S h = f, so 2 ∈ σ(A_S), whereas σ(K) = {0, (3±√5)/2}. Thus 2 is a spurious eigenvalue manufactured by the filtered compression. The abstract's 'removes spectral pollution' and Conclusion's 'remove spurious content' therefore o","section":"Section 3.1, Proposition 2; Abstract; Section 6"},{"comment":"The empirical L2(μ)-Filtered EDMD matrix A^μ_{S1,m} is central to the projection-geometry comparison in Fig. 13, but the text never defines how it is computed. Remark 5 states that implementing the L2(μ) projector 'would additionally require estimating the Gram matrix of Kg', yet no estimation scheme or algorithm is supplied. Without a precise definition of A^μ_{S1,m}, the claim that 'the L2(μ) projector approximates the limit-cycle spectrum on the same certified subspace' is not reproducible. In addition, Section 5.3 tunes competitor tolerances to maximize lattice agreement while Filtered EDMD uses 'the default numerical-rank cutoff throughout'; this asymmetric protocol makes the comparison in Fig. 10 difficult to interpret. Please specify the L2(μ) empirical construction (including Gram estimation and regularization) and use comparable parameter-selection rules, or present the projecti","section":"Section 5.3.2, Fig. 13; Remark 5"}],"minor_comments":[{"comment":"The convergence theorem is conditional on exact-rank identifiability (rank[Y;X] = dim(G+KG)) and on the numerical-rank tolerance ε in Algorithm 1. The text acknowledges this in Remark 4, but the phrase 'mild identifiability condition' understates the practical risk: if the tolerance truncates a genuine singular direction, the empirical operator converges to the wrong filtered target. A short discussion of how to choose ε or diagnose its effect would strengthen the paper.","section":"Section 4.2, Assumption 1 and Remark 4"},{"comment":"The recursion bS_{j+1,m} := row(C_j X) ∩ row(C_j Y) assumes that row(C_j Y) correctly represents the sampled forward image of bS_{j,m}. Under Assumption 1 this is true, but without it the sampled chain can diverge from the population chain; this limitation should be stated more prominently near the algorithm.","section":"Section 4.1, Eq. (12)"},{"comment":"The statement 'Filtered EDMD uses the default numerical-rank cutoff throughout' while competitors are tuned to maximize lattice agreement should be moved to the main comparison text rather than only implied by the caption; as written it gives the impression of a favorable comparison.","section":"Section 5.3, Fig. 10"},{"comment":"There are minor typographical issues: in Section 2.3 '(g,A E,m)' is used before A_{E,m} is defined; in Eq. (7) the restriction identity A^μ_S|_S = A^S_E uses S both as a subspace and as a superscript, which is confusing; and the notation bS_{0,m} := row(X) is introduced but the population analogue S_0 = G is not explicitly aligned with it.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The preservation theorem and the chain construction are sound, and the convergence proof is careful. However, the abstract and conclusion claim that Filtered EDMD removes spectral pollution, which Proposition 2 does not guarantee; the counterexample in my report shows that spurious eigenvalues can be created. This is a central claim and should be fixed by either a no-spurious theorem or a qualified restatement. The L2(μ) empirical matrix in Fig. 13 also needs a precise definition. I would not reject the paper—the core framework is valuable—but the revision must address the overclaim and the reproducibility gap before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know up front. The preservation theorem is correct, and the forward-intersection chain is a genuinely useful construction. The other thing is that the paper's abstract and title overclaim: Proposition 2 does not imply that filtered EDMD removes spectral pollution, and a simple example shows it can actually manufacture new spurious eigenvalues. So read the paper as a preservation guarantee, not a no-pollution guarantee.\n\nWhat's new and good: the PKOA framework decouples range selection from projection geometry, allowing admissible ranges that need not be invariant. The forward-intersection chain S_{j+1}=S_j∩K S_j is natural, stabilizes at the maximal invariant subspace under injectivity, and gives a hierarchy of intermediate models. I checked the main propositions and Theorem 1; the arguments are sound. The comparison of coordinate-orthogonal versus L2(μ)-orthogonal projection, especially Proposition 4's identity, is a solid addition to the EDMD literature. The Van der Pol experiment is the most interesting part: an intermediate level reveals equilibrium-lattice structure even though the invariant core is trivial, which is something invariance-based methods cannot do.\n\nThe soft spots are real. The stress-test concern holds: with H=span{f,h,h2}, Kf=2f+h2, Kh=f, Kh2=f+h2, G=span{f,h}, the admissible S1=span{f} and the projection onto S1 along h2 give A_S f=2f and A_S h=f, so 2 is an eigenvalue of A_S while σ(K)={0,(3±√5)/2}. Filtering created a spurious eigenvalue. At minimum, the authors need to restate the guarantee as preservation and add an explicit caveat, or prove a no-spurious theorem under stronger hypotheses.\n\nThe experimental section has its own gaps. The L2(μ) filtered matrix in Fig. 13 is never specified, even though Remark 5 admits a Gram estimate is needed. In the Van der Pol comparison, competitor tolerances are tuned to maximize lattice agreement while F-EDMD uses a default cutoff, which is not an apples-to-apples test. 'Certifies' in the abstract is too strong a word for a numerical observation on one system. There is no code or data, so the figures couldn't be reproduced. To the paper's credit, Section 6 is open about only implementing the coordinate projector and not quantifying noise robustness.\n\nWho should read this: people working on EDMD variants and spectral pollution. The theoretical core is worth referee time. I would send it to review, with the expectation of major revision—rescope the claims, address the no-spurious gap, define the L2 implementation, and share code/data.","headline":"A correct preservation theorem for filtered EDMD, but the paper's claim to remove spectral pollution is not supported and a valid example shows the filter can create new spurious eigenvalues.","tokens_in":25836,"tokens_out":5168,"would_cite":true,"duration_ms":54479,"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":"Projected Koopman operators remove spectral pollution from EDMD without requiring an invariant dictionary.","keywords":["Koopman operator","spectral pollution","extended dynamic mode decomposition","filtered EDMD","forward-intersection chain","coordinate-orthogonal projection","almost-sure convergence","Van der Pol oscillator"],"falsifier":"Take the Van der Pol polynomial dictionary of total degree 10 and sample initial conditions from a single short trajectory so that rank[Y;X] < dim(G+KG). Compute A^c_{S1,m} and its eigenvalues: if the spectrum still aligns with the equilibrium lattice for every such undersampled dataset, the identifiability assumption is not doing the claimed work; if the spectrum diverges or shows pollution, the assumption is genuinely load-bearing.","tokens_in":24719,"feed_emoji":"🧮","tokens_out":6431,"duration_ms":66715,"temperature":0.7,"pith_summary":"EDMD approximates the Koopman operator on a user-chosen dictionary, but when the dictionary is not exactly invariant, the projection can manufacture spurious eigenvalues. The paper claims that this spectral pollution can be removed without ever finding an invariant subspace: any projection whose range contains all Koopman-invariant content of the dictionary and lies inside G ∩ KG exactly preserves every represented Koopman eigenpair with nonzero eigenvalue. A forward-intersection chain S_{j+1}=S_j ∩ K S_j produces such admissible ranges at every level and terminates at the maximal invariant core, so intermediate levels stay spectrally faithful yet remain much richer than the core. The authors build SVD-based algorithms for a coordinate-orthogonal projector and prove almost-sure convergence of the empirical operators under noiseless i.i.d. sampling and a rank-identifiability condition. Experiments on three systems show reduced pollution, and on Van der Pol the one-step filter reveals equilibrium spectral structure that invariant-based methods cannot see.","feed_headline":"Clean Koopman spectra need no invariant dictionary","feed_subtitle":"A forward-intersection filter preserves genuine eigenpairs while cutting EDMD's spurious ones, with almost-sure convergence.","key_machinery":"The load-bearing object is the admissible generalized projection: a linear idempotent map onto a range S contained in the dictionary, defined on the Koopman image of the dictionary, and required to contain every Koopman-invariant subspace of the dictionary. The Filtered EDMD operator A_S = P_S ∘ (K|_G) is built from such a projection, so filtering happens during operator construction rather than by screening eigenvalues afterward. The forward-intersection chain S_{j+1}=S_j ∩ K S_j selects admissible ranges, and the coordinate-orthogonal projector Π^c_S = QQ^* is computed from an SVD-based compatible coefficient subspace W_S = {c : c^*Y ∈ row(CX)}, requiring no Gram-matrix estimate.","core_discovery":"The central claim is that spectral pollution in EDMD is caused by the projection geometry, not by the absence of invariance per se. If a projection has range S with V^G_max ⊆ S ⊆ G ∩ KG, then every Koopman eigenpair with nonzero eigenvalue represented in the dictionary is exactly preserved by A_S = P_S ∘ (K|_G), even though S need not be invariant. The forward-intersection chain S_{j+1}=S_j ∩ K S_j supplies such ranges canonically, and when K restricted to the dictionary is injective it stabilizes at the maximal Koopman-invariant subspace. Two projection geometries are analyzed: a coordinate-orthogonal projector that needs no function-space Gram-matrix estimate but is basis-dependent, and an","pith_inferences":["The Gram-compatibility condition H_K Π = Π H_K suggests a concrete dictionary-design criterion: choose observables whose Koopman images are nearly L2(μ)-orthonormal, so the cheap coordinate projector approximates the measure-informed projector.","The numerical-rank tolerance ε acts like a persistence horizon; a stability scan of the retained spectrum across ε could serve as an automatic model-selection rule for choosing the filtered level.","The same certified intersection spaces could be reused outside spectral estimation, for example as pre-screens for richer dictionaries or as constraints in prediction and control tasks.","The measure-invariance property of the coordinate projector hints that the equilibrium-lattice alignment seen on Van der Pol is a property of the dictionary and the flow, not of the sampling distribution; testing on other flows with multiple invariant objects would clarify how general that separation is."],"forward_implications":["Any dictionary that contains a Koopman eigenfunction with nonzero eigenvalue keeps that eigenpair at every admissible filtered level; pollution appears only in the complementary directions.","Users can pick an intermediate level of the chain to obtain a larger, more expressive model than the maximal invariant core, with a certified guarantee on the represented nonzero spectrum.","The coordinate projector's independence from the sampling measure allows comparing spectra across datasets sampled from different regions of state space on the same certified subspace.","Because only one-step compatibility (G ∩ KG) is needed, the method applies even when the dictionary contains no nontrivial invariant subspace, the regime where exact invariance-based methods return nothing useful.","Under the stated identifiability assumption, the whole family from unfiltered EDMD through intermediate filtered levels to the maximal invariant restriction is consistent with its population version almost surely."],"supporting_citations":[{"why":"Defines EDMD, the least-squares dictionary approximation that Filtered EDMD modifies by projecting the Koopman action before eigendecomposition.","marker":"[31]"},{"why":"Establishes the population EDMD operator P^μ_G ∘ (K|_G) and its empirical convergence, the baseline that Theorem 1 extends to filtered intersections.","marker":"[17]"},{"why":"Introduces symmetric subspace decomposition; its terminal invariant space is the endpoint of the forward-intersection chain and its reduced operator is compared in Remark 3.","marker":"[10]"},{"why":"Tunable SSD; its population-level reduced operator coincides with Filtered EDMD on invariant admissible subspaces and it serves as a benchmark in the numerical comparisons.","marker":"[11]"},{"why":"Recursive forward-backward EDMD; supplies the nested-subspace comparison method and the finite-iteration convergence benchmark used in the experiments.","marker":"[12]"},{"why":"Rigorous residual-based DMD certification; ResDMD is the output-screening benchmark that Filtered EDMD contrasts with its input-filtering approach.","marker":"[6]"},{"why":"Provides the polynomial system's exact finite-dimensional Koopman representation used to construct the insufficient-observable test dictionary.","marker":"[13]"},{"why":"Supplies the limit-cycle and equilibrium Koopman spectra used as ground-truth reference lattices for the Van der Pol experiments.","marker":"[24]"}],"fun_headline_variants":["Koopman spectra cleaned without invariant dictionaries","Spurious eigenvalues eliminated by new EDMD filter","Projected Koopman operator preserves true eigenpairs","Projection geometry fixes Koopman spectra, not invariance","Filtered EDMD: exact eigenpairs without invariant basis"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The samples must separate every direction in the dictionary plus its image under the Koopman operator (rank[Y;X] = dim(G+KG)), and the numerical-rank cutoff must not discard genuine spectral directions; if either fails, the computed filtered operator is not guaranteed to be the population one the theory describes.","fun_headline_variants_meta":{"raw":{"variants":["Koopman spectra cleaned without invariant dictionaries","Spurious eigenvalues eliminated by new EDMD filter","Projected Koopman operator preserves true eigenpairs","Projection geometry fixes Koopman spectra, not invariance","Filtered EDMD: exact eigenpairs without invariant basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3325,"prompt_tokens":830,"completion_tokens":2495,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":2418}},"tokens_in":574,"tokens_out":2495,"duration_ms":21747,"temperature":1.0,"reasoning_tokens":2418,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T00:46:58.415095+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Van der Pol polynomial dictionary of total degree 10 and sample initial conditions from a single short trajectory so that rank[Y;X] < dim(G+KG). Compute A^c_{S1,m} and its eigenvalues: if the spectrum still aligns with the equilibrium lattice for every such undersampled dataset, the identifiability assumption is not doing the claimed work; if the spectrum diverges or shows pollution, the assumption is genuinely load-bearing.","supporting_citations":[{"cited_title":"1995 , note =","cited_arxiv_id":null,"evidence_quote":"Defines EDMD, the least-squares dictionary approximation that Filtered EDMD modifies by projecting the Koopman action before eigendecomposition."},{"cited_title":", journal =","cited_arxiv_id":null,"evidence_quote":"Establishes the population EDMD operator P^μ_G ∘ (K|_G) and its empirical convergence, the baseline that Theorem 1 extends to filtered intersections."},{"cited_title":"On convergence of extended dynamic mode decomposition to the","cited_arxiv_id":null,"evidence_quote":"Introduces symmetric subspace decomposition; its terminal invariant space is the endpoint of the forward-intersection chain and its reduced operator is compared in Remark 3."},{"cited_title":"Chaos , year=","cited_arxiv_id":null,"evidence_quote":"Tunable SSD; its population-level reduced operator coincides with Filtered EDMD on invariant admissible subspaces and it serves as a benchmark in the numerical comparisons."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Recursive forward-backward EDMD; supplies the nested-subspace comparison method and the finite-iteration convergence benchmark used in the experiments."},{"cited_title":"Mathematical Methods in the Applied Sciences , year =","cited_arxiv_id":null,"evidence_quote":"Provides the polynomial system's exact finite-dimensional Koopman representation used to construct the insufficient-observable test dictionary."},{"cited_title":"Spectral properties of dynamical systems, model reduction and decompositions , journal =","cited_arxiv_id":null,"evidence_quote":"Supplies the limit-cycle and equilibrium Koopman spectra used as ground-truth reference lattices for the Van der Pol experiments."}],"review_version":1}