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Projected Koopman operators remove spectral pollution from EDMD without requiring an invariant dictionary.

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2026-08-05 00:46 UTC pith:XSNNYXA2

load-bearing objection 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. the 2 major comments →

arxiv 2608.02661 v1 pith:XSNNYXA2 submitted 2026-08-01 math.NA cs.ITcs.NAmath.DSmath.IT

Beyond Invariant Dictionary: Data-Driven Koopman Spectral Recovery with Filtered Extended Dynamic Mode Decomposition

Siji Chen , Igor Mezi\'c , Sui Tang This is my paper
classification math.NA cs.ITcs.NAmath.DSmath.IT
keywords Koopman operatorspectral pollutionextended dynamic mode decompositionfiltered EDMDforward-intersection chaincoordinate-orthogonal projectionalmost-sure convergenceVan der Pol oscillator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 3.1, Proposition 2; Abstract; Section 6] 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
  2. [Section 5.3.2, Fig. 13; Remark 5] 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
minor comments (4)
  1. [Section 4.2, Assumption 1 and Remark 4] 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.
  2. [Section 4.1, Eq. (12)] 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.
  3. [Section 5.3, Fig. 10] 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.
  4. [General] 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.

Circularity Check

1 steps flagged

Near-definitional preservation guarantee, but the framework's construction, projection geometry, and convergence results are independent and not circular.

specific steps
  1. self definitional [Section 3.1, Definition 4 and Proposition 2]
    "We call a subspace S admissible if V G max ⊆ S ⊆ G ∩ KG ... For an admissible S, let P S : KG → S be a projection... The Filtered EDMD operator ... A S := P S ◦(K| G). Proposition 2: ... If f∈G\{0} and Kf=λf, λ≠0, then f∈ran(P) and A P f=λf. Proof. Since λ≠0, the space span{f} is Koopman-invariant and is therefore contained in ran(P) by (C4)."

    Condition (C4) already states that every Koopman-invariant subspace of G is contained in the projection range. For an eigenfunction f with nonzero eigenvalue, span{f} is exactly such an invariant subspace, so the conclusion f∈ran(P) is a direct instantiation of the definition; A_P f = P(Kf)=P(λf)=λf then follows from linearity. The preservation 'theorem' is therefore an unpacking of the admissibility axioms rather than a derived consequence. This is a benign encoding—the paper's substantive contributions are the construction of admissible ranges via S_{j+1}=S_j∩K S_j, the projection-geometry comparison, and the LLN convergence proofs, none of which assume the conclusion—but as stated the guarantee is definitional.

full rationale

The paper's main derivation chain is self-contained. Proposition 2's preservation guarantee is indeed encoded in the definition of admissibility (C4), so that specific step is definitional rather than a substantive derivation. However, this is not a damaging circularity: the paper does not use Proposition 2 as an empirical prediction, and the core contributions—the forward–intersection chain S_{j+1}=S_j∩K S_j, its maximality (Proposition 3), the comparison of coordinate and L^2(µ) projections (Proposition 4), the SVD-based algorithms, and the almost-sure convergence under sampling identifiability (Theorem 1, Remark 7, Proposition 7)—are independent mathematical results proved from explicit assumptions. The numerical experiments compare outputs against standard reference spectra; the use of Mezić's earlier work in Appendix B is for well-known Floquet/linearization reference spectra, not to force the method's outputs. The skeptic's objection about spurious eigenvalues created by non-invariant admissible ranges is a valid correctness concern about the 'removes spectral pollution' claim, but it is not a circularity: it concerns whether the stated guarantee is sufficient for a stronger conclusion, not whether the derivation reduces to its inputs. Overall, there is no load-bearing self-citation chain or fitted-input-called-prediction; the only near-circular element is the definitional character of the preservation proposition, which does not undermine the independent content of the framework.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim rests on standard linear algebra (projection identities, SVD null-space facts), the strong law of large numbers, and three domain assumptions stated in the paper: L2-independence of the dictionary, sampling identifiability on G + KG, and injectivity of K|_G for the coefficient-basis identification. The VdP reference spectra are analytic expansions from [24], an author-cited framework. The only free parameter in the proposed pipeline is the numerical-rank tolerance in Algorithm 1, whose default value is unspecified. No new physical or mathematical entities are postulated; the forward-intersection chain and admissible ranges are derived objects built from the given dictionary and Koopman action.

free parameters (2)
  • numerical-rank tolerance ε in Algorithm 1 = unspecified default (VdP); absolute cutoff ε or relative ε·σ_max
    Defines which singular directions survive in the sampled intersections; the selected chain levels and ranks in Figs. 10-11 depend on it, and its default is undisclosed.
  • Competitor tolerances (T-SSD, RFB-EDMD, ResDMD) = tuned per system to maximize lattice agreement (VdP)
    Comparison-method parameters chosen per experiment; Filtered EDMD uses a fixed default, making the comparison asymmetric.
axioms (5)
  • domain assumption Dictionary {g_1,...,g_n} is linearly independent in L2(μ)
    Stated in Theorem 1 and Proposition 7; ensures G ≻ 0 so the coefficient representation of G is an isomorphism and EDMD population limits exist (Section 2.2).
  • domain assumption Sampling identifiability: E_m injective on G + KG (Assumption 1, eq. 14)
    Used in Proposition 6 to show intersections commute with sampling and in Remark 7 to extend almost-sure convergence to the full chain.
  • domain assumption K|_G injective when identifying Kg as a basis of KG
    Section 3.2.3 uses this to form the basis Ψ_K and make H_K positive definite; the non-injective extension is sketched via ker Ψ_K fixups.
  • domain assumption Van der Pol reference lattices Λ_lc, Λ_eq (Appendix B), built following [24]
    Experiments treat these analytic spectra as ground truth; the construction follows Mezić's spectral-expansion framework [24], an author-cited background theory not reproved here.
  • standard math Strong law of large numbers for empirical Gram matrices
    Theorem 1 and Proposition 7 Step 2 use almost-sure convergence of (1/m)XX*, (1/m)YX*, (1/m)YY* to G, B, H_K for i.i.d. samples.

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Cite this review

Pith. "Pith review of Beyond Invariant Dictionary: Data-Driven Koopman Spectral Recovery with Filtered Extended Dynamic Mode Decomposition." pith.science (2026). https://pith.science/paper/XSNNYXA2

@misc{pith2026260802661,
  author       = {Pith},
  title        = {Pith review of: Beyond Invariant Dictionary: Data-Driven Koopman Spectral Recovery with Filtered Extended Dynamic Mode Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSNNYXA2}},
  note         = {Machine review of arXiv:2608.02661}
}
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read the original abstract

The Koopman operator provides a linear framework for analyzing nonlinear dynamical systems through spectral properties. Extended Dynamic Mode Decomposition (EDMD) approximates this operator from data, but non-invariant dictionaries can introduce spurious eigenvalues. We introduce the Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators. The framework projects the Koopman action onto admissible dictionary subspaces that need not be invariant, while exactly preserving every represented Koopman eigenpair with nonzero eigenvalue. A forward--intersection chain provides a canonical hierarchy of compatible subspaces, connecting the full dictionary to its maximal invariant core while retaining useful intermediate models. We analyze two projection geometries: a coordinate-orthogonal projector, which requires no function-space Gram-matrix estimate but is basis-dependent, and a function-space orthogonal projector, which recovers population EDMD at the unfiltered level. We characterize their relationship to EDMD and existing subspace-selection methods. We also develop SVD-based algorithms for constructing sampled forward--intersection spaces and implementing the coordinate projector. Under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts. Experiments on a Kronecker flow, a polynomial system, and the Van~der~Pol oscillator demonstrate reduced spectral pollution. For Van~der~Pol, the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, whereas the $L^2(\mu)$ projector approximates the limit-cycle spectrum on the same certified subspace.

Figures

Figures reproduced from arXiv: 2608.02661 by Igor Mezi\'c, Siji Chen, Sui Tang.

Figure 1
Figure 1. Figure 1: Kronecker flow trajectories in C via z(θ) = e iθ + e iαθ: (a) rational α (periodic); (b)–(d) irrational α with increasing time (trajectory fills the curve densely). Spectral recovery and matrix structure. The EDMD matrix is 18 × 18 with full rank; its spectrum contains the 10 correct Koopman eigenvalues plus 8 spurious ones ( [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spectra of EDMD (left) and Filtered EDMD (right) on the Kronecker flow. EDMD produces 8 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Matrix heatmaps (absolute value) for the reference Koopman matrix with respect to [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reported matrix rank versus tolerance ε for the four parameterized methods (Kronecker flow). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Filtered EDMD matrices at three tolerance levels, showing progressive removal of spurious couplings [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spectra of EDMD and Filtered EDMD on the polynomial system ( [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Matrix heatmaps (absolute value) for the polynomial system. The reference Koopman matrix is [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mean relative trajectory error (%) over 1000 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Van der Pol oscillator (µ = 0.3). Left: vector field with the limit cycle (blue), the unstable equilibrium at the origin (red), and streamlines colored by flow speed. Middle and right: EDMD eigenvalues (purple) versus the limit-cycle lattice Λlc (green circles) in discrete and continuous time, respectively. Spectral comparison across methods. EDMD does not resolve the limit-cycle lattice Λlc: its 166 eigen… view at source ↗
Figure 10
Figure 10. Figure 10: Spectral outputs for the Van der Pol oscillator. Panels (a)–(c) compare T-SSD (dim = 24), [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Filtered EDMD eigenvalues along the forward–intersection chain [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Local Koopman-eigenfunction approximation from Filtered EDMD for the Van der Pol oscillator [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Spectra under three sampling measures for the Van der Pol oscillator with a polynomial dictionary [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗

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