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Avoiding spectral pollution for transfer operators using residuals

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper establishes a residual-based necessary condition that lets data-driven spectral computations reject spurious eigenvalues of transfer operators.

desk verdict Useful residual diagnostics for transfer operators, but the theorem's strict-positivity claim is false—a spurious point far outside the spectrum can pass any threshold. read the letter →

arxiv 2507.16915 v1 pith:J7PPCBV3 submitted 2025-07-22 math.DS cs.LGcs.NAmath.NAmath.SPstat.ML

April Herwig , Matthew J. Colbrook , Oliver Junge , Péter Koltai , Julia Slipantschuk This is my paper · ORCID
classification math.DScs.LGcs.NAmath.NAmath.SPstat.ML MSC 46E2246F1046N4047L5037A3037M1065P9965F99
keywords spectralpollutiontransferoperatorFrobenius–PerronKoopmankernelEDMDresidualdynamicmodedecompositionpseudospectrumHardy–Hilbertspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to give data-driven spectral computations for transfer (Frobenius–Perron) operators a way to distinguish genuine eigenvalues from spectral pollution. The authors construct a kernelized residual $k_{\mathrm{res}}(\lambda; r, M)$ that is computable from the same kernel matrices used in kernel EDMD, and prove that after taking limits it is bounded above by $\mu_1$ times the best possible $L^2$ residual of $(L-\lambda I)$. A large computed residual therefore rules out a candidate eigenvalue as spurious, while a small residual is a necessary--but not sufficient--condition for being close to the spectrum. The method is demonstrated on Blaschke product maps with known spectra and on an alanine-dipeptide molecular dynamics dataset, and it also exposes eigenvalues whose eigenfunctions lie outside the chosen function space.

What carries the argument

The workhorse is the kernelized residual $k_{\mathrm{res}}(\lambda; r, M)$, built from the kernel matrices $\mathbf{G}$, $\mathbf{A}$, $\mathbf{J}$ of kernel EDMD after a truncated singular value decomposition of the snapshot matrix. The truncation is essential: it lets the dictionary size $N$ tend to infinity before the number of data points $M$, which is the order of limits needed to recover the true residual instead of the residual of the projected operator. In practice $k_{\mathrm{res}}(\lambda)^2$ is the smallest eigenvalue of the $r\times r$ matrix $\mathbf{U} = \mathbf{J} - \lambda\mathbf{A} - \bar{\lambda}\mathbf{A}^* + |\lambda|^2\mathbf{G}$, so the whole procedure reduces to a generalized eigenvalue problem on the compressed space.

What would settle it

Run Algorithm 3.2 on an expanding Blaschke map such as $S(z)=((z-\mu)/(1-\bar{\mu}z))^2$ with $|\mu|<1/3$, where the true spectrum is known exactly, and search for any EDMD eigenvalue outside that known spectrum whose computed residual falls below the chosen tolerance; a single such false positive would show that the residual filter does not by itself eliminate spectral pollution, a possibility the theorem leaves open because it gives no lower bound.

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Extended reading notes

Core claim

The central claim is Theorem 3.2: for a Mercer kernel with largest eigenvalue $\mu_1$, the computed residual satisfies $\lim_{r\to\infty}\lim_{M\to\infty} k_{\mathrm{res}}(\lambda; r, M) \le \mu_1 \min_{\|h\|_{L^2}=1}\|(L-\lambda I)h\|_{L^2}$, and the left-hand side is strictly positive whenever the right-hand side is positive, although no uniform lower bound holds. This makes the residual a necessary condition for membership in the approximate point spectrum of the transfer operator $L$, so spurious eigenvalues produced by finite-dimensional approximations can be rejected. The same residual construction adapts the ResDMD framework of the Koopman operator to its adjoint, works when the number of snapshots is smaller than the dictionary size, and extends to reproducing kernel Hilbert spaces such as the Hardy–Hilbert space.

Load-bearing premise

The method rests on the data points forming a convergent quadrature scheme and on the kernel's reproducing kernel Hilbert space being dense in $L^2$; even under those assumptions the residual supplies only an upper bound, so thresholding can still admit false eigenvalues.

Editorial extensions

If this is right

  • Existing EDMD or kernel EDMD computations can be filtered by computing one auxiliary matrix and discarding eigenvalues whose residual exceeds a chosen tolerance.
  • The residual remains meaningful when there are fewer snapshots than dictionary functions, a regime where the standard Koopman residual vanishes identically.
  • The same machinery applies in spaces other than $L^2$, including Hardy–Hilbert and fractional Sobolev spaces, by substituting the appropriate kernel or inner product.
  • In molecular dynamics, residual filtering can flag unreliable eigenvalues before constructing metastable-state or Markov-state models from transfer-operator data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Theorem 3.2 provides no uniform lower bound, a residual below threshold cannot be used as a certificate of a true eigenvalue; the method's role is rejection, not verification.
  • By changing the kernel and observing how residuals move, one could turn the 'eigenfunction outside $L^2$' phenomenon into a systematic diagnostic for which function space the data actually supports.
  • The same truncation-and-residual construction should extend to generators of continuous-time dynamics, where spectral pollution is equally common but residual formulas would need time-derivative data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes residual-based methods for validating spectral computations of transfer (Frobenius–Perron) operators, in analogy with existing ResDMD methods for Koopman operators. The main algorithmic contribution is Algorithm 3.2, which evaluates a residual function kres(λ; r, M) for candidate eigenvalues produced by kernelized EDMD. Theorem 3.2 claims an upper bound relating this residual to the true minimal residual of L − λI, and asserts strict positivity of the double limit outside the spectrum. The authors test the method on families of Blaschke maps with known spectra and on an alanine dipeptide molecular-dynamics dataset, and they discuss the subtle role of the function space in defining the Koopman spectrum.

Significance. If the main theorem were correct, the paper would be a useful step toward certified spectral computations for transfer operators: it would provide a computable, operator-theoretically motivated residual, with code and benchmarks against analytically known spectra. The case studies, particularly the Blaschke-map examples and the rigged-Hilbert-space discussion, are informative and illustrate genuine spectral-invisibility phenomena. However, the central guarantee claimed in Theorem 3.2 is false, and the counterexample below shows that the proposed residual can vanish at points far outside the spectrum. The abstract and title claim that the algorithms compute spectra 'without spectral pollution'; the theorem as stated does not support this, and the strict-positivity statement actually contradicts the counterexample. The paper has useful heuristic content, but its advertised claim of avoiding spectral pollution is not established.

major comments (3)
  1. [Section 3.4, Theorem 3.2] The assertion that the left-hand side of (3.12) is strictly positive whenever min_{‖h‖=1}‖(L−λI)h‖>0 is false. Let S(x)=2x mod 1 on the unit circle, and let L be the transfer operator on L2(T); then σ(L) is the closed unit disk. Take λ=2, so min_{‖h‖=1}‖(L−2I)h‖ ≥ 1. Choose a Mercer kernel with Mercer features ψ_n=μ_n e_n, where e_n(z)=z^n is the Fourier basis and μ_n→0 (e.g. μ_n=ρ^{|n|}). For fixed r, the empirical PCA span converges as M→∞ to span{e_n : |n|≤N(r)} with N(r)→∞. For each r choose h_r=e_{2^{K(r)}} with 2^{K(r)}≤N(r) and K(r)→∞. Then (L−2I)h_r=e_{2^{K(r)-1}}−2e_{2^{K(r)}}, so Σ_j |⟨ψ_j,(L−2I)h_r⟩|^2 = μ_{2^{K(r)-1}}^2 + 4μ_{2^{K(r)}}^2 → 0. Hence lim_{r→∞} lim_{M→∞} kres(2;r,M)=0, while the right-hand side of (3.12) is at least μ_1>0. The proof only shows positivity for each fixed r and does not control the infimum over r; this counterexample shows the gap is fatal. Consequently, a spurious point far outside the spectrum can pass any fixed residual threshold, so the claimed pollution control is not delivered.
  2. [Appendix A, Lemma 3.1] The proof of (3.10) assumes that the quadrature estimate (A.2) holds uniformly over h ∈ span eΨ, but eΨ itself depends on M through the truncated PCA in (3.6). No argument is given that the empirical PCA subspaces converge to the r-dimensional Mercer eigenspace in a way that allows the quadrature error to be uniform along this M-dependent family. Since eΨ appears on both sides of (3.10), this is a genuine gap; a precise assumption on the data-generating quadrature and on the spectral gap of the kernel integral operator is needed before the limit can be taken.
  3. [Abstract and Section 1] The claim that the algorithms compute spectra 'without spectral pollution' is stronger than anything the theorem can support. The paper's own Interpretation paragraph correctly states that the result gives only a necessary condition and no uniform lower bound. The counterexample in the first major comment shows that even the strict-positivity part of Theorem 3.2 is false, so thresholding kres cannot certify that an accepted point is close to σ(L). The title, abstract, and introduction should be revised to present the residual as a heuristic diagnostic with a necessary-condition interpretation, rather than as a method that provably avoids spectral pollution.
minor comments (4)
  1. [Algorithm 3.2, line 6] The matrix name pU is inconsistent with the derived matrix eU in (3.13); renaming it to eU would avoid confusion with the pG, pA, pJ notation.
  2. [Equation (3.9)] The norm in the definition of kres(λ) is taken in C^M (or ℓ2^M), not in an infinite-dimensional ℓ2 space; this should be stated explicitly for clarity.
  3. [Figure 1.1 caption] The caption says 'see subsection 4.2' although the figure appears in Section 1; the cross-reference should point to Section 4.2.
  4. [Section 4.2, Figure 4.5 caption] The caption says 'See subsection 4.2 for analysis' but the figure is already inside subsection 4.2; this self-reference should be corrected or removed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the residual functional is derived from first principles and its convergence is proven independently of fitted parameters.

full rationale

The paper's central contribution is the residual kres defined in (3.9) directly from the data matrices pG, pA, pJ and the truncated SVD. Lemma 3.1 (Appendix A) proves that its M -> infinity limit is the minimum over the PCA span of sum_j |<psi_j,(L - lambda I)h>|^2. That target is not the spectrum and is not defined in terms of the conclusions of Theorem 3.2; the theorem's upper bound follows by Parseval and the decay of mu_j, with no fitted parameter entering the argument. The strict positivity/absence-of-uniform-lower-bound part is an explicit mathematical claim, and the paper honestly states in Section 3.4 that no sufficient L2 condition can be obtained, so thresholding is only a necessary test. Self-citations (e.g., [19] for the ad hoc residual ykres, [5] for the analytic Blaschke spectrum, [24] for ResDMD) are used as motivation, benchmarks, or background; none is the premise of the convergence proof. No equation in the derivation is equivalent to its input by construction, and no fitted quantity is renamed as a prediction. The reviewer's concern about Theorem 3.2's strict-positivity statement is a correctness issue, not a circularity issue.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard spectral theory, Mercer's theorem, and prior results on Blaschke maps. No new physical entities are introduced. The free parameters are algorithmic choices, not fitted to the target result.

free parameters (3)
  • kernel width c = c^2=0.01 for Blaschke maps, c=0.09 for alanine dipeptide
    The Gaussian kernel width is chosen by hand or set to a data-dependent value (the 2-norm of the empirical covariance matrix). The convergence theorem holds for any Mercer kernel, but the numerical residuals and detected spectrum depend on this choice.
  • compression rank r = chosen per experiment, not reported precisely
    The algorithm truncates the SVD to r principal components. Convergence requires r tending to infinity, and finite r is a user choice affecting the residual computation.
  • tolerance epsilon = threshold in figures (e.g., 10^-p levels)
    The residual threshold used to accept or reject eigenvalues. It is a user-defined parameter and not fitted to data.
assumptions (5)
  • domain assumption Known spectral structure of transfer operators for expanding circle maps (Bandtlow, Just, Slipantschuk, 2017)
    The paper uses the analytic spectrum of L on H2(Ar) as ground truth in the Blaschke examples. This is a prior result by the authors, not derived in this paper.
  • standard math Mercer's theorem: a continuous symmetric positive definite kernel admits an expansion in an orthonormal basis
    Used to define the kernelized dictionary and the RKHS in Sections 2.3 and 3.4.
  • domain assumption The RKHS generated by the chosen kernel is dense in L2
    Required for the limit (3.12) in Theorem 3.2. Not true for all kernels and not verified for the Gaussian kernels used in the experiments.
  • domain assumption Snapshots are generated by a convergent quadrature scheme
    The proof of Lemma 3.1 assumes deterministic or random quadrature convergence, as in equations (A.2) and (A.4). In the protein-folding example, the data are trajectory samples and this property is not established.
  • domain assumption Gelfand triple H2(Ar) ⊂ L2(T) ⊂ H2(Ar)* with the isometric isomorphism to Xr
    Used in Section 4.1 to interpret the dual space and compute residuals in the Hardy-Hilbert space. This relies on [5].

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Pith. "Pith review of Avoiding spectral pollution for transfer operators using residuals." pith.science (2026). https://pith.science/paper/J7PPCBV3

@misc{pith2026250716915,
  author       = {Pith},
  title        = {Pith review of: Avoiding spectral pollution for transfer operators using residuals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7PPCBV3}},
  note         = {Machine review of arXiv:2507.16915}
}
read the original abstract

Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.

Figures

Figures reproduced from arXiv: 2507.16915 by the authors.

Figure 1.1
Figure 1.1. Detection of spectral pollution using the proposed methods. The right subfigure is a zoomed-in view of the left for emphasis. Black dots: true eigenvalues of the Koopman operator for a mixing dynamical system (in a space larger than L 2 ); see subsection 4.2. Orange crosses: EDMD approximation of selected leading eigenvalues in the bespoke space. Contour lines approximate boundaries of ϵ-pseudospectra (logarithmic s… view at source ↗
Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figures from the paper (12 more)
Figure 4.1
Figure 4.1. Figure 4.1: Spectrum of L|H2(Ar) (black dots), spectrum of the EDMD matrix L (orange crosses), residuals calculated using Algorithm 2.2 (contours are logarithmically scaled). the ResDMD Algorithm 2.2 to compute pseudospectra of K [PITH_FULL_IMAGE:figures/full_fig_p015_4_1.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 4.2
Figure 4.2. Figure 4.2: Left: residuals calculated using Algorithm 2.2 with the inner product on H2 (Ar) ∗ . Right: spectrum of Kp with residuals calculated using Algorithm 3.2, In both cases, black: true spectrum of L|H2(Ar) , orange: spectrum of the approximated matrix (left: L using the …
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 4.3
Figure 4.3. Figure 4.3: Residuals calculated using Algorithm 2.2 on the fractional Sobolev spaces Hs (T). Left: On H−1 (T), right: on H−6 (T). In both cases, black: true spectrum of L|H2(Ar) , orange: spectrum of L. Contour lines are logarithmically scaled. 17 [PITH_FULL_IMAGE:figures/full…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 4.4
Figure 4.4. Figure 4.4: “Deviation from normality” of the operator [PITH_FULL_IMAGE:figures/full_fig_p018_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Spectrum of L (and hence also of K = L ∗ ) with residuals calculated using Algorithm 2.2 in the fractional Sobolev space H−4 (left) and in the Hardy–Hilbert space H2 (Ar) ∗ with r = 0.755 (right). In both cases, black: true spectrum of L|H2(Ar) , orange: spectrum of …
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 4.6
Figure 4.6. Figure 4.6: Alanine dipeptide molecule skeleton [31]. The dihedral angles φ and ψ are the primary determining factors of the shape and chemical reaction properties of the molecule. Among many other approaches [42], recently, kernel-type methods have been proposed for a discretiz…
Figure 4.7
Figure 4.7. Figure 4.7: Spectrum of Kp with residuals computed by Algorithm 3.2. Contour lines are logarithmically scaled. For completeness, we also show the eigenfunctions at the two eigenvalues close to 1 projected onto the two dihedral angles, cf. [40, 41], in [PITH_FULL_IMAGE:figures/f…
Figure 4.8
Figure 4.8. Figure 4.8: Left: First two nontrivial eigenfunctions of [PITH_FULL_IMAGE:figures/full_fig_p021_4_8.png]

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