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REVIEW 3 major objections 6 minor 26 references

Beyond DSA: Conjugacy-based Comparison of Dynamical Systems

T0 review · 3 major / 6 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Orthogonal alignment of Koopman operators is neither necessary nor sufficient for dynamical conjugacy; alignments must come from state-space bijections.

desk verdict Clean theoretical fix to DSA: orthogonal Koopman alignment is neither necessary nor sufficient for conjugacy; CSA is the right object when you can supply a conjugacy family. read the letter →

arxiv 2607.04493 v1 pith:7B553QHX submitted 2026-07-05 q-bio.NC math.DS

Prakhar Godara , Pang Shiang Tay , Marcelo G. Mattar This is my paper · ORCID
classification q-bio.NCmath.DS MSC 37M1037C1547B33
keywords topologicalconjugacyKoopmanoperatorDynamicalSimilarityAnalysiscompositionEDMDHankel-DMDobservabledictionariescomputationalequivalence
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 asks when two dynamical systems compute the same way even if their coordinates or measurements look different. The natural mathematical answer is topological conjugacy: a state-by-state relabelling that makes their transition rules identical. Dynamical Similarity Analysis (DSA) tries to detect this by checking whether finite Koopman matrices can be aligned by an orthogonal change of basis. The authors show that this test is the wrong one: genuine conjugacies can induce non-orthogonal transfer matrices that DSA cannot find, and non-conjugate systems can still have orthogonally equivalent operators that DSA cannot tell apart. They introduce Conjugacy-based Similarity Analysis (CSA), which only admits alignments that come from candidate state-space bijections, and prove that the matrix CSA fits is the finite-data projection of the composition operator induced by that bijection. Controlled experiments with explicit dictionaries and with delay-embedding bases show that the distinction changes which systems are judged equivalent.

What carries the argument

The composition (pullback) operator Ch induced by a candidate bijection h, whose finite-data projection onto the observable dictionaries is the transfer matrix P heta that CSA optimizes; the CSA score is then the residual of the intertwining relation PθF ≈ GPθ.

What would settle it

Find a pair of genuinely conjugate systems, give CSA a conjugacy family that does not contain the true map (or any close approximation), and check whether CSA still reports a small residual while a ground-truth-aware method would; or exhibit non-conjugate systems whose projected composition operators still intertwine under the supplied family.

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Extended reading notes

Core claim

Orthogonal similarity of finite-dimensional Koopman matrices is neither necessary nor sufficient for topological conjugacy. The correct intertwiner is the composition operator induced by a state-space bijection (or its finite-data projection onto the chosen observable dictionaries). CSA recovers that object by construction; DSA does not.

Load-bearing premise

CSA only works if the user supplies a family of candidate relabellings that already contains, or closely approximates, the true conjugacy; the paper’s experiments hand it that family.

Editorial extensions

If this is right

  • A small DSA distance no longer licenses a claim that two systems implement the same computation, and a large distance no longer licenses the claim that they do not.
  • Any conjugacy-aware Koopman comparison must search over alignments that can be interpreted as projected pullbacks of state-space maps, not over arbitrary orthogonal matrices.
  • When observable bases are learned implicitly (Hankel/SVD pipelines), the basis-transfer problem cannot be ignored; orthogonal alignment alone conflates coordinate artefacts with dynamical differences.
  • Claims of shared dynamics across recording sessions, animals, models, or architectures must make the admissible class of relabellings explicit.
  • Weaker notions of similarity (spectral match, task-relevant subspace match) remain useful, but must not be conflated with conjugacy.

Reading between the lines

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

  • If the candidate-bijection family is misspecified, CSA can fail silently; practical deployments will need diagnostics that flag empty or poorly covering families rather than reporting a residual alone.
  • The same pullback-construction idea can be ported to continuous-time flows and to stochastic generators without changing the core argument that the intertwiner must be composition-induced.
  • Cross-subject or cross-session BCI alignment that currently uses orthogonal Procrustes-style maps may be systematically under- or over-estimating dynamical equivalence when the true coordinate change is non-volume-preserving.
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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 / 6 minor

Summary. The paper argues that Dynamical Similarity Analysis (DSA) is neither necessary nor sufficient as a test of topological conjugacy between dynamical systems, because DSA aligns finite-dimensional Koopman matrices by arbitrary orthogonal similarity, whereas conjugacy induces a specific composition (pullback) operator that need not be unitary and is not the only source of unitary equivalence. The authors formalize Conjugacy-based Similarity Analysis (CSA), which restricts alignments to matrices P_θ obtained by least-squares projection of candidate state-space bijections h_θ, and prove (Propositions 2–4) that the fitted P_θ is the empirical/population projected composition operator. Controlled EDMD and Hankel-DMD experiments on logistic maps and linear contractions, under known conjugacies and dictionary changes, show that CSA preserves conjugacy structure in regimes where DSA does not.

Significance. If the necessity/sufficiency critique and the projection interpretation hold—and the classical counter-examples (Appendices A–C) and Propositions 2–4 make a strong case—the paper supplies a needed clarification for a method already used in systems neuroscience and deep learning. The distinction between orthogonal operator alignment and conjugacy-induced pullbacks is load-bearing for any claim that two circuits or models implement the same computation. Strengths include: (i) clean operator-theoretic framing of conjugacy as intertwining by a composition operator; (ii) classical Bernoulli-shift and non-unitary conjugacy counter-examples; (iii) explicit finite-data projection theorems for P_θ; (iv) controlled experiments that isolate alignment class rather than confounding approximation error. The work is more a principled template and diagnostic than a ready high-dimensional pipeline, which the Discussion largely acknowledges.

major comments (3)
  1. Appendix G.5 / §5.2: CSA-Krylov constructs Φ2 by evaluating delay-embedded Krylov features via the known maps f_i at arbitrary grid points (ϕ_i(x)=V_{i,r}^⊤(x,f_i(x),…)). This is not available in a purely data-driven setting where only trajectories are observed. The abstract and §5 claim that the distinction matters when dictionaries are chosen “implicitly from data”; the Hankel instantiation as written still requires the dynamics to build the conjugacy-induced transfer. Either restate the Hankel results as requiring known (or separately identified) maps, or provide a trajectory-only construction of P_θ and re-run the conjugate-pair sweeps.
  2. §5 (opening paragraph and all reported panels): every CSA experiment is given a parametric family that contains the ground-truth bijection. That design correctly isolates the alignment-class question, but it leaves the empirical claim that “CSA correctly identifies conjugate systems” untested under family misspecification. A single negative control—e.g., CSA restricted to a family that excludes h_θ, or to a wrong parametric form—would show whether the method fails gracefully (large residual) rather than spuriously reporting conjugacy. Without that, the experiments support only the weaker claim that the right alignment class works when the true map is supplied.
  3. §3.1 and Appendix B: the Bernoulli-shift argument shows unitary equivalence of infinite-dimensional Koopman operators without conjugacy, which correctly establishes that orthogonal/unitary alignment is not sufficient in the infinite-dimensional limit. The finite-dimensional DSA criterion (Eq. 10) is a different object. A short remark on how the finite-rank EDMD/Hankel truncations of those shifts behave under DSA would close the gap between the classical counter-example and the numerical method actually used in practice.
minor comments (6)
  1. Figure 1 caption: typo “comaprison” → “comparison”; also “Pairwise (comaprison) distance matrices”.
  2. Eq. (10) writes Q F Q^{-1} with Q∈O(m); for orthogonal matrices this is Q F Q^⊤. Stating the transpose form would match standard DSA notation and avoid implying a general GL(m) search.
  3. §4, Eq. (12): the normalisation ∥P_θ∥_F is sensible but not unique; a brief note on scale invariance (or lack thereof) under rescaling of dictionaries would help readers compare CSA residuals across dictionary sizes.
  4. Figure 5b–c: plotting d_DSA/√r alongside CSA is convenient for display, but the main text should state once that conjugacy-invariance conclusions use the raw residual behaviour, not the 1/√r factor.
  5. References: Ostrow et al. (2023) is the central foil; a short related-work paragraph on other conjugacy/isomorphism tests in applied Koopman theory (beyond Mezic 2020 and Korda–Mezić 2018) would situate CSA more clearly.
  6. Appendix I Algorithm 1: the returned d_CSA is √OBJECTIVE, while Eq. (12) is already a normalised Frobenius residual; confirm consistency between the algorithm and the displayed formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: conjugacy intertwining, non-containment of orthogonal vs composition classes, and projection theorems are self-contained operator-theory arguments; controlled experiments openly supply the true bijection family.

full rationale

The paper's load-bearing claims do not reduce to their inputs by construction. Topological conjugacy is defined as h◦f=g◦h (Eq. 1); the composition operator Ch and the intertwining Ch Kg = Kf Ch (Eqs. 2–4) and matrix form PF=GP (Eqs. 6–9) follow by direct substitution, not by fitting. Necessity/sufficiency failure of orthogonal alignment is established by independent classical counter-examples: a unitary non-multiplicative map on span{1,x} (Appendix A), unitarily equivalent but non-conjugate Bernoulli shifts with unequal periodic-point counts (Appendix B, citing Rédei–Werndl 2012), and non-measure-preserving conjugacies such as hθ(x)=x+θsin(πx) whose pullbacks are non-unitary (Appendix C). Propositions 2–4 identify Pθ,M as the empirical L2 projection of Chθ onto the dictionaries and prove a.s. convergence to the population projection and strong-operator recovery under complete bases—standard least-squares/EDMD arguments, not circular redefinitions. CSA's distance (Eq. 12) is an optimization residual over a declared candidate family Θ; the experiments explicitly give CSA a family containing the ground-truth map to isolate the alignment-class distinction, and the Discussion acknowledges family misspecification as a limitation rather than hiding it. No self-citation chain, uniqueness theorem imported from the authors, or fitted parameter renamed as prediction supports the central claim. Score 0 is appropriate.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The paper rests on standard dynamical-systems and operator-theoretic background plus the modeling choice that a tractable parametric family of bijections is available. No new physical entities are postulated; CSA is a computational procedure, not an ontological claim. Free parameters appear only in the synthetic experiments (dictionary size, rank, conjugacy-family parameters) and do not enter the theoretical claims.

free parameters (2)
  • candidate conjugacy family parameter θ (and a,n for ha,n)
    In experiments the true map is drawn from a known parametric family that is also supplied to CSA; the recovered θ̂ is reported but the family itself is chosen by hand to contain the ground truth.
  • observable dictionary size m / SVD rank r / delay dimension d
    Finite-dimensional truncation parameters (m=25, r up to 99, d=100) chosen for the controlled examples; convergence is checked but the values remain free design choices.
assumptions (4)
  • domain assumption Topological conjugacy is the intended notion of 'same computation' (h∘f = g∘h for a bijection h).
    Stated in Section 1–2; other equivalences (spectral, input–output) are acknowledged as weaker but not used as the target.
  • standard math Koopman operators act by composition on L2 observables and finite EDMD/Hankel-DMD matrices approximate them.
    Standard background (Koopman 1931, Williams et al. 2015, Arbabi & Mezic 2017) used throughout Sections 2–5.
  • standard math A composition operator is multiplicative and is unitary only when the inducing map is measure-preserving.
    Used in Section 3 and Appendices A–C to separate orthogonal from composition-induced alignments.
  • ad hoc to paper A tractable parametric family of candidate bijections containing the true conjugacy is available to CSA.
    Explicitly assumed in Section 5 experiments and discussed as a limitation in Section 6; without it CSA cannot search.
invented entities (1)
  • Conjugacy-based Similarity Analysis (CSA) distance independent evidence
    purpose: Restricts the alignment search to projected composition operators induced by candidate state-space bijections, yielding a residual that can support conjugacy claims.
    Defined in Section 4 (Eqs. 11–12); independent evidence is the controlled recovery of known conjugacies, but the entity is a method, not a physical postulate.

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Cite this review

Pith. "Pith review of Beyond DSA: Conjugacy-based Comparison of Dynamical Systems." pith.science (2026). https://pith.science/paper/7B553QHX

@misc{pith2026260704493,
  author       = {Pith},
  title        = {Pith review of: Beyond DSA: Conjugacy-based Comparison of Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7B553QHX}},
  note         = {Machine review of arXiv:2607.04493}
}
read the original abstract

Comparing whether two dynamical systems implement the same computation despite differences in coordinates or measurements is a central problem in neuroscience and machine learning. Dynamical Similarity Analysis [DSA; Ostrow et al., 2023] addresses this problem by aligning finite-dimensional Koopman approximations through an orthogonal similarity transformation. Here we show that orthogonal alignment is neither necessary nor sufficient for topological conjugacy: conjugate systems may require a non-orthogonal basis-transfer matrix that DSA cannot capture, while non-conjugate systems may have orthogonally equivalent Koopman operators that DSA fails to distinguish. We use this observation to formulate Conjugacy-based Similarity Analysis (CSA), which restricts alignments to those induced by candidate state-space bijections rather than arbitrary orthogonal matrices. We prove that CSA's fitted alignment is the finite-data projection of the composition operator associated with the candidate bijection, and use controlled examples to show why this distinction matters when observable dictionaries are chosen explicitly or implicitly from data. These results clarify what Koopman-based similarity measures must ensure to support claims of identifying conjugacies between computational systems.

Figures

Figures reproduced from arXiv: 2607.04493 by the authors.

Figure 1
Figure 1. State-space transformations with fixed dictionary. Pairwise (comaprison) distance matrices for the two methods - CSA (a) and DSA (b), when one copy of the logistic-map family is transformed by hθ(x) = x + θ sin(πx) and both Koopman operators are represented in the same Fourier dictionary for both the methods. Scatter plots for reference and comparison matrix entries for CSA (c) and DSA (d). (a) (b) (c) 0.0 0.1 0.2 0… view at source ↗
Figure 2
Figure 2. Robustness to state-space transformations. (a) nRMSE as a function of θ for the two methods - CSA (red), DSA (blue). (b) Correlation between entries of the comparison and matrix entries as a function of θ. (c) CSA’s recovered conjugacy parameter ˆθ as a function of θ. 5.1 EDMD-based implementation We instantiate the EDMD pipeline on the logistic-map family fk(x) = k x(1 − x), x ∈ [0, 1], k ∈ [2, 4], sweeping k acros… view at source ↗
Figure 3
Figure 3. Cross-dictionary comparison. Pairwise (comparison) distance matrices for the two meth￾ods - CSA (a) and DSA (b), when the same logistic-map family is represented in different dictio￾naries (Fourier (horizontal axis) vs. Legendre (vertical axis)), θ = 0. Scatter plots of reference and comparison matrix entries for CSA (c) and DSA (d). (a) (b) 0.0 0.2 0.4 Basis distance 0.0 0.2 0.4 0.6 0.8 nRMSE DSA CSA 0.0 0.2 0.4 Ba… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Robustness to basis mismatch. (a) nRMSE as a function of basis distance (Appendix E) for the two methods - CSA (red), DSA (blue). (b) Correlation between entries of the comparison and matrix entries as a function of distance between the bases. known map hθ, but represe…
Figure 5
Figure 5. Figure 5: Hankel-DMD/Krylov comparison under smooth conjugacies. (a) Deviations ha,n(x)− x for a = 0.8. (b) Conjugate-pair distances under sweeps over n at fixed a = 0.8 and over a at fixed n = 4, at rank r = 20. (c) Conjugate-pair distances as a function of rank at (a, n) = (0.…
Figure 6
Figure 6. Figure 6: Pairwise distance matrices of within transformation pairs for both DSA (top) and CSA [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: State-prediction residual ρx versus truncation, by representation. (a) Fourier-EDMD on fk; lighter shades denote larger nb. (b) Legendre-EDMD on fk. (c) Hankel-DMD on the smooth-bijection setup of Section 5.2 (linear contractions fr and their smooth-conjugate twins g (…

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Reviewed July 11, 2026 · model on record in the stance chip above.