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Unfolding Generative Flows with Koopman Operators: Trajectory-Preserving Linearization

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

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Pith's one-line read The paper claims that a single linear operator in a learned lifted space reproduces the full non-linear trajectory of a pre-trained flow-matching model, so sampling becomes one step and the operator's spectrum becomes a tool for…

desk verdict The consistency loss is a real step beyond endpoint distillation, but the paper's 'trajectory-preserving' claim currently rests on latent-space evidence; worth a serious review with a request for pixel-space trajectory metrics. read the letter →

arxiv 2506.22304 v3 pith:DJDGOQHX submitted 2025-06-27 cs.LG cs.CV

Erkan Turan , Aristotelis Siozopoulos , Louis Martinez , Julien Gaubil , Emery Pierson , Maks Ovsjanikov This is my paper
classification cs.LGcs.CV
keywords KoopmanoperatortheoryContinuousNormalizingFlowsConditionalFlowMatchingtrajectory-preservinglinearizationone-stepgenerativesamplingspectralanalysisofdynamicssimulation-freeconsistencylossmodelinversionandediting
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

This paper tries to show that the non-linear, time-dependent dynamics of a pre-trained Continuous Normalizing Flow (CFM) can be replaced by a single fixed linear operator without losing the teacher's behavior along the way. Its claim is that a learned encoder lifts each state into a 1024-dimensional Koopman coordinate space in which the teacher's vector field becomes exactly the linear law $\mathrm{d}z/\mathrm{d}t = Lz$, matching the teacher infinitesimally at every point of the trajectory and not merely at its endpoints. If that holds, sampling collapses to one step — encode the noise, apply one matrix exponential, decode — and the eigenvalues and modes of $L$ become handles for analyzing, editing, and inverting the generative process. This matters because existing one-step distillation methods treat the teacher as a black box and discard its intermediate dynamics, whereas a trajectory-preserving linearization keeps those dynamics and makes them readable through linear algebra alone.

What carries the argument

The central object is a learned finite-dimensional Koopman representation: a diffusion-style SongUNet encoder–decoder pair that maps the data space to a 1026-dimensional lifted space (1024 observables plus time and a constant coordinate), together with a dense affine generator matrix $L$ whose constrained block structure forces $\dot{t} = 1$ and $\dot{1} = 0$. The load-bearing identity is the Koopman generator equation $Lg = \nabla g \cdot v_t$, adapted from the autonomous-system generator formula to non-autonomous flow matching; the consistency loss is exactly the square of this equation averaged over the marginal path distribution. The mechanism that makes the objective trainable is the unbiased-estimator identity of Proposition 3, which rewrites the expectation over the unknown marginal path as an expectation over data samples and conditional paths — the simulation-free trick that makes CFM itself trainable, applied here with the teacher's known marginal velocity. Finally, the real Schur decomposition of $L$ splits the flow into independent exponential and spiral modes, and the paper orders these modes by their eigenvalues to reveal a coarse-to-fine hierarchy in generation.

What would settle it

Run the identical pipeline on a teacher whose dynamics are known to have no finite-dimensional Koopman invariant subspace — for instance a chaotic or polynomial flow of the kind discussed in the literature the paper cites on finite-dimensional Koopman forms — and record the consistency-loss residual and trajectory MSE at a large latent dimension. If the residual stays bounded well away from zero while endpoints still match, the trajectory-preserving claim is refuted: the method would be fitting boundaries and only approximating interiors. A complementary test that applies to the paper's own models is to compare intermediate marginals: integrate the teacher to a time $t \in (0,1)$ and compare the distribution of $x_t$ with one-step Koopman rollouts $\exp(tL)$ from the same noises; agreement at $t=1$ with divergence in the middle would falsify full-trajectory linearization.

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Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that a trajectory-preserving linearization of a pre-trained Conditional Flow Matching model exists and is learnable. For a teacher with velocity field $v_t$, the paper constructs an encoder $g_\phi$, a decoder $g_\psi^{-1}$, and a fixed generator matrix $L$ such that the lifted coordinate $z_t = [1, t, g_\phi(t, x_t)]$ evolves linearly, $\mathrm{d}z/\mathrm{d}t = Lz$, with the block structure of $L$ forcing time to advance at unit rate. The decisive ingredient is the consistency loss $L_{\mathrm{cons}} = \mathbb{E}\,\lVert L g(t,x_t) - \nabla g(t,x_t)\cdot v_t(x_t)\rVert^2$, which is an unbiased, simulation-free estimate of the true marginal objective (Proposition 3); the paper proves that the naive conditional substitute carries a positive gap (Proposition 2) and that the decoder's reconstruction loss fixes the coordinate freedom left by linear gauge invariance (Proposition 1). With this loss, one-step rollouts $x_1 \approx g^{-1}(\exp(L)z_0)$ reproduce teacher trajectories to mean squared error near $5\times10^{-6}$, whereas the same pipeline without the loss matches endpoints only, with error near $1.3\times10^{-3}$. The paper further reports that the spectrum of $L$ then carries meaning: Schur modes arrange coarse-to-fine, single-mode perturbations yield coherent attribute edits (sunglasses coherence 0.97), and images can be inverted back to noise through $\exp(-L)$.

Load-bearing premise

Everything rests on the assumption that the teacher's flow admits a finite-dimensional invariant Koopman subspace — that roughly 1024 learned coordinates are enough to make the teacher's dynamics exactly linear. The paper itself flags this as a strong condition that need not hold for arbitrary non-linear flows; if it fails, the consistency loss cannot approach zero and the linearization is only approximate.

Editorial extensions

If this is right

  • Sampling from a CFM becomes one step — encode the noise, apply the matrix exponential once, decode — with the paper reporting FID 10.1 on FFHQ and 16.7 on CIFAR-10 at about 37 ms per image.
  • The linearization is not a boundary fit: Koopman rollouts track the teacher's full trajectories to mean squared error near $5\times10^{-6}$ with the consistency loss, versus roughly $1.3\times10^{-3}$ without it.
  • The spectrum of $L$ becomes a control surface: moving along a single Koopman mode edits attributes (CLIP coherence 0.97 for sunglasses, 0.94 for brown hair), and discovered directions transfer back into the original CFM's noise space.
  • Inversion becomes a matrix exponential: $\exp(-L)$ sends images back to the teacher's noise space, which the paper uses for reconstruction and for downstream inpainting, super-resolution, and denoising.
  • Because the consistency estimator needs only the teacher's velocity and marginals, the same linearization extends to any flow with an evaluable marginal velocity, including diffusion models via their probability-flow ODE.

Reading between the lines

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

  • Editorial inference: if real-world flow models admit such faithful finite-dimensional Koopman representations, then editing, interpolation, and inversion of generative models reduce to linear algebra in one fixed basis — a learned Fourier analysis of generation — and the eigenvalue ordering itself becomes a compressed description of how a dataset's images are produced.
  • Editorial inference: the consistency-loss residual is a measurable 'Koopman defect' per model, so it could be used as a diagnostic that ranks teachers or datasets by how far their dynamics are from linearizable; the paper builds this instrument but does not deploy it that way.
  • Editorial inference: a direct testable extension is to run the same linearization inside the latent space of a high-resolution diffusion transformer, where the operator dimension stays near 1024 while image resolution grows; the paper's quality gap on harder datasets would then either close, supporting the invariant-subspace story, or persist, pointing to the encoder as the bottleneck.
  • Editorial inference: the class-conditioned spectral signatures indicate that modes split into shared coarse structure and class-specific fine structure, suggesting a control scheme — steering class or identity by rescaling mode coefficients — that the paper does not itself propose.
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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

4 major / 4 minor

Summary. The paper proposes a Koopman-operator framework that linearizes the full generative dynamics of a pre-trained Continuous Flow Matching (CFM) model. The authors learn an encoder g_phi, a decoder g_psi^{-1}, and a linear generator matrix L such that the lifted coordinates evolve as dg/dt = L g, with an affine augmentation to handle the non-autonomous time dependence. The training objective combines phase, target, reconstruction, and a trajectory consistency loss; the consistency loss (Eq. 13) is claimed to be a simulation-free, unbiased estimator of the marginal consistency objective. Sampling is one-step via x1 = g^{-1}(exp(L) g(0, x0)). The paper further demonstrates applications in spectral mode analysis, semantic editing, inversion, and class-conditional spectral signatures, and reports competitive FID scores on MNIST, FFHQ, and CIFAR-10.

Significance. If the central trajectory-preservation claim is fully verified, this would be a meaningful advance: it would provide a single linear surrogate for the non-autonomous dynamics of a generative flow, enabling one-step parallel sampling while retaining fidelity to the teacher's intermediate states, and it would make spectral-analysis and control tools available for generative models. The paper is transparent about its main theoretical limitation (finite-dimensional invariant Koopman subspace), releases code, and includes an ablation clearly showing that the consistency loss is responsible for trajectory fidelity in the learned latent space. However, the current empirical verification is incomplete in a way that is load-bearing for the main claim: trajectory fidelity is only measured in the learned latent space, and the theoretical estimator is unbiased with respect to a mixture path rather than with respect to the teacher's actual ODE rollouts. The significance is therefore conditional on additional pixel-space trajectory evaluation.

major comments (4)
  1. [Sec. 6.2, Table 6] The trajectory-fidelity claim is supported only by latent-space comparisons: the authors encode teacher rollouts g(t, x_t) and compare them to exp(tL) g(0, x0) in the learned Koopman space. Because the reconstruction loss L_recon (Eq. 10) is applied only at t=1, no reported quantity constrains the decoder's output at intermediate times. A model could have small latent MSE while decoding to pixel-space states that do not match the teacher's intermediate images. Since the abstract and introduction explicitly claim full-trajectory preservation, the manuscript must report pixel-space trajectory fidelity, e.g., the MSE or LPIPS between g^{-1}(exp(tL)g(0,x0)) and the teacher ODE rollout x_t on shared noise seeds, at several times t (and with and without the consistency loss). Absent such a measurement, the central claim is not empirically established.
  2. [Sec. 4.3, Eq. (13)] Proposition 3 shows that Eq. (13) is an unbiased estimator of the marginal consistency loss with respect to the mixture path p_t(x) = ∫ p_t(x|x1) q(x1) dx1. However, the teacher's actual ODE trajectories are generated by the vector field v_t, and unless v_t is exactly the marginal velocity of p_t, the conditional-path samples are not distributed as teacher rollouts. For a pretrained CFM this holds only approximately, and it is precisely the approximation error that the linearization must tolerate. The paper should either justify that Eq. (13) controls the residual along true teacher rollouts, or quantify the discrepancy by evaluating the consistency residual on stored teacher trajectories and comparing it with the conditional-path estimate. Without this, 'trajectory-preserving' is only established relative to a surrogate path.
  3. [Sec. 6.3] The paper acknowledges that a finite-dimensional invariant Koopman subspace is 'a strong condition that need not hold for arbitrary nonlinear flows' and cannot be guaranteed a priori. This assumption is load-bearing: without it, the consistency loss cannot reach zero and the linearization is approximate. The current evidence (latent rollout MSE ~ 5e-6, competitive FID, and an intermediate optimal dimension in Fig. 5) is suggestive but indirect. A direct diagnostic would strengthen the claim significantly: report the held-out consistency-loss value L_cons (Eq. 13) and, ideally, the residual norm evaluated on actual teacher rollouts. If these residuals are not near zero, the paper should soften the trajectory-preservation claim accordingly.
  4. [Sec. 5.2 and Sec. 6.4, Table 2] The semantic-mode evaluation is partially circular: modes are selected using CLIP embedding similarity, and then coherence (Eq. 19) is measured with the same CLIP embeddings. This selection-evaluation loop inflates the reported coherence scores, especially for the 'with consistency' model. The interpretability claims should be validated with an independent attribute classifier or human evaluation, or the paper should explicitly state that the reported coherence is relative to the CLIP space used for discovery rather than an independent semantic measure.
minor comments (4)
  1. [Eq. (11)] The notation in Eq. (11) omits the time argument in the Jacobian: it should read ∇_x g_phi(t, x_t) · v_t(x_t), consistent with Eq. (13) and the surrounding text.
  2. [Fig. 3] The caption says 'N=400, 800, 100, full'; the third value is likely meant to be 1000 rather than 100.
  3. [Sec. 6.3 vs Fig. 5] The text states that 'the optimal FID value [is] at an intermediary Koopman dimension as seen in Figure 5', but the Figure 5 caption says 'the higher the dimension, the lower the FID'. These statements are inconsistent and should be reconciled.
  4. [Sec. 4.1, Eq. (7)] The block structure in Eq. (7) would be clearer if the dimensions of the blocks (b_g, A_gt, A_gg) were specified, since the augmented state mixes scalar time with vector-valued observables.

Circularity Check

2 steps flagged · score 4.0 of 10

CLIP-selected 'semantic modes' are scored by the same CLIP coherence metric, and latent-space trajectory fidelity is the training residual; the core Koopman linearization remains externally supervised by the teacher velocity field.

  1. fitted input called prediction [Section 5.2 (Eq. 19), Section 6.4 (Table 2), Appendix F.2, Appendix E.3]
    "Binary attribute labels (e.g., smiling vs. not smiling, eyeglasses vs. no eyeglasses) are obtained via CLIP classification using natural-language prompts. ... these modes were not manually selected; rather, they were automatically identified by ranking all eigenmodes according to their CLIP coherence scores with respect to each attribute prompt. ... C^β_k = 1/N Σ_i sign(⟨CLIP(z'_k),CLIP(p_β)⟩−⟨CLIP(z),CLIP(p_β)⟩)"

    Appendix F.2 discovers semantic directions from CLIP labels, and Appendix E.3 selects modes by ranking the Eq. 19 CLIP-coherence score. Table 2 then reports the maximum coherence for the same attributes and prompts. Because the mode-selection objective and the reported evaluation metric are the same CLIP embedding and the same attribute prompts, high coherence is partly guaranteed by the selection procedure; it is a restatement of the fitting criterion rather than independent evidence that the modes encode semantics.

  2. fitted input called prediction [Section 4.3 (Eq. 11 and 13), Section 6.2 (Table 1), Section 6.3]
    "Lcons = E_{t, x_t∼p_t(x_t)} ||Lg_ϕ(t, x_t) − ∇_x g_ϕ(x_t)·v_t(x_t)||^2. ... We encode a teacher's trajectory {x_t}_{t∈[0,1]} in the latent space and compare this ground truth path z_t = g_ϕ(t, x_t) against the analytical linear trajectory from our model, z̃_t = exp(L̃t) z̃_0. We show the results in Table 1."

    The reported trajectory MSE is an integrated version of the exact residual minimized by Lcons, evaluated in the same learned latent coordinates g. If the consistency residual is zero, then d/dt g(t,x_t) = Lg(t,x_t), so the rollout z̃_t = exp(tL)z_0 matches g(t,x_t) by construction; hence near-zero Table 1 MSE primarily reports convergence of the training objective. Section 6.3 then uses this same 'trajectory reproduction to within MSE ~10^-6' as evidence that a finite-dimensional Koopman representation exists, making that existence argument partly circular. Since L_recon is applied only at t=1, no independent image-space trajectory metric verifies intermediate-state preservation.

full rationale

The core linearization is not circular: Eq. 13 is a simulation-free estimator whose target is the pretrained teacher velocity field v_t, and Proposition 3's unbiasedness is a law-of-iterated-expectation identity, not a re-derivation of the teacher. The method is thus externally supervised rather than self-defined. No load-bearing self-citations appear. Two evaluation steps are partially circular, however. First, semantic modes are selected by ranking CLIP coherence (Eq. 19) and then reported as having high CLIP coherence for the same prompts, so the interpretability numbers in Table 2 are partly an artifact of selection. Second, 'trajectory fidelity' is measured in the learned latent space with a metric that is the integrated training residual Lcons; the paper's Section 6.3 even relies on this residual to argue that a finite-dimensional Koopman representation exists. The manuscript itself concedes in Sections 6.3 and 7 that finite-dimensional Koopman representability is not guaranteed a priori and that the quality gap widens on complex datasets, which is an honest limitation but does not remove the circularity of using the fitted residual as evidence. Overall, the central linearization claim retains independent content, but the interpretability and trajectory-preservation evidence are partially reduced to their own fitting criteria, yielding a score of 4.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The core derivation (Props 1-3) is self-contained given standard Koopman theory, but the empirical success rests on the unproven existence of a learnable finite-dimensional invariant subspace, plus hand-tuned loss weights and latent dimension.

free parameters (5)
  • Koopman latent dimension = 1026 (1024+2)
    Ablated in Fig. 5 as optimal for FID; the paper states 1026 is optimal and higher dimensions do not improve quality.
  • Target loss weight with consistency = 0.01 (versus 1.0 without consistency)
    Table 3 shows the target weight is reduced when consistency is added; this reweighting was chosen by hand to make the composite loss trainable.
  • Edit strength alpha = 3.0 (Table 8)
    Used for semantic editing evaluations; the choice of scale affects reported CLIP and LPIPS numbers.
  • Number of pre-generated teacher pairs = 10^4 to 10^6 depending on dataset
    Training data for boundary losses; the quantity varies per dataset and affects distillation quality.
  • Unreported loss weights for phase and reconstruction terms = not specified
    Section 4.2 defines lambda_phase, lambda_target, lambda_recon, lambda_cons but Table 3 only gives the target weight; these undisclosed hyperparameters affect training.
assumptions (4)
  • ad hoc to paper The Koopman generator L has a finite-dimensional invariant subspace of observables for the teacher CFM dynamics, and this subspace is learnable by the chosen encoder and decoder.
    Needed for the central claim; explicitly not guaranteed in Sec. 6.3 and the Limitations section.
  • domain assumption The teacher marginal vector field v_t is deterministic and can be evaluated at conditional samples x_t ~ p_t(·|x1).
    Used in Eq. 13 to build an unbiased estimator of the marginal consistency loss.
  • standard math Standard Koopman theory: composition of observables with the flow is linear, and the generator is the Lie derivative.
    Background from Sec. 3.2; no new content.
  • standard math The augmented affine lift z = [1, t, g] with the block-structured generator in Eq. 7 enforces the time coordinate evolves as ?t = 1.
    Standard state-space augmentation for non-autonomous systems (Strogatz).
invented entities (1)
  • Koopman latent space and Koopman modes
    purpose: Learned representation where the CFM dynamics become linear and spectrally decomposable.
    The latent space is a learned mathematical construct, not a physically observable entity. Its semantic content is claimed but only demonstrated with self-selected CLIP evaluations; no external falsifiable handle is provided.

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Cite this review

Pith. "Pith review of Unfolding Generative Flows with Koopman Operators: Trajectory-Preserving Linearization." pith.science (2026). https://pith.science/paper/DJDGOQHX

@misc{pith2026250622304,
  author       = {Pith},
  title        = {Pith review of: Unfolding Generative Flows with Koopman Operators: Trajectory-Preserving Linearization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DJDGOQHX}},
  note         = {Machine review of arXiv:2506.22304}
}
read the original abstract

Continuous Normalizing Flows (CNFs) enable elegant generative modeling but remain bottlenecked by their iterative nature requiring costly sampling and lacking interpretability of the intermediate states. Recent approaches accelerate sampling by straightening trajectories or distilling endpoints, yet they treat the original generative process as a black box, discarding the teacher's intermediate dynamics. We propose a fundamentally different perspective: globally linearizing flow dynamics via Koopman theory to achieve trajectory-preserving linearization. By lifting a pre-trained Conditional Flow Matching (CFM) model into a higher-dimensional Koopman space, we represent its evolution with a single linear operator. Crucially, unlike boundary-only distillation, our method enforces infinitesimal consistency with the teacher's vector field along the full generative path. We derive a practical, simulation-free training objective that ensures this global alignment and yields two key benefits. First, sampling becomes one-step and parallelizable. Second, because the linearization is faithful to the dynamics, the Koopman operator provides unique insights on the generation. We demonstrate that this structure enables novel applications unavailable in prior approaches, including discovery of semantically coherent editing directions, inversion with a teacher-aligned linear operator and class-conditional spectral signatures. Empirically, our approach achieves competitive sample quality, while enabling spectral analysis and control of the entire trajectories of generative flows.

Figures

Figures reproduced from arXiv: 2506.22304 by the authors.

Figure 1
Figure 1. Overview of our method: We propose to learn a Koopman latent space of “observables”, where the nonlinear dynamics of a CFM teacher becomes linear (Section 4.). Any point in the trajectory is now accessible with a single matrix exponential step of the Koopman operator L, including noise-to-image generation and image-to-noise inversion. The operator L can be decomposed and used to analyze the CFM teacher (Section 5.) … view at source ↗
Figure 2
Figure 2. t-SNE visualization of CFM and Koopman trajectories in the embedding space on FFHQ (test set). The consistency loss makes Koopman rollouts (dotted) follow the teacher dynamics (continuous) more closely. This is seen both in the proximity of trajectories and in the alignment of their endpoints. Circles mark starting points and squares mark end points. the teacher’s velocity field vt, not from the linearization itself… view at source ↗
Figure 4
Figure 4. CFM-based semantic editing comparison. We port identified semantic directions from the Koopman latent space to the CFM noise-space via inversion as explained in F.3, here for sunglasses. Notably, we can see that recovered direction in the purely distilled model provides unreliable edits on generated images, as it comes with noticeable noisy artifacts, as opposed to our consistent model [PITH_FULL_IMAGE:figures/full… view at source ↗
Figures from the paper (15 more)
Figure 5
Figure 5. Figure 5: FID score as a function of Koopman dimension on the FFHQ dataset. The higher the dimension, the lower the FID. 6.3. Ablation Koopman space dimension. As shown in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Per-class spectral deviation on CIFAR-10. How does the teacher acquire its dynamics during train￾ing? Similarity matrices shown in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Eigenmodes similarity matrices comparing early and mid-training checkpoints against the fully trained model. 20 40 60 80 100 Mode index k 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Cumulative similarity Early (20k) Mid (80k) Late (160k) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Cumulative average diagonal similarity shows progres￾sive mode acquisition during training Outlook The ability to pose these questions, and to answer them with linear algebra tools, rests on the non-trivial global linearization afforded by Koopman theory. We regard the…
Figure 9
Figure 9. Figure 9: Training losses on CIFAR-10 and FFHQ datasets across different loss components. C.2. Trajectory Fidelity with and without Consistency Loss To visualize trajectory fidelity in an interpretable coordinate system, we project dynamics onto the Schur basis of the learned ge…
Figure 10
Figure 10. Figure 10: Trajectory comparison in Schur coordinates of the learned Koopman generator. With consistency, learned Koopman modes (dashed) accurately track CFM dynamics (solid). Without consistency, endpoints match but intermediate trajectories diverge, indicating the learned mode…
Figure 11
Figure 11. Figure 11: Uncurated samples from our Koopman generative model across three datasets. All samples are obtained via our one-step strategy. E. Interpretability and Spectral Analysis E.1. Koopman mode structure [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Left: Mean coefficients |ci| 2 projected on the generator modes ordered by corresponding eigenvalue magnitude |λi| 2 . Top corresponds to the spectrum along the modes obtained from training with consistency and bottom to those obtained from training without consistenc…
Figure 14
Figure 14. Figure 14: a [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]
Figure 13
Figure 13. Figure 13: Progressive mode reconstruction sorted by Re(λ). Slow modes (negative Re(λ)) capture coarse structure, while fast modes (positive Re(λ)) add fine details. The learned Koopman spectrum provides an interpretable hierarchy reflecting the multi-scale nature of the generat…
Figure 14
Figure 14. Figure 14: a Sunglasses (Mode 1019) — with consistency [PITH_FULL_IMAGE:figures/full_fig_p026_14.png]
Figure 15
Figure 15. Figure 15: a With consistency loss xopt (Optimized Noise) x1 (Original) x1 (Reconstructed) [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: Consistency loss enables stable semantic editing. Edits via optimized noise perturbations x0 + α∆x0 integrated through CFM (α ∈ [0, 3], increasing left-to-right). With consistency loss (left), edits remain coherent and identity-preserving. Without (right), large α cau…
Figure 17
Figure 17. Figure 17: Additional semantic directions. Same setup as [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: a Inpainting [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.