One-Step Offline Distillation of Diffusion-based Models  via Koopman Modeling
Authors: Nimrod Berman
Ilan Naiman
Moshe Eliasof
Hedi Zisling
Omri Azencot
Venue: NeurIPS 2025 poster ()
Keywords: Koopman theory
diffusion models
offline distillation
one-step generation
dynamical systems
Primary area: deep_learning
TL;DR: We propose the Koopman Distillation Model (KDM), a novel offline distillation method for diffusion models that leverages Koopman theory to enable single-step generation with strong semantic consistency and state-of-the-art FID performance.

Abstract: Diffusion-based generative models have demonstrated exceptional performance, yet their iterative sampling procedures remain computationally expensive. A prominent strategy to mitigate this cost is *distillation*, with *offline distillation* offering particular advantages in terms of efficiency, modularity, and flexibility. In this work, we identify two key observations that motivate a principled distillation framework: (1) while diffusion models have been viewed through the lens of dynamical systems theory, powerful and underexplored tools can be further leveraged; and (2) diffusion models inherently impose structured, semantically coherent trajectories in latent space. Building on these observations, we introduce the *Koopman Distillation Model* (KDM), a novel offline distillation approach grounded in Koopman theory - a classical framework for representing nonlinear dynamics linearly in a transformed space. KDM encodes noisy inputs into an embedded space where a learned linear operator propagates them forward, followed by a decoder that reconstructs clean samples. This enables single-step generation while preserving semantic fidelity. We provide theoretical justification for our approach: (1) under mild assumptions, the learned diffusion dynamics admit a finite-dimensional Koopman representation; and (2) proximity in the Koopman latent space correlates with semantic similarity in the generated outputs, allowing for effective trajectory alignment. Empirically, KDM achieves state-of-the-art performance across standard *offline distillation* benchmarks - improving FID scores by up to 40% in a single generation step.

One-Step Offline Distillation of Diffusion-based Models via Koopman Modeling
Nimrod Berman" Ilan Naiman"
Moshe Eliasof 1,2
Hedi Zisling'
Omri Azencot
Ben-Gurion University of the Negev University of Cambridge
https://sites.googl/ www/view////eeccccc.ccc......aat wwwpooooonnaninlaaaiiiiiioaaaaee
Abstract
Diffusion-basel generative models have demonstrated exceptional performance, yet their iterative sampling procedures remain computationally expensive. A prominent strategy to mitigate this cost is distillation, with offline distillation offering particular advantages in terms of efficiency, modularity, and flexibility. In this work, we identify two key observations that motivate a principled distillation framework: (1) while diffusion models have been viewed through the lens of dynamical systems theory, powerful and underexplored tools can be further leveraged; and (2) diffusion models inherently impose structured, semantically coherent trajectories in latent space. Building on these observations, we introduce the Koopman Distillation Model (KDM), novel offline distillation approach grounded in Koopman theory a classical framework for representing nonlinear dynamics linearly in transformed space. KDM encodes noisy inputs into an embedded space where learned linear operator propagates them forward, followed by a decoder that reconstructs clean samples. This enables single-step generation while preserving semantic fidelity. We provide thcoretical justification for approach: (I) under mild assumptions, the learned diffusion dynamics admit a dimen-dimensional Koopman representation; and (2) proximity in the Koopman latent space correlates with semantic similarity in the generated outputs, allowing for effective trajectory alignment. KDM achieves highly competitive performance across standard offline distillation benchmarks.
Introduction
Diffusion based generative models [72], including score-based and flow-matching variants [30, 47], have become ubiquitous across wide range of domains ffrm images and videos to audio and time series [18, 31, 41, 16, 59, 21, 26]. These models now surpass approaches such as GANs and VAEs in terms of sample quality, while also exhibiting greater training stability. Despite these advantages, one of their key limitations remains the high computational cost associated with sampling [63, 68, 51]. Generating single high-fidelity sample typically requires executing lengthy iterative process, progressively refining random noise through dozens or even hundreds of model evaluations [30].
A widely adopted strategy to address this involves minimizing the number of inference steps [73], for example by leveraging improved numerical solvers or designing more expressive noise schedules. enabling high-quality generation with significantly fewer denoising iterations. Another increasingly popular alternative is distillation [68], where student model learns emulate the behavior of teacher model. Distillation approaches vary in supervision and setup: in online distillation, the student directly learns from the teacher's predictions during training, with the teacher providing trajectory information across time steps. In contrast, offline distillation relies on precomputed noise noise-image pairs or teacher-gen generrted samples only, where the student learns without any further access to the teacher model, including its weights, evaluations, or internal states.
Equal contribution
39th Conference on Neural Information Processing Systems (NeurIPS 2025).

---

CN
D
Ee
E+
TT
Figure 1: A Gaussian sample (bottom left) evolves into a clean image (bottom right) via nonlinear dynamics Pr. Leveraging the Koopman framework, learned encoders Eg and Es transform noise and image into an embedding space where the evolution becomes linear under C₁ (top).
Offline methods inherently offer key advantages over online tools. Training on precomputed, teacher- generated noise-image pairs avoids multi-stage training and repeated teacher queries, reducing both neural function evaluations and memory footprint [24]. It also enables pre-filtering or anonymization of potentially privacy violations [22]. Consequently, once the dataset is produced, the teacher can be discarded, shrinking the attack surface and supporting secure collaboration (c.g.. outsourcing training without exposing proprictary information) [12, 32, 61]. Beyond privacy and efficiency, offline distillation is model-agnostic and architecture-genee general (as evidenced in our experiments), and it scales flexibly with parameter budgets. On the limitation side, offline setups require upfront generation and storage costs for the precomputed corpus and, lacking teacher access beyond the noise image pairs, may constrain model expressivity and adaptability. By contrast, online methods may warm-star start from teacher weights or query the teacher for stepwise supervision. Despite these trade-offs, offline distillation remains practical, private, and sustainable approach to generative- distillation. The two paradigms can be considered complementary, and the optimal choice depends on factors such as privacy requirements, computational cost, and deployment constraints.
Towards developing distillation method, we identify two key observations that guide our approach: (I) While denoising-based methods leverage dynamical systems perspectives for both general generative modeling [76, 47] and distillation [89, 13], wide range of powerful tools from dynamical systems theory remain underexplored; (2) Trained diffusion models impose structured organization on the latent space, coherently mapping noisy inputs toward clean data. Specifically, our empirical analysis suggests that semantically related images tend to originate from nearby points the noise space.
Building on these observations, we propose leveraging a dynamical systems approach that has not yet been explored in the context of distillation: Koopman operator theory. This theory classical yet recently revitalized approach for representing nonlinear systems with linear dynamics in an embedded space [42, 67]. The core idea that by mapping the system into a suitable embedding space, its evolution becomes linear. Thus, learning the diffusion process then reduces to jointly learning these embeddings and the linear operator. While Koopman operators are typically prove that an exact finite-dine dimeesidimnii representation exists under mild assumptions [33]. Moreover, prior work suggests that the Koopman operator preserves the structure inherent certain dynamical systems [28, 9]. This motivates the use of Koopman theory to harness the observed coherent semantic structure diffusion dynamics, offering a principled foundation for accurate teacher model distillation. To this end, we formally connect the observed structure in diffusion models to the Koopman framework by proving that this structure is preserved within the learned dynamical representation.
we
In practice, we propose KDM (Koopman Distillation Model), framework for modeling diffusion processes through linear evolution learned embedding space. Given noisy input, an encoder maps it into an embedding space where a learned Koopman operator evolves it forward, and decoder reconstructs the clean sample. A schematic illustration of our framework is presented in Fig. 1. The training objective encourages semantic preservation (via reconstruction loss), accurate latent linear dynamics (via Koopman consistency), and fidelity in output space (via prediction and adversarial losses). This design naturally yields fast, single-step generation, while preserving structural coherence distillation. We evaluate our method on the offline distillation benchmark, while adding two higher-resolution datasets, and achieve highly competitive results in both unconditional

---

and conditional generation. These results demonstrate the effectiveness of our Koopman-based based formulation in bridging the performance gap with online methods, offering a fast and high-fidelity offline diffusion model distillation. Our key contributions are:
1. Observing diffusion dynamics. We identify, analyze and formalize two key observations: (i) tools from dynamical systems theory, specifically Koopman operator theory, remain underexplored in diffusion modeling and distillation; (ii) training induces coherent semantic organization in noise space, These insights motivate a structure- proettotigg dynamics approach to distillation.
2. A principled and practical Koopman- -aaaaad distillation framework. We develop a theoretical foundation proving the existence of finite siite-dimensooal Koopman representations and semantic structure preservation under mild assumptions and introduce a simple, scalable encoder-linear dynamics-decoder architecture that preserves the teacher generation dynamics semantic structure.
3. Highly competitive results. We conduct comprehensive unconditional and conditional evaluation showing that our method outperforms prior offline distillation approaches, achieving FID improvement, while enabling efficient, single-step generation.
2 Related Work
Generative Modeling. Deep generative models have made significant progress in recent years, with diffusion models emerging as leading framework for high-quality image, audio, and molecular generation. Building on principles from nonequilibrium thermodynamics and score matching [72, 76], these models define forward process that gradually corrupts data with noise and learn reverse via a neural network-bard based denoising process. Variants like DDPMs [30] and score-based generative models [76] have shown competitive performance across wide range of benchmarks, often surpassing VAEs and GANs [40, 27] in sample fidelity and diversity.
Accelerating Diffusion Sampling. Diffusion models generate high-quality samples but are slow due iterative sampling. One approach improves integration using higher-order solvers and expressive noise schedules [73, 49], achieving high-quality results in significantly fewer steps. Another class of approaches seeks to distill a student model from a pre-trained teacher, reducing the sampling process to a single or few steps. Online distillation techniques [50, 68] supervise the student directly with teacher predictions during training, but require continuous access to the full teacher model, resulting in high memory and compute overhead. Recent methods demonstrate high-quality one-step generation across CIFAR-10, ImageNet, and text-to-image benchmarks [86, 93], using objectives based on distribution matching [86], score identity [93], adversarial training [69, 71], and consistency models [75, 38] among others [13, 52, 62, 94, 6]. Offline distillation methods [24] rely on seacher-generater noise-data pairs and decouple student training from the teacher. Aside from the concurrent work [82], current distillation methods largely unexplored soopman-based perspectives and their relation to the teacher's underlying dynamics.
Koopman- Based Modeling. Koopman operator theory provides a powerful framework for modeling
nonlinear dynamical systems via linear operators acting on observable functions [42, 67]. This view has been applied to tasks such as deep learning of dynamical systems [79, 53, 84, 65, 58, 60], sequential disentanglement [7, 5]. and control [29]. A related strand of work focuses on identifying coherent sets regiong of the state space that evolve coherently over
Koopman- based embeddings [56, 23]. While these works exploit the Koopman structure for interpretability or stability, work focuses on using Koopman- bbased based modeling for distilling diffusion models. In this context, demonstrate that the learned embedding space in diffusion models encodes semantically meaningful structure, and we leverage this to design principled offline distillation framework.
3 Mathematical Background
Deterministic Difffusion Sampling. We consider deterministic sampling in diffusion models, as introduced in the EDM framework [35]. A pre-trained diffusion model defines a time-indexed denoising vector field fe : IR. x [0, T] - R", which maps noisy latent state d, E IR" time t e T] toward the data manifold (at time 0). Sampling is performed by solving the reverse-time ordinary differential equation (ODE).
dx: dt fo( IT ~~0,000,
(1)
3

---

integrated backward from Gaussian sample 2T at time T to a clean sample xy at time t = 0. The function fo is typically trained to approximate the score function V... log pt (2,) ) or a reparamettriization thereof, such as the probability flow [76]. These deterministic formulations enable sampling via high-order ODE solvers and provide a tractable setting for distillation and trajectory analysis [35].
Koopman Operator Theory. Let 42 : R + R"" define a (possibly time-dependent) dynamical
system mapping states forward in time, ie.. Zy (x0)) The Koopman operator K2 [42] is a linear operator that acts on scalar-valued observables g 1 + C via composition,
Kgg = g D..
(2)
mapping each observable to its pullback along the dynamics. While the original system o may be nonlinear, KG is linear on the space of observables (.²) (with respect a suitable measure) [20]. In classical Koopman theory, this operator is dimensional subspace O span(gi. ................................ gd} C. (C") that is invariant under K, for all E [o, T). then the dynamics of 4, can be exactly represented in this lifted space via a family of linear operators C. Raadd E [o. T), such that for all g E O, Kig = Cig. This allows modeling nonlinear dynamics through linear evolution in learned observable space, principle that underpins our Koopman- based distillation framework.
However, if there exists finite-
4 Koopman Distillation of Diffusion Models
We propose KDM, a principled offiin distillation framework for diffusion models based on Koopman operator theory. The key idea is to model the dynamics of the generative process, specifically the transformation from noisy sample øT to its clean counterpart Zo, as single-step linear evolution in a learned embedding space. This approach enables fast and dtructure-prssevve forward pass, without requiring access to the full diffusion trajectory or teacher model at training time.
generation in single
Dynamical System Perspective on Diffusion Models. We build upon two key observations regarding diffusion models. First, diffusion can be understood as dynamical system, where the evolution of a noisy data point z; is governed by a dime-dependent velocity field, see Eq. (1). This continuous-time formulation admits an integrated flow map &, such that 22 = & (2o), where t evolves the initial condition zo forward in time. This formulation makes it natural to propose adopting a Koopman operator perspective, a novel approach in the context of distillation, which enables the lifting of nonlinear dynamics into linear evolution in function space under suitable choices of observables, see Eq. (2).
The second observation considers the structure of the trained diffusion dynamics. Score-based diffusion models learn the score Zz, log Pe (arz) of corrupted data at multiple noise levels and then reverse that specific corruption process using Langevin dynamics [76] or learned reverse kernels [30] to transform Gaussian noise into images. This framework has been unified through the lens of stochastic differential equations (SDEs) by [76]. Flow matching models [47] learn continuous velocity field that transports a simple reference distribution directly into the data distribution, where sampling is typically performed by integrating the learned ordinary differential equation (ODE). In what follows, we study the structure of the dynamical mapping arising from diffusion- based learning.
We pose the following question: Is there semantic structure in the generative process of diffusion models? In general, prior training, there is no assurance of semantic structure between the noise and data distributions (see Fig. 2, left). However, after training, we observe the emergence of mapping coherent and semantically meaningful mapping from the Gaussian noise distribution the data
distribution (see Fig. 2, right). Specifically, by utilizing a simple 2D checkerboard toy dataset, we train an EDM [35] model and generate 50,000 samples, recording both the initial noise vectors and their corresponding generated outputs on the checkerboard. We then color-code each sample by its target checker cell and assign the same color to its originating noise vector. This visualization reveals that samples originating from the same semantic region in data space tend to cluster together in noise space, indicating structured and coherent dynamics. Notably, we observe a similar pattern in both flow matching model and our KDM. This emergence of coherent structure, together with prior evidence that the Koopman operator preserves structure in certain dynamical systems [28, 9], highlights the relevance of Koopman theory diffusion dynamics. Indeed, our theoretical analysis (Sec. 5) establishes a formal link between this observed structure and Koopman theory, motivating its adoption as principled framework for modeling these dynamics.

---

untrained
trained
Figure 2: Untrained noise distributions semantically unstructured (left). Training EDM on the data reveals the emergence of coherent clusters in noise space, i.e., at time (right).
Koopman-based Distillation. Our Koopman distillation framework is designed by the following principle: given pairs of noisy and clean data (xy, zu), where zo Φr (2x) for some unknown nonlinear map 2 we seek model this transformation using an embedding space, following Koopman theory. To achieve this, we introduce observable functions encoder networks Es, E.p for Io, JT, respectively, that map states from the original data space into a latent observable space. In this space, we assume the dynamics can be modeled linearly by a finite-din ditte-dimensinall Koopman operator linear layer, C₃- Finally, corresponding decoder network D maps the evolved observables back to the data space. The overall modeling objective becomes:
linear operator in
{g1, ................................ 9d.. parameterized
To R D& (C,, Ex(T))
This formulation distills the potentially complex, nonlinear generative process into a simple one-step linear map in the embedded observables space, while maintaining fidelity to the original dynamics.
Koopman's 's Loss Functions. In what follows, d(-,-) denotes a suitable distance function, such as mean squared error (MSE) or a perceptual loss (e.g., LPIPS [88]). To effectively learn the components Eg, De, and Cy, we define a composite loss comprising three terms (see similar objectives in [53, 4)):
Kooppann Frec Lia + Epred
The first term, the reconstruction loss Liee, ensures that the encoder-decoder pair (Eş, D can faithfully reconstruct no!
Lree = d(xo, D (E
This term regularizes the embedding by encouraging Eg to preserve sufficient semantic information about the data, while not encoding any dynamic features. The second term, the latent dynamics loss Lia, enforces consistency of the linear evolution in the observable space:
List dl (F))) Ci,y Fap(ar))
(6)
where C. is the Koopman operator latent space. This term ensures that applying C, to the embedded noisy state Esp (xT) approximates the embedded clean state Eg(10). C, is either implemented with a regular learning linear layer or by a new approach we develop that factorizes the matrix, which enables eigenspectrum control (KDM-F). We describe the decomposition and its merits in App. C.
Finally, the prediction loss Gpedd evaluates the full end-to-end prediction after encoding, applying the Koopman operator, and decoding:
Spred = d(zo, D&s (G,C,,
encouraging the system accurately predict vy from or in single forward pass through the Koopman pipeline. Together, these terms provide complementary supervision signals.
Adversarial Loss. Adversarial frameworks are widely employed in distillation methods [70, 38, 77, 78]. For example, [38] leverages an adversarial loss to enhance the fidelity of one-step generation. Motivated by these approaches, we introduce an auxiliary adversarial loss to further refine the distillation process. Specifically, we introduce a discriminator network D,, parameterized by 7, and define the adversarial loss as:
Ladv (bb, w. 7) Ezo [log DT, + Ber [og(1 - D,,(t)))
(8)
5

---

where i0 (((T)) is the Koopman-predicted sample. Thus, the final training objective is:
C = Koopman + Xadv
(9)
where And balances the adversarial component. In practice, throughout our experiments, we choose Audv = 0.01. Importantly, utilized simple D, with four convolutional layers. By enhancing the perceptual quality of the predictions without altering the fundamental Koopman modeling, this adversarial augmentation strengthens the overall generation performance while preserving the structured distillation at the heart of our method [8]. We present losses and more ablation studies in App. D.I. Additionally, A pseudo-code of our KDM is provided in App. E.I.
Control Modeling for Conditional Dynamics. A crucial aspect of generative modeling with denoising models is the ability to condition generation on auxiliary signals or labels [18]. Koopman theory provides natural framework to incorporate such conditioning through control signals that influence the Koopman operator [3, 11]. Formally, let e denote the control signal or conditioning variable (e.g.. class label) that want the generative process to be conditioned on. Our goal is to model the conditional dynamics & ((xr, c). In the Koopman framework, we extend the observable and evolution operators to incorporate c, and formulate the generation
Do (CTT)) + Cu(c), c) ~ ((Fr,c),
where C)(c) is control-dependent modulation term (e.g.. linear transformation of c). This formulation enables our model to adapt its generative trajectory based on the conditioning signal, seamlessly integrating the conditioning mechanism into the Koopman dynamics. We apply this control modeling in our conditional generation experiments.
5 Theoretical Properties of KDM
We now state the first key theoretical result, establishing the existence of an exact finite-dimensional Koopman representation under mild conditions. Specifically, for an analytic dynamical system, is possible to approximate its nonlinear evolution arbitrary accuracy by a finite-dimensions linear operator acting on lifted coordinates. In particular, a small number of polynomial observables can capture the system's 's dynamics with controllable approximation error. This result is significant because it means we can faithfully replicate the complex behavior of the teacher model using a simpler, linear system in transformed space. In practical terms, ensures student learn to follow the teacher's 's dynamics with high accuracy.
Theorem 5.1 (Finite Koopman Operator for Analytic Dynamics). Let 4 RT - I be an analytic map, and let xT ~ (0, I.). Then, for any O, there exists linear operator C' Baxd such that
(IIE(((ETT) CE(+T)]13] Sc.
(10)
where & : denotes the observable lifting map. Moreover. the required dimension d grows at most polynomially with 1/e, and the mapping t can be approximated arbitrarily well using multiaariate polynomials up to degree d.
The proof builds on the analytic formulation of Φ, allowing a convergent multivariate Taylor expansion [81], and exact finite-dimensional Koopman operators for polynomial dynamics [33], see App. A.
Our second theoretical result concerns the structure induced by the reverse diffusion dynamics, under a set of assumptions established by Thm. 5.1. Under these conditions, the theorem states that proximity in the Koopman coordinate representation at the initial noise time implies semantic similarity between the corresponding samples at data time. In practice, the theorem says that if two inputs start out close together in the transformed space, the images they generate will be semantically similar. This provides a theoretical foundation for the structure- preccurig behavior of our model, as observed empirically in Sec. 6.1.
Theorem 5.2 (Semantic Proximity via Koopman-Invariant Coordinates). Let &. .R + R be the reverse diffusion flow of a trained model (from time T to 0), and (2((R)) be fimite-dimensional subspace such that:
1. KGO O for all E [O,
2. 3C, 1 - d such that Krt o, = Cr all O.

---

0.05
0.1
0.2
0.3
0.4
0.5
0.6
O=0
0.05
0.1
0.2
0.3
0.4
0.6
Figure Visualization of vicinity comparison across different noise levels o. Left: EDM [35]. Right: our method. Each image corresponds to small perturbation around the original noise sample.
3. Tr is locally Lipschitz.
For any wt, 2 E RR, define o6 == (x for j 1, and let &: IR" + IRd be the vector-valued observable {(x) == (z)(.)...... Ea(x)) O. Then:
| =لل < L- 11 ( S L ||Cr|I (1((1) E(27))
The proof builds on the observation that diffusion model dynamics tend to preserve semantic similarity between inputs and outputs. By leveraging the Koopman operator's spectral properties, we construct a finitedimensional space of functions that remains stable under the model's evolution. Within this space, the Koopman coordinates & provide locally bi-Lipschitz representation, while the reverse diffusion map Tr is Lipschitz continuous. Together, these properties ensure that semantically similar inputs remain close throughout the diffusion process. Note that the theorem holds also for := id, ie., the identity map [54, 19]. The full proof is given in App. B.
Importantly, we observe in practice an average spectral norm of the Koopman operator Cr of 0.88 + 0.18, indicating a reasonably tight bound and supporting the practical utility of our theory. Moreover, our framework enables explicit control over this norm via eigenvalue penalties the Koopman operator. As detailed in App. C, we factorize Cr to permit direct constraints on its spectrum (c.g., radius or banding), providing principled regularizer on ||Cr|]-
6 Experiments
We begin by further investigating the emergent dynamics structure that map noise to images, and its presence in KDM (Sec. 6.1.. We then extensively assess our method on the offline distillation benchmark (Sec. 6.2), including analyses of computational complexity, data and parameter scaling, and denoiser-agnostic utility. Additionally, an ablation study of the loss functions and adversarial losses stability is provided in App. D.1. Our code is in https: h/tp:://giii httgggttubb...oo aww.cote-totooooo eeeencot-group// KDDM.
Setup and evaluation. We follow the protocol of GET [24] for generating noisy-clean pairs and for evaluation, using the CIFAR-10 10 [43]. FFHQ 64 x4 [37]. and AFHQv2 64 64 [15] datasets. Our training setup matches that of PD [68] and GET [24]. For evaluation, we report image quality using Fréchet Inception Distance (FID) and Inception Score (IS), computed over 50k samples. Both encoders E.g and Eg. as well as the decoder D&. are implemented as compact UNets with reduced output channels to ensure a similar parameter size to the other single-step methods. The discriminator consists of four simple Conv2D layers. Full implementation details are provided in App. E.3.
6.1 Observations of Emergent Latent Structure
Real-worid data dynamics: local structure analysis. Extending the 2D toy experiment from Sec. 4, we now explore the high-dimensional FFHQ 64x64 dataset. Since dimensionality reduction can obscure fine-grained patterns, we instead analyze the local neighborhood of a single latent sample IT (O I.). We generate perturbed versions îT ZT + E-E- with E ~ (O I,), and normalize each êt using its own mean and standard deviation. This setup tests whether semantic structure emerges locally around xT. We vary a E (0, 0.05, 0.60} and visualize the outputs Fig. (left). For all a < 0.3, the images remain semantically aligned with the original image (when o = 0), indicating local smoothness and manifold adherence. As a grows, semanties diverge but retain global coherence even at a 0.60. The same experiment KDM (Fig. 3, right) reveals similar behavior,
7

---

Figure 4: Sampled images using 79 NFEs with EDM [35] (left), and using NFE with KDM (right).
affirming that KDM preserves the local semantic structure of the original EDM. Our results show that both models organize latent space into meaningful neighborhoods, even in high-dimensional settings.
Visual analysis of dynamics distillation. Beyond preserving local structure, KDM also approxi- mately reproduces the noise-to-image mapping of EDM. When starting from the same noise vector IT ~ (0, In), sampled using seeds not seen during training, both EDM (with 79 NFEs) and KDM (with single NFE) generate perceptually similar outputs (see Fig. 4). This suggests that KDM generalizes well beyond its training data and effectively distills the mapping from noise to data. We observe this consistency across multiple datasets and include further qualitative results in App. D.3. To quantify this, we generate 50k samples using both methods and compare outputs using LPIPS and MSE. KDM achieves scores of 0.13 (LPIPS) and 0.009 (MSE), in contrast to 0.48 and 0.57 when comparing randomly permuted images, demonstrating the similarity between the models' generation. To complement quantitative metries, we conduct human study where participants choose the more realistic image collage between our model and EDM (Fig. 4; details in App. E.4). Evaluators showed clear preference: 55% of 300 choices favored our model, highlighting its strong fidelity.
Outliers and their relation to the dynamics structure. We analyze recurring failure mode in both the teacher and distilled models: the generation of outlier samples that fall outside the designated checkerboard regions. These citoo--distribution points often appear in the final outputs and reflect a limitation in capturing the intended data distribution. To investigate their origin, we trace these outliers back through the generation process and identify two primary sources: (1) the tails of the Gaussian prior and (2) regions near decision boundaries between adjacent checkerboard cells. Fig. 5C shows this behavior for the distilled model (black crosses denote outliers); additional comparisons with the teacher are provided in App. E.7. These findings suggest that the model's 's dynamics are more error-prone in nigh-uncertainty regions, offering potential directions for improvement. Integrating more robust sampling techniques or leveraging self-supervised signals may help mitigate such failures. This insight could also inform future efforts uncertainty estimation.
6.2 One-step Generation Results
Unconditional and conditional generation with CIFAR-10. We evaluate the standard offline distillation benchmark of [24], which distills a pre-trained EDM on CIFAR-10. Whereas the EDM teacher samples with 35 function evaluations (NFEs), our student attains comparable quality with single NFE. We also adapt two recent SOTA baselines, IMM [91] and RF-2++ [45], to the same strictly offline setting, where only teacher-generated data are available and the original model weights (and online queries) are nol accessible (the RF-2++ variant corresponds to its "w/o pre-train" configuration. All methods are evaluated under identical protocols. Tab. reports unconditional generation results,
FID
FID 4
B
11
Ours GET
Ours
7
5
4
20
38
0.25
0,50
1.00
Model Size (M)
Data Size (M)
Figure 5: A) Model size vs FID. B) Data Size vs FID. C) Outlier analysis
8

---

Table 1: Generative performance uncondi-
Table 2:
Generative performance class-
tional CIFAR-10.
conditional CIFAR-10.
Models ( Metrics (++)
NFE 4 L
Models (4) /Merrcs (-+)
NFE FID 4 IS
Diffusion Models
GANs
DDPM [30]
1000
3.17
9.46
BigGAN [10]
14.73
9.22 10.14
Score SDE [73]
2000
9.89
StyleGAN2-ADA [36]
2.42
EDM [37]
35
2.04
9.84
Diffusion Distillation
Continuous Flows
Guided Dist. 0.3) [55]
7.34
8.90
Flow Matching (Diffusion) [46]
183
ECM [25]
3.81
Flow Matching (OT) [80]
50
4.94
Offline Distillation
2-rf++ [45]
1
3.07
GET [24]
6.25
9.40
GANs
KDM (EDM)
3.56
9.54
StyleGAN-XI, [36]
1
1.85
KDM-F (EDM)
3.24
9.68
Diffusion Distillation
Table 3: Generative performance on un-
PD [68]
1
9.12
DFNO [90]
1
4.12
conditional generation on FFHQ 64x64 and AFHQv2 64x64. *Indicates our reproduction.
iCD-EDM (LPIPS) [74]
1
2.83
9.54
CTM (LPIPS) [74]
I
1.98
ECM [25]
4.54
Madels (4) Metrics (-) NFE FID
IS
Offline Distillation
FFHQ 64x64
GET [24]
6.91
9.16
EDM*
79 1
2.47
3.37
2-rf++ w/o pre-train [45]
9.01
KDM (EDM)
6.54
3.12
IMM [91]
4.81
9.16
AFHQV2 64x64
KDM (EDM)
1
4.65
9.21
KDM (FM)
1
5.91
8.86
EDM*
79
2.02
9.04
KDM-F (EDM)
4.68
9.08
KDM (EDM)
4.85
8.25
where our method achieves highly competitive performance, surpassing the previous offline baseline in both FID and IS. In particular, we observe ~ 25% reduction in FID over GET. Tab. 2 shows results for class-conditional generation, where our model yields 40% improvement in FID and superior IS compared to GET. Although offline distillation offers advantages in efficiency, privacy, and flexible deployment [22, 24], a performance gap with online methods remains. This may partly be because online approaches leverage the full generation trajectory, whereas, in the offline pairwise setup, methods only observe two endpoints or just the generated data, placing them natural disadvantage. Our results demonstrate substantial step toward closing this gap.
Furthermore, unlike prior methods that rely on architectural assumptions [25, 45], our approach is denoiser-agnostic and successfully distills both EDM and flow matching (FM) models (see Tab. 1), demonstrating broad applicability. Finally, we implemented a factorized KDM (KDM-F). Since every matrix C, is diagonalizable up to arbitrarily small perturbation of the entries [2], we can write: C P- where P Cdxd is an invertible matrix and A diag(A Aa) E Caxd While enabling efficient implementation of spectral penalties on A (see complexity analysis below), this approach preserves strong performance and accuracy.
Unconditional generation with FFHQ and AFHQy2. To assess the scalability of KDM to higher- resolution data, evaluate method FFHQ and AFHQv2, distilling each into single-step generator and assessing performance both quantitatively (Tab. 3) and qualitatively (App. D.3). Our method yields FID scores approximately 2.5x higher than EDM, similar the degradation observed on CIFAR-10. Due to the high computational cost, we omit GET results: training a single model takes 5-7 weeks on an A6000 GPU, which is technically infeasible in computational resources. We also exclude ImageNet, as large-scale training with EDM [35] is computationally infeasible for us.
Scalability in parameters and data. Offline distillation inherently supports architectural indepen- dence from the teacher model, allowing flexible scaling of the student meet diverse deployment needs [24, 25). To assess this, we evaluate our method against GET across three model sizes-201 38M, and 84M parameters (Fig. 5A). Our method consistently outperforms GET in FID scores at all

---

scales, with especially notable gains at higher capacities (e.g.. 4.89 vs. 7.19 at 84M), demonstrating its effectiveness for both lightweight and high-capacity settings. In parallel, we investigate how performance scales with training data size, key factor in offline distillation. Following GET [24], we train on 250K, 500K, and IM samples and report FID scores on unconditional CIFAR-10 Fig. 5B, Results show that more data yields better generation quality. For instance, increasing the dataset from 250K to 1M samples results in a Re 33% FID reduction, showing the importance of data scale. While our method remains data-efficient, additional data can enhance performance.
Complexity analysis. Offline distillation can offer advantages over online methods, both memory- wise and time- timewwise (see App. E.5). KDM further improves over GET. As shown in Tab. 4, it matches GET in parameter count but achieves 4x speedup per training iteration and over 8x faster sampling. Notably, KDM-F (App. C) maintains this efficiency even with spectral regularization. Specifically, we evaluate KDM and KDM-F with an additional eigen-loss (EL) term, observing that KDM-F + EL requires training time comparable to KDM (without EL), whereas KDM + EL incurs longer training duration. These results underscore the practical scalability of our approach.
7 Discussion
In this work, we present two key observations. First, we reinterpret the distillation process through the lens dynamical systems theory. Second, uncover the semantical structure in the mapping from latent space to data space in trained diffusion dynamics. Motivated by these insights, we propose a Koopman-basee based frame- work for distilling diffusion models, bridging operator theory and generative modeling. We theoretically establish the existence of a Koopman operator that captures the underlying generative process, and demonstrate, both theoretically and empirically, that our method preserves its structure. This perspective also offers insights into potential failure modes. Beyond its conceptual contributions, our approach is practical and efficient, achieving highly competitive single-step offline distillation. We further analyze its robustness and scalability. We hope this work lays the foundation for future directions, including leveraging richer training signals and extending distillation techniques to advanced text-to-image image generative models. See App. for further discussion.
Table 4: Parameters in millions, time in seconds.
RTX 4090 GPU used with batch size 128.
Metric
GET
KDM
KDM EL
KDM-F EL
Params (M)
62
62
62
Train (s)
3.60
0.91
4.82
Sampling 937
114
114
116
Acknowledgments
This research was partially supported by the Lynn and William Frankel Center of the Computer Science Department, Ben-Gurion University of the Negev, an ISF grant 668/21, an ISF equipment grant, and by the Isracli Council for Higher Education (CHE) via the Data Science Research Center, Ben-Gurion University of the Negev, Israel.
10

---

References
[1] M. S. Albergo, M. Goldstein, N. M. Boffi, R. Ranganath, and E. Vanden-Eijnden. Stochastic interpolants with data-dependent couplings. arXiv preprint arxiv:2110.03725, 2023, 36
[2] S. Axler. Linear algebra done right. Springer, 2015. 9, 25
[3] O. Azencot, M. Ben-Chen, Chazal, and M. Ovsjanikov. An operator approach to tangent vector field processing. In Computer Graphics Forum, volume 32. pages 73 82. Wiley Online Library, 2013, 6
O. Azencot, N. Erichson, Lin, and M. Mahoney. Forecasting sequential data using consistent Koopman autoencoders. In International Conference on Machine Learning, pages 475-485. PMLR, 2020. 5
[5] T. Barami, N. Berman, I. Naiman, H. Amos, R. Ezra, and O. Azencot. Disentanglement beyond static vs. dynamic: henchmark and evaluation framework for multi-factor sequential representations. In Advances in Neural Information Processing Systems (NeurIPS) 39, Datasets and Benchmarks Track, 2025. 3
[6] N. Berman, A. Hallak, and A. Shocher. Who said neural networks aren't linear? vrXiv:2510.055770 2025.
arXiv preprint
[7] N. Berman, I. Naiman, and O. Azencot. Multifactor sequential disentanglement via structured Koopman autoencoders. In The Eleventh International Conference on Learning Representations, ICLR, 2023, 3
[8] Y. Blau and T. Michaeli. The penception-distortoon tradeoff. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2228-2337, 2018. 6
[9] Boullé and M. Colbrook. Multiplicative dynamic mode decomposition. arXiv preprint arXiv:2405.0338 2024. 2,4
[10] A. Brack, Donahue, and K. Simonyan. Large scale gan training for high fidelity natural image synthesis. arXiv preprint 1809 1096, 2018. 9
[11] S. L. Brunton and J. N. Kutz. Modern Koopman Theory for Dynamical Systems. SIAM, 2022. 6
[12] N. Carlini, J. Hayes, M. Nast, M. Jagielski, V. Sehwag, F. Tramer, B. Balle, D. Ippolito, and E. Wallace. Extracting training data from diffusion models. In 32nd USENIX security symposium (USENIX Security
23), pages 5253-5270, 2023. 2
[13] D. Chen, 7. Thou, J.PP. Mei, C. Shen, C. Chen, and C. Wang. A geometric perspective on diffusion models. arXiv preprint 1rXiv:2305.19947 2023. 2,
[14] Y. Chen and U. Vaidya. Sample complexity for nonlinear dynamics. arXiv preprint arXiv: 18X0v18110.11743 2018. 36
[15] Y. Choi, Y. Uh. Yoo, and J.WW. Ha. Stargan v2: Diverse image synthesis for multiple domains. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 3188-8197, 2020. 7
[16] A. Coletta, Gopalakrishnan, D. Borrajo, and S. Vyetrenko. On the constrained time-scries generation problem. Advances in Neural Information Processing Systems, 36:61048-6168 61059, 2023.
[17] V. De Bortoli, J. Thornton, J. Heng, and A. Doucet. Diffusion schrödinger bridge with applications score-based generative modeling. Advances Neural Information Processing Systems, 14:17695-17709, 2021. 36
[18] P. Dhariwal and A. Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780- 8794, 2021. 1.
[19] A. S. Dogra and W. Redman. Optimizing neural networks via koopman operator theory. Advances in Neural Information Processing Systems, 33:2087-2097, 2020.
[20] T. Eisner, B. Farkas, M. Haase, and R. Nagel. Operator theoretic aspects of ergodic theory, volume 272. Springer, 2015, 4
[21] G. Fadlon, I. Arbiv, N. Berman, and O. Azencot. diffusion model for regular time series generation from irregular data with completion and masking. In Advances in Neural Information Processing Systems (NeurIPS) 39, 2025.
[22] V. Fernandez, P. Sanchez, W. H. L. Pinaya, G. Jacenków, S. A. Tsaftaris, and J. Cardoso. Privacy distillation: reducing re-ideniification risk of multimodal diffusion models. arXiv preprint arXiv:2306.0132,, 2023. 2,

---

[23] G. Froyland, S. Lloyd, and A. Quas. Coherent structures and isolated spectrum for froben-srobenius cocycles. Ergodic Theory and Dynamical Systems, 30(3):229-756. 2010.
[24] Z. Geng, A. Pokle, and J. Z. Kolter. One-step diffusion distillation via deep equilibrium models. Advances in Neural Information Processing Systems, 36:41914 41931, 2023. 2, 9, 10, 32, 33,
[25] Z. Geng, A. Pokle, W. Luo, Lin, and Z. Kolter. Consistency models made easy. arXiv preprint arXiv:2406, 14548, 2024. 9
[26] T. Gonen, Pemper, I. Naiman, N. Berman, and O. Azencot. Time series generation under data scarcity: unified generative modeling approach. In Advances in Neural Information Processing Systems (NeurIPS) 39, 2025.
[27] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu. D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems, pages 2014. 3
[28] P. Goyal, S. Yildiz, and P. Benner. Deep learning for structure-preserviing universal stable koopman-inspired embeddings for nonlinear canonical hamiltonian dynamics. Machine Learning: Science and Technology, 6(1):015063 2025. 2, 4
[29] M. Han, J. Euler-Rolle, and R. K. Katzschmann. Desko: Stability-assured robust control with a deep stochastic koopman operator. In International conference on learning representations (ICLR), 2022. 3
[30] J. Ho, A. Jain, and P. Abbeel. Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, volume 33, pages 6840-6851, 2020. 1, 4,9
[31] J. Ho. T. Salimans, A. Gritsenko, W. Chan, M. Norouzi, and D. J. Fleet. Video diffusion models, Advances in Neural Information Processing Systems, 35:8633-8646, 2022.
[32] H. Hu and J. Pang. Memhership inference of diffusion models. arXiv preprint arXiv: 2023. 2
[33] L. C. Iacob, M. Schoukens, and R. Tóth. Finite dimensional koopman form of polynomial nonlinear systems. IFAC-Pupess,nnLine, 56(2))6423-6428, 2023. 2, 6,
[34] S. Jayasumana, Ramalingam, A. Veit, D. Glasner, A. Chakrabarti, and S. Kumar. Rethinking fid: Towards better evaluation metric for image generation. arXiv preprint orXiv:2401 0096003, 2024. 34
[35] T. Karras, M. Aittala, T. Aila, and S. Laine. Elucidating the design space of diffusion-basse generative models. Advances in neural information processing systems, 35:26565-26577, 2022. 3, 4, 7, 8, 9, 23, 27, 28, 29, 30, 31, 32, 33
[36] T. Karras, M. Aittala, J. Hellsten, S. Laine, J. Lehtinen, and T. Aila. Training generative adversarial networks with limited data, Advances in neural information processing systems. 12104-121144 2020.
[37] T. Karras, S. Laine, and T. Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 4401 4410, 2019. 7.9
[38] D. Kim, C. Lai, W. Liao, N. Murata, Y. Takida, T. Uesaka, Y. He, Y. Mitsufuji, and S. Ermon. Consistency trajectory models: Learning probability flow ODE trajectory of diffusion. In The Twelfth International Conference on Learning Representations. ICLR. OpenReview.net, 2024. 3, 5,36
[39] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In Y. Bengio and Y. LeCun, editors, 3rd International Conference Learning Representations, ICLR 2015, San Diego CA, USA, May 7-9. 2015, Conference Track Proceedings. 2015. 33
[40] D. Kingma and M. Welling. Auto-eneoding variational bayes. In International Conference on Learning Representations (ICLR), 2014. 3
[41] Z. Kong, W. Ping, Huang, K. Zhao and B. Catanzaro. Diffwave: versatile diffusion model for audio synthesis. arXiv preprint 20Xiv:2009.09700 2020. 1
[42] B. O. Koopman. Hamiltonian systems and transformation in hilbert space. Proceedings of the National Academy of Sciences, 17(5)315-318, 1931. 2,
[43] A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Toronto, ON, Canada, 2009. 7
12

---

[44] T. Kynkäänniemi, T. Karras, M. Aittala, T. Aila, and J. Lehtinen. The role of imagenet classes in fréchet inception distance. arXiv preprint arXiv:2203.0602 06026, 2022. 34
[45] S. Lee, Z. Lin, and G. Fanti. Improving the training of rectified flows. Advances in Neural Information
Processing Systems, 37:63082-63109, 2024. 8, 9
[46] Y. Lipman, R. T. Chen, H. Ben-Hamu. M. Nickel, and M. Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02744 2022. 9, 32
[47] Y. Lipman, E. Hajiramezanali, M. Mahdavi, A. Benson, A. Vahdat, S. Koyejo, and D. Yang. Flow matching for generative modeling. In International Conference Machine Learning (ICML), 2023. 1, 4, 33
[48] A. Lazano-Durán and G. Arranz. Information-thherrtiicc formulation of dynamical systems: causality, modeling, and control. Physical Review Research, 4(2)x02395,, 2022. 36
[49] C. Lu, Y. Zhou, F. Bao, J. Chen, C. Li, and J. Zhu. Dpm-solver. A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775-5787, 2022. 3
[50] E. Luhman and T. Luhman. Knowledge distillation in iterative generative models for improved sampling speed. arXiv preprint arXiv:2001. 02388. 2021.
[51] S. Luo, Tan, L. Huang. J. Li, and H. Zhao. Latent consistency models: Synthesizing high-resolution images with few-step inference. arXiv preprini arXiv:3310004378, 2023.
[52] W. Luo, T. Hu, S. Zhang, J. Sun, Z. Li, and Z. Zhang. Diff-instruct: A universal approach for transferring knowledge from pre-trained diffusion models. Advances in Neural Information Processing Systems, 36:76525-76566, 2023. 3
[53] B. Lusch, N. Kutz, and S. L.. Brunton. Deep learning for universal lincar embeddings of nonlinear dynamics. Nature communications, 9(1):4950, 2018. 3,5
[54] A. Mauroy, Y. Susuki, and I. Mezić. Introduction to the Koopman operator in dynamical systems and control theory. The koopman operator in systems and control: concepts, methodologies, and applications, pages 3 33, 2020. 7
[55] C. Meng. R. Rombach, R. Gao, D. Kingma, S. Ermon, J. Ho, and T. Salimans. On distillation of guided diffuusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition. pages 14397-14306, 2023.
[56] I. Mezić. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41:309-325, 2005. 3
[57] D. Muthirayan and P. P. Khargonekar. Online learning robust control of nonlinear dynamical systems. arXiv preprint 106. 04092, 2021. 36
[58] I. Naiman and O. Azencot. An operator theoretic approach for analyzing sequence neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 37, pages 9268 9276, 2023. 3
[59] I. Naiman, N. Berman, I. Pemper, I. Arbiv, G. Fadlon, and O. Azencot. Utilizing image transforms and diffusion models for generative modeling of short and long time series. Advances in Neural Information Processing Systems, 172121699-117799 2024. 1
[60] I. Naiman, N. B. Erichson, P. Ren, M. W. Mahoney, and O. Azencot. Generative modeling of regular and irregular time series data via Koopman VAEs. The Twelfth International Conference on Learning Representations, 2024. 3
[61] M. NasT, N. Carlini, J. Hayase, M. Jagielski, A. F. Cooper, D. Ippolito, C. A. Choquette-Chon F. Wallace, F. Tramer, and K. Tee. Scalable extraction of training data from (production) language models. arXiv preprint arX1::331..17355 2023. 2
[62] T. Nguyen and A. Tran. Swiftbrush: One-step text-to-image diffusion model with variational score distillation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7807-7816 16, 2024.3 3
[63] A. Q. Nichol and P. Dhariwal. Improved denoising diffusion probabilistic models. In International conference machine learning, pages $162-8171 PMLR, 2021. 1
13

---

[64] M. Otani, R. Togashi, Y. Sawai, R. Ishigami, Y. Nakashima, E. Rahtu, J. Heikkilä, and S. Satoh. Toward verifiable and reproducible human evaluation for text-to-image generation. In Proceedings the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pages 12345-12354, 2023, 34
[65] S. Otto and W. Rowley. Linearly recurrent autoencoder networks for learning dynamics. SIAM Journal on Applied Dynamical Systems, 18(1):558-593, 2019.
[66] F. Philipp, M. Schaller, K. Worthmann, S. Peitz, F.d Nueske, Error bounds for kernel-based approximations of the koopman operator. arXiv preprint M.Xiv:3301006637 2023. 36
[67] C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson. Spectral analysis of nonlinear flows. Journal of fluid mechanics, 641:115-127, 2009. 2,3
[68] T. Salimans and J. Ho. Progressive distillation for fast sampling of diffusion models. arXiv preprint arXiv:2202.005/2 12, 2022. 7.9
[69] A. Sauer, F. Boesel, T. Dockhorn, A. Blattmann, P. Esser, and R. Rombach. Fast high-resolution image synthesis with latent adversarial diffusion distillation. In SIGGRAPH Asia 2024 Conference Papers, pages 1-11, 2024. 3
[70] A. Sauer, D. Lorenz, Blattmann, and R. Rombach. Adversarial diffusion distillation. arXiv preprint arXiv:2311.17022 2023.
[71] A. Sauer, D. Lorenz, A. Blattmann, and R. Rombach. Adversarial diffusion distillation. In European Conference on Computer Vision, pages 87-103. Springer, 2024. 3
[72] J. Bohh--Dickttein, E. Weiss, N. Maheswaranathan, and S. Ganguli. Deep unsupervised learning using nonequilibrium thermodynamies. In International conference machine learning, pages 2256 2265. pmlr, 2015. 1,3
[73] J. Song, C. Meng, and Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010002500 2020. 1, 3.9
[74] Song and P. Dhariwal. Improved techniques for training consistency models. arXiv:2310.14189 2023. 9
arXiv preprint
[75] Y. Song, P. Dhariwal, M. Chen, and I. Sutskever. Consistency models. In International Conference on Machine Learning, ICMI, volume 202 of Proceedings of Machine Learning Research, pages 32211-32252. PMLR, 2023.
[76] Y. Song. J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations. In International Conference Learning Representations (ICLR). 2021. 2, 3,
[77] A. N. Specified. Fast high-resolution image synthesis with latent adversarial diffusion distillation. arXiv preprint Xiv 12015, 2024. 5
arXiv preprint [78] A. N. Specified. Sdxl-lightning: Progressive adversarial diffusion distillation. arXiv:2402. 13929, 2024. 5
[79] N. Takcishi, Y. Kawahara, and T. Yairi. Learning koopman invariant subspaces for dynamic mode decomposition. Advances in neural information processing systems, 30, 2017. 3
[80] A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang. J. Rector-Brooks
Brooks, G. Wolf, and Y. Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. arXiv preprint arXiv:200..00482, 2023.
[81] L. N. Trefethen. Approximation theory and approximation practice, extended edition. SIAM, 2019, 23
[82] E. Turan, A. Siazopoulos, and M. Ovsjanikov. Unfolding generative flows with koopman operators: Fast and interpretable sampling. arXiv preprint arXiv:2506.22300 2025.
[83] R. Vershynin. High-dimensional probabilit: An introduction with applications in data science, volume 47,
Cambridge university press, 2018. 23
[84] E. Yeung, S. Kundu, and N. Hodas. Learning deep neural network representations for koopman operators of nonlinear dynamical systems. In 2019 American Control Conference (ACC), pages 4832 4839, IEEE, 2019. 3
14

---

[85] T. Yin. M. Gharbi, T. Park, R. Zhang, E. Shechtman, F. Durand, and B. Freeman. Improved distribution matching distillation for fast image synthesis. Advances in neural information processing systems, 37:47455-477 474477, 2024, 36
[86] T. Yin, M. Gharbi, R. Zhang, E. Shechtman, F. Durand, W. T. Freeman, and T. Park. One-step diffusion with distribution matching distillation. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 6613-6623, 2024. 3
[87] B. J. Zhang. T. Sahai, and Y. Marzouk. A koopman framework rare event simulation in stochastic differential equations. Journal of Computational Physics, 456:111025, 2022. 36
[88] R. Zhang, Isola, A. Efros, E. Shechtman. and O. Wang. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and puitern recognition, pages 586-595, 2018. 5
[89] Y. Zhang, C. Deck, A. Tagliasacchi, J. Sohl-Dickstein, P. Jaini, and T. Zhang. Fast training of diffusion models via operator learning. In International Conference on Learning Representations (ICLR), 2023. 2
[90] H. Zheng, W. Nie, A. Vahdat, K. Azizzadenesheli, and A. Anandkumar. Fast sampling of diffusion models via operator learning. In International conference on machine learning, pages 42390-422444 422422. PMLR, 2023.
[91] L. Zhou, S. Ermon, and J. Song. Inductive moment matching, arXiv preprint arXiv:5503.07565, 2025. 8. 9
[92] L. Zhou, A. Lou, S. Khanna, and S. Ermon. Denoising diffusion bridge models. arXiv preprint arXiv:2309. 2023. 36
[93] M. Zhou, H. Zheng. Z. Wang, M. Yin, and H. Huang. Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation. In Forty-first International Conference Machine Learning, 2024.
[94] Z. Zhou, D. Chen, C. Wang, C. Chen, and S. Lyu. Simple and fast distillation of diffusion models. Advances in Neural Information Processing Systems, 37:40831 -00600 2024.
15

---

NeurIPS Paper Checklist
The checklist is designed encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
You should answer [Yes], [No], or [NA].
[NA] means either that the question is Not Applicable for that particular paper or the relevant
information is Not Available.
Please provide short (1-2 sentence) justification right after your answer (even for NA).
The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include (after eventual revisions) with the final version of your paper, and its final version will published with the paper.
The reviewers of your paper will be asked to use the checklist as one of the factors their evaluation. While "[Yes] is generally preferable to "[No] ", it is perfectly acceptable to answer "[No] provided proper justification is given (e.g., "error bars are not reported because would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No] or "[NA]" is not grounds for rejection. While the questions are phrased in binary way, we acknowledge that the true answer is often more nuanced, please just your best judgment and write justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to question, in the justification please point to the section(s) where related material for the question can be found.
IMPORTANT, please:
Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist",
Keep the checklist subsection headings, questions/answers and guidelines below.
Do not modify the questions and only use the provided macros for your answers.
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
Justification:
Contributions are clearly state at the end of the intro
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results be expected to generalize to other settings.
is fine include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
We dedicate discussion in the appendix for improvements and limitations
16

---

Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, c.g., if the approach was only tested few datasets with few runs. In general, empirical results often depend on implicit assumptions, which should articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification:
We provide in the appendix full proof's
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch provide intuition.
Inversely, any informal proof provided the core of the paper should complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification:
We provide technical details in the appendix and code in the supplementary
Guidelines:
The answer NA means that the paper does not include experiments.
17

---

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility w be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate the research performed.
While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g.. large language model), then there should either be way to access this model for reproducing the results or a way to reproduce the model (e.g.. with open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: Yes, in the supplementary material we provide code and link to anonymous drive with the data
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips.cr cc/ mublic/uidess/Cdeee wwt.Subbbisiiooosiisoiooooio for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.c
for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
18

---

Providing as much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the puper specify all the training and details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer:
[Yes]
Justification: The code provided reconstructs the exact experimental setting
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer:
[Yes]
Justification: results are reported by the standard protocol for generative models evaluation Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
should be clear whether the error bar is the standard deviation or the standard error of the mean.
OK to report 1-sigma error bars, but one should state The authors should preferably report 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that out of range (e.g. negative error rates).
If error bars reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Answer:
[Yes]
Justification:
We provide complexity analysis and hardware information.
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
19

---

The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.
The authors should make sure preserve anonymity (e.g.. if there is special consideration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly. harms that could arise when the technology being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, scraped datasets)?
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper poses such risks.
20

---

Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re- re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
this information not available online, the authors are encouraged to reach the asset's s creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification:
See codebase
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsoureing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
21

---

Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods this research? Note that if the L.M used only for writing, editing, formatting purposes and does not impact the core methodology. scientific rigorousness, or originality the research, declaration is not required.
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs any important, original, non-standard components.
Please refer to our LLM policy http:///urripssccco/eeeeeeeeeee for what should or should not be described.
22

---

A Proof of Theorem 5.1
For the reader's s convenience, restate the theorem.
Theorem (Finite Koopman Approximation for Analytic Dynamics), Let tb : + R be an analytic map, and let 2t (0, In). Then, for any 0, there exists linear operator C RRdxd such that
E [II(((((TT) CE( <,
where & _R R denotes the observable lifting map. Moreover, the required dimension d grows at most polynomially with I/e,, and the mapping & can be approximated arbitrarily well using multivariate polynomials up degree d.
Proof. Since : 12 + R" is analytic, for every and every compact set C R,, there exists multivariate polynomial map R + _ such that
sup ( p(=)ll ~
zek
which follows from the theory of approximation by Taylor polynomials for analytic functions [81]. Let Ty (O, In). Since æf has exponentially decaying tails [83], for any rj there exists such that
P(rT & BR) S 17:
where By = {a E R" |||||| < R). Restricting attention to Bn, we consider the polynomial approximation p of o with error most 5.
Now, following the result of lacob et [33], for the polynomial map p. there exists a finite-dimensional lifting m R. + Rd consisting of monomials up to some degree, and a linear operator C e Rg such that
<(p(x)) Cς(x)
for all x R", In particular, on Br.
[I{(*())) C((x)]] s [((((())) {((()))),
since {((()) = Cf(x) exactly.
Since is a smooth (polynomial) map, there exists Lipschitz constant L depending on the derivatives of over BR, such that
[l)()())) E(p())) s (z) - p(x)] s L5
for all & E BR.
Thus, for E BR.
[[(()()))) - CE(xT)) Lg.
Outside BR. the trivial bound
[I((((xT)) - C((xT)] < 2M
holds for some finite constant M > 0, due to the polynomial growth of & and the Gaussian decay of FT.
Hence, the expected Koopman error satisfies
E (+ C((xT) < (1 (1- + n(2M))²
Given any > 0, we can choose v sufficiently small so that 17(2M² S e/2,, and then sufficiently small that (1 L2 < e/2.
Notably, the degree of the polynomial map needed, and hence the number of monomials in E. grows at most polynomially with 1/5 and thus with 1/c, concluding the proof.
We further note that the function implemented by DhariwalUNet each module composing the network-including convolutions, SiLU activations, residual connections, attention mechanisms, and group normalization with e 0-is an analytic function. Since compositions of analytic functions remain analytic, it follows that the entire forward pass & is analytic on I
[35] analytic. Specifically,
23

---

B Proof of Theorem 5.2
We restate the theorem for the readers' convenience.
Theorem (Semantic Proximity via Koopman-Invarian Coordinates), Let 9; IR. R be the reverse diffiusion flow of a trained model (from time to t)0), and O C I,² (RR) be a dintte-iimnnsonaa subspace such that:
1. K,O for all E [o,
2. 3C, : + R such that Krf o = for all O.
3. 4T locally Lipschitz.
For any wt 22 RR, define un = PT for j 1,2, and let & IR R& be the vector-valued
observable {(x) = (((()),)(((()))) E O. Then:
1110 2 s L. (x)) (- S L. ||Cr|| (1((x)) E(
Proof. Let p, R - RR x denote the reverse diffusion dynamics. Based on the proof of Thm. 5.1, there exists linear operator Ct Rd -> IR satisfying
K& { 4, = C
for every E and E [0, T).
Define the vector-valued map for the basis {&,} of O
E(x) (z) ... Ea(x)) Rd.
Since the are smooth and capture independent semantic directions, & is locally bi-Lipschitz near typical zy. That is, there exist constants C1 C2 > 0 such that
C1l(((+)) {( s | 2 S Call((a)) E(.
Because 4 is locally Lipschitz, there exists L > such that
(2) Φ (= || - L|| 1
Substituting the bi-Lipschitz inequality yields
| 4r (x+) LCa1l((a)) E(
Moreover, by Koopman linearity,
{( CrE(xt), j= 1,2,
and therefore
(( ( C ((x)))
Thus, proximity in ((xr) coordinates implies proximity of final samples z/j and 23 under Py.
24

---

C Factorized Koopman Matrix for Fast Eigenspecturm penalty
To enable control over cigenvalues of the Koopman operator, we describe an efficient implementation of our Koopman-baser Distillation Method (KDM), which we term KDM-F, alleviating the need for computation of the eigendecomposition during training. Since every matrix == Cy diagonalizable up to an arbitrarily small perturbation of the entries [2], we can write:
C = PAP-1
where P Cdxd is an invertible matrix and - diag(AI,..... Aa) E Cdxd, This way enables efficient implementation of spectral penalties on A, without the need compute it on the fly during training.
To represent this decomposition within a real-valuee neural network framework, we learn the real and imaginary components of the Koopman eigenvectors separately. Specificallyy two orthonormal matrices, Pre and Pin. represent the real and imaginary parts of the eigenvectors. These are combined into a real-valued block matrix using the standard transformation for complex-to to-reel matrix representation:
P
Pre P
-
Pim
E ER2. 2dxdd
Pre
Rather than explicitly computing the inverse P-1, we parameterize separate learnable matrix Pin- with the structure, constructed analogously from unconstrained real matrices. This improves flexibility and avoids numerical instability associated with matrix inversion.
The eigenvalues of the Koopman operator are encoded using polar coordinates. Specifically, the modulus and phase of each eigenvalue are parameterized as
\ = e exp()s) e0 e- exp(x;) (cos(0,) + siin()))),
where Vy and Oj are learnable parameters. Let Ase (real (4y)): Am (imag(Aş)) E Rd be the aggregated vector of real and imaginary parts of the eigenvalues, respectively. We assemble these values into real-valued block-diagone matrix:
diag( (Treg(Ar)) diag Xim)
diag( Aim) diag( Are)
A
R 1d
At inference time, the latent state vector Rd is first augmented 2d2d by concatenating vector: 2 = O]. The evolution under the Koopman operator is then computed as
Znext
P AP..........
Finally, discard the imaginary part and retain only the first d entries return real-valued output of the same dimensionality as the input.
This formulation enables to simulate rich, expressive dynamics through fully linear evolution in the lifted Koopman space, while maintaining computational tractability and interpretability. Moreover, this decomposition is particularly amenable to distillation: the structure of the transformation ensures that semantic and temporal coherence is preserved during student training, as the model explicitly disentangles dynamics into magnitude and phase components.
25

---

D Additional Experiments and Analysis
D.1 Ablation Studies
Loss Ablation Study, We conduct an ablation study to evaluate the contribution of each loss component. As shown in Tab. 5, using only ered any pairwise combination of the Koopman-related losses still enables the model to learn, as indicated by moderate FID and IS scores. However, incorporating all three losses together, including Koopmann creates a synergistic refectt-reducng the FID from around 10 to 7.83 and achieving a higher IS than any partial combination. Finally, we highlight the importance of incorporating an adversarial setup, which further improves both FID and IS scores by enhancing perceptual quality.
Note, integrating adversarial losses into alternative distillation methods for diffusion models is often non-trivial and presents significant challenges. Many existing approaches lack the architectural flexibility training stability required to effectively incorporate such objectives, unlike our proposed KDM. Notably, we extended GET with adversarial training, which improved its FID from 6.91 I to 6.13. However, KDM still achieves a superior FID of 4.68, demonstrating its effectiveness beyond prior methods.
Table 5: Koopman losses ablation; bold is best and underscore is second-best
Eped
Eree
Cia + Lyeed
Lree + freed
Liee + Lix
Kouupmun
C
GET + Adv
FID
487
11.2
457
9.53
10.7 8.26
11.16 8.68
7.83
4.97
6.13
IS
1.00
8.23
1.00
8.82
8.83
9.22
8.98
Ablation Loss Weighting and Andy Sensitivity We investigate the effect of different weightings for the components of the Koopman loss, namely Lrees Llut and fpred, as well as the influence of the adversarial weight Aady- The results indicate that the model is robust to moderate changes in these loss weights, with FID scores remaining within a narrow range (4.94-5.03). For the adversarial term, varied Andv E {0.5, 1.0, 0.25) and observed that Aadv 1.0 achieved the best trade-off between fidelity and stability, yielding an FID of 4.57 compared to 4.98 for Aadv = 0.5 and 4.67 for the default Andy 0.25 setting used in the main paper. These findings suggest that KDM remains stable under different loss scalings and that its performance is not overly sensitive to the choice of Agdv, demonstrating strong robustness to training hyperparameters.
Empirical Investigation of Koopman Matrix Size. We evaluate the impact of varying the size of the Koopman matrix. As shown in Tab. 6, smaller matrices tend to generalize better than larger ones. This result is somewhat counterintuitive, as the scaling experiments in Sec. 6.2 demonstrate clear trend where increasing the number of parameters typically improves performance. We hypothesize that smaller matrices generalize more effectively, and we leave deeper understanding of this phenomenon through the lens of dynamical systems theory and empirical analysis for future research.
Table 6: FID scores across matrix sizes on the CIFAR- 10 unconditional task; bold is best.
Size
1024
4096
9216
FID
5.08
7.24
9,41
D.2 Training Stability and Adversarial Loss Convergence
To evaluate the training stability of KDM, we compare the generator and discriminator loss dynamics of KDM and GET across uniformly sampled training steps. Despite the inclusion of an adversarial component in KDM, both losses exhibit stable and convergent behavior throughout training, indicating that the adversarial term does not introduce significant instability practice. Fig.6 illustrates generator and discriminator loss trajectories for both KDM and GET. We discard the 0% step due to corrupted values the raw logs. Overall, the results confirm that KDM maintains stable optimization behavior comparable GET, while benefiting from enhanced perceptual quality.
26

---

Generator and Discriminator Loss Convergence for KDM and GET
Loss Value
10
20
30
40
50
60
70
80
90
100
Training Progress (%)
KDM Gen. Loss
KDM Disc. Loss
GET Gen. Loss GET Disc. Loss
Figure 6: Training loss convergence comparison between KDM and GET. Despite including
adversarial loss, KDM maintains stable generator and discriminator convergence comparable to GET.
D.3 Additional Qualitative Results
We include additional qualitative results our method, extending the evaluation presented in the main text more datasets and testing scenarios.
Additional Comparison of Student vs Teacher. In Figs. 8, 9, and 10, we present generations from model alongside those of the original EDM model. Corresponding blocks indicate samples generated from the same initial noise.
D
Figure 7: Unconditional Generation of FFHQ. Sampled images using 79 NFEs with EDM [35] (left), and using NFE with KDM (right).
27

---

Figure 8: Unconditional Generation of FFHQ. Sampled images using 79 NFEs with EDM [35] (left),
and using NFE with KDM (right).
Figure 9: Unconditional Generation of CIFAR-10. Sampled images using 35 NFEs with EDM [35]
(left), and using NFE with KDM (right).
28

---

Figure 10: Conditional Generation of CIFAR-10. Sampled images using 35 NFEs with EDM [35] (left), and using NFE with KDM (right).
29

---

Additional Local Structure Analysis. We extend the experiment from Sec. 6.1 to AFHQV2. conditional CIFAR-10, 10, and unconditional CIFAR-10 10 datasets, as shown in Fig. 11, Fig. 12, and Fig. 13, respectively. The results reinforce the conclusions of the main experiment: both EDM and our distilled model exhibit local semantic coherence, preserving features such as position, color, identity, and more even when traversing farther from the original point in noise space.
Figure 11: Unconditional AFHQv2: Comparison of neighborhood exploration across different noise levels a. Left: EDM [35]. Right: our method. Each image corresponds to small perturbation around the original noise sample.
30

---

0.05
0.2
0.3
0.4
0.5
0.6
0.05
0.2
Conditional CIFAR-10: Comparison of neighborhood exploration across different noise levels a. Left: EDM [35]. Right: our method. Each image corresponds to small perturbation around the original noise sample.
Figure 12:
0.05
0.1
0.2
0.3
0.4
0.5
0.6
a=0
0.05
0.2
Unconditional CIFAR-10: Comparison of neighborhood exploration across different noise levels a. Left: EDM [35]. Right: our method. Each image corresponds to small perturbation around the original noise sample.
Figure 13:
31

---

E
Additional Details
E.1
Training and Sampling Procedures Psudo-codes
The training process of the model follows Koopman-inspired backward mapping approach, augmented with adversarial supervision. Given clean-noisy pair (To, the model learns latent Koopman operator that evolves the noisy encoding backward to approximate the clean encoding, while a decoder reconstructs the input from the latent space. The model also supports class-conditional dynamics via Koopman control.
Algorithm Training Procedure
Require: Training pairs (T), labels y (optional), model Egg. Eng, Dy discriminator D.
1: for each training step do
2: Encode 20 -- (x 0,
3: Encode 22 - (x .V)
4: Add noise: žy + 2a co, 2T t- 2r + er, where ex, co ~ (00.0.44
5: Push via Koopman: 20 +- (θT)
6: if conditional Koopman then
Add control: push f push + Cy( end if
7:
8:
9:
Decode: 20 + Dv( 0,3)
Decode pushed: atpun + (20 push 00,3
10:
Compute latent loss: Lia ||00 20 pusap 1|2
12:
Compute reconstruction loss: Lroe + d(Io, Lo)
Compute full pass loss: Epred + d(zo, af
13:
Compute model adversarial loss: LG CI,1,, (,,
15:
Compute discriminator loss: C + C,,L, + CEEOO D, model Combine losses:
Liaal Live + Speed
List + Wadv C
17: Update Esg, Exp, C.p. Cu. Du. and D,, via backpropagation end for
is a distance measure such as MSE or LPIPS. We utilize LPIPS. CE is cross-entropy loss. At inference time, the model can sample clean images by reversing from random Gaussian noise ø7, applying the Koopman operator in latent space, and decoding to the image space.
Algorithm 2 Sampling Procedure
Require: Koopman model: Eg, Cn. Cu. Dw,, labels y (optional)
1: Sample noise xT ~ N/ o²)
2: Encode: 2T +- (xTT T,y)
3: Push: 2gush +- (2r)
4: if conditional Koopman then push
5: Add control: 20
+ 20
push
+ (3)
6: end if
7: Decode: 2g + (2f 0, 3)
8: return Clean sample go
F.2 Datasets Collection
For the Checkerboard dataset, we sampled using 10 NFEs with EDM [35] or flow matching [46]. Following [24], we sampled CIFAR-10 with 35 NFEs from EDM. For FFHQ and AFHQv2, we used 79 NEEs using EDM, as recommended.
32

---

E.3 Method Implementation Details
We provide the full code implementation for the Checkerboard and CIFAR-10 (both conditional and unconditional) datasets in the supplementary material. Due memory constraints, the imple- mentations for FFHQ and AFHQv2 will be released shortly. this section, we briefty describe the implementation details of each method, while we encourage the reader refer to the codebase for exact configurations. We train our models for 800k iterations with the Adam optimizer [39] at a fixed learning rate of 3e-4. We do use warm-up, weight decay, or any learning-rate schedule. All runs are on single NVIDIA A6000 GPU or RTX4090. Note that our method can fully support distributed training.
Checkerboard. For this dataset, we build upon the implementation of [47]. Both zo and aT are encoded using the same denoising architecture described in their work. The decoding of zo is performed using the inverse this module. The Koopman operator implemented as a simple linear layer. Complete implementation details are available in the shared code.
CIFAR-10, FFHQ, and AFHQv2. We adopt the Song architecture from [35] for encoding To and æy, as well as for decoding To reduce latent dimensionality, the output channels of the network are set 1, effectively reducing the latent space size by factor of The Koopman operator is implemented either as a standard linear transformation or using our custom Fast Koopman module. To maintain architectural fairness, we downscale all networks to match the total parameter count of the original Karras network [35] and the GET distillation model [24]. The FFHQ and AFHQv2 configurations follow a similar encoder-decoder structure with one key difference: a linear projection layer is appended to the end of the encoder and the beginning of the decoder further control the latent space dimensionality. All other aspects remain consistent across datasets. Note that the discriminator is trained using separate optimizer, akin to the Koopman module's optimizer.
Hyperparameters and Training. We provide complete training scripts along with plug-and-pian configuration files in the supplementary material. An overview of the key hyperparameters used for each dataset is summarized in Tab. 7. Overall, our method requires minimal hyperparameter tuning and we did not perform any hyperparameter search.
Unet Out Channels: Controls the number of output channels Song
Unet Model Channels: Determines the internal channel width, affecting the model's capacity.
Noisy Latent Injection:
A Gaussian noise term (0, 1) added latents (as described in the pseudocode) to improve generalization and mitigate numerical instabilities.
Adversarial Weight: Applied the adversarial loss term in generator loss; used for the discriminator loss directly.
Linear Module: Indicates the presence of additional linear layer for compressing the latent space, applied at the output of the encoder and the input of the decoder.
Table 7: Model Hyperparameters Across Datasets
Configuration
Checkerboard
CIFAR-10
FFHQ
AFHQv2
Batch Size
4096
128
64
64
Learning Rate (LR)
0.0003
0.0003
0.0003
0.0003
Iterations
800k
800k
800k
800k
Unet Out Channels
1
1
1
Unet Model Channels
64
64
32
32
Noisy Latent Injection
0.4
0.4
0.4
0.4
Adversarial Weight
0.01
0.01
0.01
0.01
Linear Module
No
No
Yes
Yes
Latent Dim Size
1024
1024
512
512
33

---

Table 8: Human Evaluation of Image Realism. The reported ratio reflects the number of times participants selected the model-generated collage as more realistic, out of a total of 300 evaluations.
EDM (Teacher)
KDM (Student)
Ratio
45%
55%
E.4 Human Evaluation Experiment
While the Fréchet Inception
Distance (FID) is widely used to evaluate the quality of images generated by generative models, recent studies have highlighted its limitations in aligning with human judgments of image realism. For instance, Jayasumana al. [34] demonstrate that FII) can contradict human evaluations, particularly in assessing perceptual quality and semantic alignment, due to its reliance on Inception-v3 features and the assumption of Gaussian distributions. Similarly, Otani et al. [64] emphasize that automatic metries like FID often fail to capture the nuanced aspects of human perception, underscoring the need for standardized human evaluation protocols. Furthermore. Kynkäänniemi et al. [44] reveal that FID can be manipulated by aligning top-N class histograms without genuinely improving image quality, indicating potential biases in the metric. These findings suggest that while FID provides useful quantitative signal, it may not fully encapsulate human perceptions of image realism, highlighting the importance of incorporating human evaluations when assessing generative models.
Therefore, in the main paper, we report the results of a human evaluation experiment designed to compare our method against EDM. Specifically, we curated 10 image collages (6x6) of human faces generated by the teacher model (EDM, 79 NFEs) and 10 collages generated by our student model (1 NFE), using samples from the FFHQ dataset. Each comparison slide presents one collage from each model, displayed side-by-side as illustrated in Fig. 4, with the order randomized. We then asked 30 human raters to evaluate 10 such slides and choose, for each, which collage appeared more realistic. both appeared equally realistic, they were instructed to make a random choice. This process yielded 300 total decisions. We report in the main paper and in Tab. 8, the proportion of times each model was preferred. Finally, we attach to the appendix both the slides that we sent to the practitioners and the Excel that aggregates the results. The results indicate that decisions random, suggesting that participants had difficulty distinguishing between the student and teacher generations. This implies that, despite existing FID gaps, human evaluators found it hard to perceive meaningful differences. Participants reported that both collages included highly realistic images, as well as images with noticeable artifacts that appeared unrealistic, in their judgment.
made almost at
E,5
Cost and Efficiency of Offline Distillation
An important consideration in offline distillation is the total computational cost, which includes both the training of the student model and the one-time cost of generating the teacher-produced noise-data pairs. While this pair generation step incurs initial overhead, it is performed only once and be reused across multiple student architectures or experiments, effectively amortizing its cost. Following the analysis of [24], which explicitly incorporates the cost of data-pair generation into its complexity assessment, the overall computational load of offline distillation remains favorable compared to online approaches. This primarily because online distillation requires repeated access to the full teacher model throughout training, resulting in substantially higher neural function evaluations and memory usage. our experiments on the main paper, we observe that even when accounting for the data generation step, our method achieves a markedly better trade-off between cost and performance than prior offline frameworks such as GET. These results collectively support that, despite the initial preprocessing overhead, offline distillation constitutes an efficient and scalable alternative for practical deployment. To see the full analysis between online and offline approaches, please refer to [24].
E.6 Parameter Scaling
We report the full results of the scaling experiment shown in Sec. 6.2, including IS score from the parameter scaling comparison in Tab. 9. The results demonstrate that our approach consistently outperforms GET in terms of FID across all model sizes, except for in 20M, indicating superior perceptual quality. Notably, the improvement is more pronounced as the model size increases, with
34

---

our 84M model achieving an FID of 4.89 compared to 7.19 for GET. In terms of IS, our method matches or surpasses GET, with particularly strong results at larger scales, These results underscore the scalability and effectiveness of our approach.
Table 9: FID and IS scores for different model sizes, comparing approach with GET.
20M
38M
84M
Ours
GET
Ours
GET
Ours
GET
FID
10.2
10.72
5.04
8.00
4.89
7.19
IS
8.50
8.69
9.12
9.03
9.11
9.09
E.7 Extended Outlier Analysis
In Sec. 6.1, we present the results of our model's component (left side) in the experiment shown in Fig. 14. Outlier detection was performed using the DBSCAN algorithm with a distance threshold of 0.15. Interestingly, our model exhibits a slightly increased number of outliers. We hypothesize that this may be due to early stopping during training, as our primary objective performance on this particular dataset, but rather to gain insight and intuition into the nature of the learned dynamics. Finally, we observe that the outlier phenomenon is inherited from the teacher model (right side), suggesting that it is more general characteristic of the generative dynamics rather than an artifact introduced by the distillation process.
not maximize
16 xx
Figure 14:
Outlier analysis generated checkerboard patterns. The two left images are our model (student), and the two images on the right are from the flow matching model (teacher).
F Additional Discussion of Future Work and Current Limitations
In this study, we took significant step toward leveraging dynamical systems theory for offline distillation, demonstrating its strong potential in this domain. In this section, we outline several future research directions that we believe are both promising and intellectually compelling.
Standalone Koopman Generative Model. Koopman theory enables modeling nonlinear dynamical systems in a transformed space where they can be analyzed linearly. This raises an intriguing question: Can the Koopman framework serve as standalone generative model, capable of learning data distributions from scratch? While the question remains open, exploring this direction could yield novel and theoretically grounded framework for generative modeling.
Utilization of Full Trajectory Information. Our current offline setup relies solely on noise-data pairs extracted from pre-trained teacher models. However, the diffusion process inherently includes intermediate steps that are discarded. We hypothesize that leveraging this full trajectory information could enhance trajectory alignment and improve student performance. Moreover, Koopman theory naturally accommodates multi-step signals, making this a promising direction for further investigation. Additionally, the current framework does not support quality-versus-suee pressu--aapliigg-tery trade-off, which some online methods do enable. Future work incorporating the full diffusion trajectory could integrate this flexibility into our approach.
35

---

Incorporating Broader Koopman Theory Literature. The connection between Koopman theory and diffusion model distillation opens the door to rich body of theoretical and practical tools. On the theoretical side, works on sample complexity and error bounds for Koopman operators [66, 14, 57] may provide insights into convergence guarantees. Information-theeretie perspectives [48] could offer new modeling constraints or regularization schemes. On the practical side, methods for rare-event sampling [87] may improve long-tail generation capabilities. Finally, while the current mathematical formulation supports discrete Koopman evaluation, the underlying theory naturally extends to continuous settings as well. Exploring these directions could substantially extend and generalize the impact of our framework.
Enhancing the Adversarial Setup. While adversarial components have been shown to be critical in many distillation methods [85, 38), our current implementation uses basic discriminator architecture with standard loss functions and training schemes. We believe that integrating more advanced adversarial paradigms could significantly boost performance, potentially closing the gap with state-of-the-ar online distillation methods.
Decision Boundary Insights. In our synthetic experiments, we identified common cause underlying failure modes or "outliers" in the generative dynamics. Further investigation is needed to determine whether this phenomenon generalizes development of strategies such as uncertainty estimation, selective sampling to avoid generating low-quality outputs, incorporating structural constraints into the latent space prevent such failures, or introducing auxiliary self-supervised objectives-such as contrastive learning- enhance the model's ss ability distinguish between semantically similar groups.
more complex datasets. If it does, it could guide the
Toward Broader Applications. Due to limited computational resources, we were unable to extend our offline distillation framework more complex tasks, such as text-to-image generation, that require large-scale compute infrastructures. It will be interesting to explore how our method can scale or adapt other domains and tasks. Additionally, because our method makes no assumptions about the initial and terminal states of the underlying dynamical system, we believe it is applicable to a broader class of generative models, including DDBMs [92], stochastic interpolants [1]. and others [17].
36


Reviews:
[{"review_id": "AKxhpiv6EY", "reviewer": "Reviewer_FqCX", "summary": "This paper proposes the Koopman Distillation Model (KDM), a novel offline distillation framework designed for one-step generation. The proposed method is grounded in Koopman operator theory, a classical mathematical framework that seeks to represent complex nonlinear dynamics as a globally linear operator acting on a transformed space of \"observable\" functions. In practice, KDM employs an encoder-linear-decoder architecture. A neural network encoder maps a noisy input into a high-dimensional latent space; a learned, finite-dimensional Koopman matrix then propagates the latent state forward in a single, linear step; finally, a decoder network reconstructs the clean, final sample from the evolved latent state. The authors provide theoretical justification for this architecture, proving that the learned diffusion dynamics admit a finite-dimensional Koopman representation under mild assumptions.", "questions": "1. Forgive me if I have missed it, but I would like to see the relationship between the number of pre-constructed noise-data pairs and the distillation performance. One big problem with offline distillation is the possibility of prior holes.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 1, "originality": 3, "strengths_and_weaknesses": "**Strength**\n1. Novel application of Koopman operator theory to the problem of offline diffusion model distillation.\n2. The paper provides a solid theoretical foundation for its method. Key contributions include a proof that the learned diffusion dynamics can be represented by a finite-dimensional Koopman operator under mild assumptions, which justifies the core architectural choice. \n3. The writing is clear and easy to follow. \n\n**Weakness**\n1. The proposed method cannot really utilize more inference-time compute for better generation, unlike Consistency Models.\n2. I do not think that the community can agree over the statement on offline distillation methods are more scalable and modular. Generally, the community has moved away from this family of approaches.\n3. The empirical results are very weak. The paper only conducts experiments on toyish datasets like CIFAR and FFHQ. The results are lackluster. Without adversarial training, it is worse than the only offline distillation baseline (7.83 vs 6.91). It is not at all comparable to the likes of DMD and CM.", "was_revised": true, "final_justification": "My original main complaint was on the restrictive nature of the setting, and the above response has provided improvement in that aspect. In all, I appreciate the development of the method itself, and offline distillation might be a reasonable setting.", "rebuttal": "Thank you for your constructive and thorough review; it's helped us significantly improve our work. We really appreciate your recognition of the novelty of our Koopman operator-based distillation approach, the strength of our theoretical framework, and the clarity of the paper. We've addressed your questions and concerns below and will incorporate your suggestions into the final revision.\n\n---\n\n> W1: The proposed method cannot really utilize more inference-time compute for better generation, unlike Consistency Models.\n\nYou're right that, in its current form, KDM doesn't support inference-time compute scaling in the same way as Consistency Models. However, we briefly discuss a potential simple extension of this idea in Appendix F. Specifically, since our model is based on the Koopman operator, its infinitesimal version theoretically allows for simulating the full diffusion trajectory via integration. This offers a path to better adhere to the teacher’s dynamics and leverage additional NFEs at inference time. We view this as a promising direction for future work and appreciate you highlighting its relevance.\n\n---\n\n> W2: I do not think that the community can agree over the statement on offline distillation methods are more scalable and modular. Generally, the community has moved away from this family of approaches.\n\nTo avoid vague terminology like \"scalable and modular,\" we'll revise that sentence in the related work section to more clearly communicate the concrete benefits of our approach.\n\nThat said, we believe the offline setting remains highly valuable, particularly in scenarios where training without access to the teacher model is desirable, computational and memory resources are limited, or precomputed supervision needs to be reused across multiple student models. These advantages, outlined in Lines 36–42, are especially relevant in resource-constrained environments or where privacy is a key concern. Please also see our response to Reviewer `mkU1` for W1, where we analyze the complexity trade-offs between our method and state-of-the-art online distillation approaches. While the community has indeed focused more on online distillation, we believe that offline distillation offers inherent advantages that can be highly meaningful for many use cases and people in the broader research community.\n\nFor instance, from a privacy perspective, offline methods offer a clear advantage by reducing the risk of inversion and membership inference attacks. Unlike traditional online distillation, where the student repeatedly queries the teacher (e.g., during denoising steps), increasing exposure to sensitive information, offline distillation relies on a fixed set of noise–data pairs. Once generated (and optionally anonymized or filtered), the teacher model can be discarded or kept private, significantly reducing the attack surface. This makes it computationally infeasible for either the student or an adversary to recover the original training distribution beyond what is provided. A practical use case of this setting is outsourcing training to third parties without disclosing sensitive model parameters or proprietary data. The data curator (e.g., a hospital or financial institution) can sanitize training samples to remove personally identifiable information (PII), and since the external trainer does not need access to the full generative model or its sampling process, they remain isolated from internal correlations that may otherwise leak private content. This enables privacy-preserving collaboration while retaining model utility.\n\nIn general, while our KDM doesn't yet match the performance of leading online techniques, it significantly narrows the gap between offline and online approaches. This reveals the potential of offline methods and highlights a promising, yet underexplored, direction. We hope our work will motivate further research attention in this area, where our results already demonstrate tangible improvements.\n\n---\n\n> W3: The empirical results are very weak. The paper only conducts experiments on toyish datasets like CIFAR and FFHQ. The results are lackluster. Without adversarial training, it is worse than the only offline distillation baseline (7.83 vs 6.91). It is not at all comparable to the likes of DMD and CM.\n\nOur experimental setup closely follows the standard evaluation protocol established by GET, the most recent and relevant offline distillation baseline. Unfortunately, scaling to large-scale datasets such as ImageNet is currently infeasible for us due to computational resource limitations at our institute. That said, we have incorporated two medium-sized, high-quality datasets, FFHQ and AFHQv2, that were not included in the original GET benchmark, thereby contributing to the expansion of the offline distillation evaluation landscape and showing the models' capability to scale to higher resolution datasets. Within this benchmark, KDM achieves state-of-the-art performance in the offline setting. While our method doesn't yet match the performance of online or simulation-free approaches, we are transparent about this limitation and clearly frame our contributions within the offline distillation paradigm, which has fundamental differences from the online setting.\n\nAdditionally, integrating adversarial losses into alternative distillation methods for diffusion models is often non-trivial and presents significant challenges. Many existing approaches lack the architectural flexibility or training stability required to effectively incorporate such objectives, unlike our proposed KDM. Notably, we extended GET with adversarial training, which improved its FID from 6.91 to 6.13. However, KDM still achieves a superior FID of 4.68, demonstrating its effectiveness beyond prior methods. Furthermore, the DMD method is also a method that utilizes a generator and fake vs. real training setup as part of its components, sharing similar outlines to our method. Therefore, we believe it would be somewhat misleading or unfair to exclude the adversarial component, as it's an integral part of our method.\n\nWe would also like to highlight that our model is reasonably comparable to Consistency Models (CM). CM achieves an FID of 3.55, while our method attains a score of 4.68, a relatively small margin of just 1.13 FID points. However, CM requires 8 A100 GPUs over several days for training, whereas our method achieves its performance using a single A100 GPU over the same duration, demonstrating significantly greater efficiency. Additionally, the DMD method has only been evaluated on conditional generation with CIFAR-10, achieving an FID of 2.62. On the same task, our model achieves an FID of 3.24, reflecting a small performance gap of only 0.62 points. Therefore, we respectfully disagree with the assertion that our results are not at all comparable. And, we believe that in addition to offering the inherent advantages of offline approaches (e.g., privacy and memory efficiency), it can provide a trade-off between performance and computational cost.\n\n---\n\n> Q1: Forgive me if I have missed it, but I would like to see the relationship between the number of pre-constructed noise-data pairs and the distillation performance. One big problem with offline distillation is the possibility of prior holes.\n\nWe do analyze the effect of the number of precomputed noise-data pairs in our study (see Lines 341–346 and Fig. 5B). As shown there, increasing the dataset size from 250k to 1M samples leads to a substantial improvement in FID, approximately a 33% reduction, highlighting the importance of broad coverage in the training distribution. While we agree that further scaling would provide deeper insight into the limits of offline distillation and potential “prior holes,” we were constrained by available computational resources and leave larger-scale studies to future work."}, {"review_id": "ZWSPzZwofO", "reviewer": "Reviewer_mkU1", "summary": "This work proposes the Koopman distillation model for offline distillation, in order to compress diffusion models. The authors observed that there is lack of theoretical adoption of dynamic systems theory to diffusion models. It is also empirically found that there is semantic coherence in latent spaces induced by diffusion models. From these two observations, Koopman distillation model (KDM) is proposed and theoretically analyzed. It distills diffusion models via linear evolution in a learned embedding space which is generated by an encoder given a noisy input. From the embedding, a decoder then reconstructs the clean samples. The authors also theoretically proved the existence of finite-dimensional Koopman representations and semantic structure preservation under mild assumptions. Experimental results also demonstrate its performance compared with baselines.", "questions": "1. *Training loss*: The design of the loss function contains the adversarial loss in order to enhance the quality of the generated images. However, it may introduce the instability of the training process and the difficulty to converge. Would the authors demonstrate the training loss convergence curve in their experiments and compare it with GET and CTM?\n\n2. *Memory cost*: In experiments, the quality of generated images is compared on the same scale of parameters across KDM and GET. However, as claimed in the introduction that KDM as an offline distillation method is more efficient than diffusion distillation methods such as CTM. Hence could the comparison include CTM on the parameter efficiency or other metrics to demonstrate the efficiency of KDM over CTM?\n\n3. *Higher resolution dataset*: The experiments are conducted on CIFAR-10, FFHQ 64×64 and AFHQv2 64×64 datasets. Would the authors present the experiments on ImageNet-1k to evaluate the performance of KDM on higher resolution datasets?\n\n4. *Clarification*: A minor question is on the notations. Could the authors clarify and explain further on the differences between $C_\\mu$ and $C_\\eta$?\n\n5. *Quality*: The quality and motivations should be stronger. The current prominent compression method for diffusion models are consistency models and their variants. Although the privacy is mentioned as one of the advantages, experiments do not demonstrate and support it. It would improve the quality to include them and articulate them in experiments briefly.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. *Originality*: This work novelly proposed the one-step Koopman distillation model (KDM) as an offline distillation of diffusion models. It is of originality to consider the Koopman operator in distillation of diffusion models, which did not appear in past research.\n\n2. *Clarity*: The presentation is clear and logically linear. The whole work is easy to follow. The motivations and methods are clearly articulated in terms of the current limitations of diffusion models and the mechanism and pipeline of KDM. \n\n3. *Significance*: The excessive computational cost is a current pressing issue of diffusion models as a powerful visual content generator. The proposed KDM is important to solve this problem which is also vital.\n\n**Weaknesses**\n\n1. *Quality*: The performance of the proposed KDM does not of the state-of-the-art among all the presented compression methods for diffusion models. Although KDM is an offline distillation method, the experimental results did not demonstrate the advantages of it over CTM as diffusion distillation methods.", "was_revised": true, "final_justification": "All my questions are well addressed by the authors. Hence I raised my score from 4 Borderline Accept to 5 Accept.", "rebuttal": "Thank you for your thoughtful and constructive review. We appreciate your recognition of the novelty, clarity, and significance of our proposed KDM, as well as your detailed feedback. Your comments on evaluation, training stability, efficiency comparisons, and clarity are insightful. We address your questions below and, if given the opportunity, will incorporate your suggestions into the final revision.\n\n---\n\n> W1: Quality: The performance of the proposed KDM does not of the state-of-the-art among all the presented compression methods for diffusion models. Although KDM is an offline distillation method, the experimental results did not demonstrate the advantages of it over CTM as diffusion distillation methods.\n\nWhile KDM does not yet match the performance of online or simulation-free methods like CTM, we are transparent about this gap throughout the paper and use it to contextualize our contributions. Our aim is not to claim superiority over all alternatives, but to establish a strong baseline within the offline distillation setting and demonstrate its practical viability.\n\nSpecifically, although CTM reports strong results, it is an online distillation method that requires repeated teacher queries during training, resulting in approximately 6.5x higher computational cost than KDM (5.2M vs. 0.8M NFEs using batches of 128 samples on CIFAR-10). In contrast, KDM operates entirely in the offline setting using precomputed supervision, making it more scalable and resource-efficient. Additionally, KDM is significantly lighter in terms of model size, using only $\\approx65$M parameters (our Koopman and GAN models), while CTM requires $\\approx180$M parameters, comprising both a full EDM teacher and student, as well as an auxiliary state-of-the-art GAN ($\\approx30$M parameters). Furthermore, while KDM employs a simple GAN framework, CTM uses a state-of-the-art GAN that is more than 15x larger in size compared to ours. Though both methods benefit from adversarial losses (as shown in CTM’s Table 3), they are not direct competitors due to these fundamental differences in training setups, supervision, compute, and model size.\n\nFinally, we would like to emphasize the privacy advantage of KDM over CTM as an offline distillation method. Offline distillation is limiting vulnerability to inversion or membership inference attacks. In traditional online distillation, especially when teacher models are repeatedly queried (e.g., denoising at various timesteps), there's a risk that an adversary could recover training data or determine whether a sample was in the original dataset.\n\nIn contrast, offline distillation relies only on a fixed dataset of noise–data pairs. Once these are generated (and potentially anonymized or filtered), the teacher model can be discarded or kept private. This one-time generation significantly reduces the attack surface, as no further queries are made to the teacher, and thus it becomes computationally infeasible for the student model, or an adversary accessing it, to reconstruct the original training distribution beyond what the static examples reveal.\n\nAs a simple example of a use case, it's possible to outsource the training process to a third party without exposing sensitive model parameters or proprietary intellectual property. The curator (e.g., the data owner) can pre-select and sanitize the training samples to remove personally identifiable information (PII) or domain-sensitive content. Since the third-party trainer doesn’t need access to the full generative model or the underlying sampling process, they are not exposed to unintended internal correlations or behaviors learned by the teacher model, which may inadvertently reveal private training data. As a result, sensitive institutions (like hospitals, finance firms, or proprietary content holders) can leverage external compute without compromising confidentiality.\n\n---\n\n> Q1: Training loss: The design of the loss function contains the adversarial loss in order to enhance the quality of the generated images. However, it may introduce the instability of the training process and the difficulty to converge. Would the authors demonstrate the training loss convergence curve in their experiments and compare it with GET and CTM?\n\nTo assess the training stability of KDM, we provide a comparison of the generator and discriminator loss curves for both KDM and GET across uniformly sampled training steps (see the table below). Despite the inclusion of an adversarial loss in KDM, the generator and discriminator losses exhibit stable behavior and convergence, suggesting that the adversarial component does not introduce significant instability in practice.\n\nThe CTM framework computes the adversarial loss differently and on a different scale, as it employs a distinct discriminator architecture and adversarial setup. CTM released their adversarial loss values, which we also report in the table below for completeness. While the fidelity is less accurate compared to our method and GET since the loss is combined for both the generator and the discriminator, the reported loss exhibits similarly stable behavior, as expected, showcasing the robustness of their model.\n\n| Step           | 0%     | 10%    | 20%    | 30%    | 40%    | 50%    | 60%    | 70%    | 80%    | 90%    | 100%   |\n| :------------- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- |\n| KDM Gen. Loss  | 1.98   | 0.74   | 0.65   | 0.71   | 0.75   | 0.84   | 0.68   | 0.79   | 0.68   | 0.89   | 0.84   |\n| KDM Disc. Loss | 1.00   | 1.15   | 1.12   | 1.08   | 1.17   | 1.05   | 1.02   | 1.14   | 1.15   | 1.09   | 1.17   |\n| GET Gen. Loss  | 1.95   | 0.89   | 0.75   | 0.77   | 0.72   | 0.69   | 0.74   | 0.88   | 0.85   | 0.94   | 0.91   |\n| GET Disc. Loss | 1.01   | 1.09   | 0.99   | 1.05   | 1.15   | 1.06   | 1.17   | 1.12   | 1.07   | 1.13   | 1.19   |\n| CTM Gen. Loss  | -      | -      | -      | -      | -      | -      | -      | -      | -      | -      | -      |\n| CTM Disc. Loss | .025   | .009   | .008   | .009   | .010   | .008   | .008   | .011   | .010   | .012   | .012   |\n\n*Table: Convergence of training loss over uniformly sampled steps.*\n\n---\n\n> Q2: Memory cost: In experiments, the quality of generated images is compared on the same scale of parameters across KDM and GET. However, as claimed in the introduction that KDM as an offline distillation method is more efficient than diffusion distillation methods such as CTM. Hence could the comparison include CTM on the parameter efficiency or other metrics to demonstrate the efficiency of KDM over CTM?\n\nWe have addressed this comparison in our response above, where we clarify that KDM uses significantly fewer parameters ($\\approx65$M, corresponding to our Koopman+GAN models), whereas CTM requires $\\approx180$M parameters due to the inclusion of both a teacher and student EDM, as well as an auxiliary GAN ($\\approx30$M). These distinctions underline the efficiency advantages of the offline KDM framework in both memory aspects.\n\n---\n\n> Q3: Higher resolution dataset: The experiments are conducted on CIFAR-10, FFHQ 64x64 and AFHQv2 64x64 datasets. Would the authors present the experiments on ImageNet-1k to evaluate the performance of KDM on higher resolution datasets?\n\nDue to the significant computational resources required for training on ImageNet-1k at scale, we were unfortunately unable to include these experiments in the current work. Our focus was instead on widely used, moderately sized benchmarks such as CIFAR-10, FFHQ 64x64, and AFHQv2 64x64, which still offer meaningful diversity in image content and structure. We view evaluation on higher-resolution datasets as an important direction for future work, particularly as greater resources become available.\n\n---\n\n> Q4: Clarification: A minor question is on the notations. Could the authors clarify and explain further on the differences between $C_\\mu$ and $C_\\eta$?\n\nIn our framework, $C_\\eta$ denotes the primary linear (Koopman) operator that governs the evolution of latent inputs through the dynamical system. On the other hand, $C_\\mu$ is a distinct linear operator that is learned to act solely on the auxiliary condition or context variable, not on the inputs themselves. This structure reflects a modular design often seen in linear control systems, where different operators handle different aspects of the system dynamics and exogenous inputs.\n\n---\n\n> Q5: Quality: The quality and motivations should be stronger. The current prominent compression method for diffusion models are consistency models and their variants. Although the privacy is mentioned as one of the advantages, experiments do not demonstrate and support it. It would improve the quality to include them and articulate them in experiments briefly.\n\nOur goal is to highlight the unique advantages of offline distillation, particularly its flexibility, efficiency, and broader reuse of precomputed supervision signals, as a complementary direction to online approaches. As noted in Lines 36–42 of the introduction, our discussion of privacy and other benefits refers to general advantages of offline distillation as a paradigm, supported by prior work (e.g., \\[Fernandez et al., 2023; Geng et al., 2023]) rather than specific claims about our method. We agree that incorporating explicit empirical evaluations of privacy would strengthen the contribution, but this was outside the scope of our current study. We have revised the text to clarify that these benefits are general to the offline setting and not directly assessed in our experiments."}, {"review_id": "deq1lYUwf5", "reviewer": "Reviewer_dXrx", "summary": "This paper introduces the Koopman Distillation Model (KDM), a novel method for offline distillation of diffusion models. The main innovation is its use of Koopman operator theory to represent the non-linear process of generating an image from noise in diffusion models as a simple linear transformation in a learned embedding space. This provides  a principled way to achieve single-step generation.\n\nThe authors provide theoretical proofs showing that the diffusion process can be accurately represented by a finite-dimensional Koopman operator and that this representation preserves the semantic structure of the generated images. Empirically, KDM outperforms previous offline distillation methods on standard benchmarks like CIFAR-10, FFHQ, and AFHQv2, improving FID scores by up to 40% in a single generation step. The method is not only fast, achieving a 4x speedup in training time compared to prior work, but also scales effectively with model size.", "questions": "I would appreciate it if the authors can answer the below questions from the weaknesses section above:\n\n- Point 2. from \"Contribution of the $L_{\\text{Koopman}}$ loss\".\n- Point 1. from \"Weighting of the $L_{\\text{Koopman}}, L_{\\text{adv}}$ losses\".\n- Point 1. from \"Practical relevance of the bound in Theorem 5.2\".", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths:\n\n### **Quality**:\nOverall, the paper is of high quality and supports its claims with extensive empirical validation, which includes experiments on standard benchmark datasets (CIFAR-10, FFHQ, AFHQv2) using standard evaluation metrics (FID, IS) as well as a human evaluation study.\n\n### **Clarity**:\nThe paper is well-written and clear. The paper is logically structured and the method is described in detail, including the specific loss function and model architecture.\n\n### **Significance**:\nBy developing a method for single-step offline distillation that is not only fast but also achieves state-of-the-art results, the paper provides\nprovides a practical tool that both researchers and practitioners are likely to use and build upon. The efficiency gains are significant, with a reported 4x speedup in training and over 8x faster sampling compared to the previous offline SOTA.\n\n### **Originality**:\nThis paper presents a original idea: it uses Koopman operator theory to linearize the complex generation dynamics of diffusion models, leading to a novel approach for offline distillation. Moreover, the insight that the semantic structure in the noise-to-image mapping of diffusion models is preserved in the learned embedding of the Koopman operator is an original and valuable contribution in its own right.\n\n---\n\n# Weaknesses:\n\n### **Contribution of the  $L_{\\text{Koopman}}$ loss**:\n\n- It's not clear to what extent the $L_{\\text{Koopman}}$ loss contributes to the success of the approach. The authors include an adversarial loss \"to further refine the distillation process\", however without the adversarial component (see Table 5) their method actually performs worse than their main point of comparison (GET).\n- Moreover, GET itself is not adversarially trained so the comparison between an adversarially trained KDM and a non-adversarially trained GET is not totally fair. The authors mention in Appendix D.3 that they struggled to train GET with an adversarial loss - it would be helpful to also see the results/loss curves for these attempts.\n\n### **Weighting of the $L_{\\text{Koopman}}, L_{\\text{adv}}$ losses**:\n\n- Following up from the previous comments, I wonder if the authors tried different weightings for the $L_{\\text{Koopman}}$ and  $L_{\\text{adv}}$ losses? Currently, the weighting on the different components of $L_{\\text{Koopman}}$ is uniform, even though they may potentially lie on different scales during training. Moreover, there is no ablation for different values of $\\lambda_{\\text{adv}}$. It would be nice to see some ablations for this to better understand the effect of the adversarial loss and also the different components of $L_{\\text{Koopman}}$.\n\n### **Practical relevance of the bound in Theorem 5.2**:\n\n- Thm. 5.2 establishes that semantic proximity is preserved, showing that the distance between two generated images is bounded by the distance between their latent representations. However, clearly the quality of this bound depends on the norm $||C_\\eta||$ - can the authors show what kind of norms $||C_\\eta||$ actually achieves in practise?", "was_revised": true, "final_justification": "The authors have adequately answered all of my questions and I have therefore decided to retain my score. Upon reading the comments of other reviewers (in particular comments about significance of offline distillation and empirical results) I have decided to revise my confidence.", "rebuttal": "Thank you for your thoughtful and constructive review. We appreciate your recognition of the novelty, clarity, and potential impact of our work, as well as your detailed feedback on areas for improvement. We address your questions below and, given the opportunity, will incorporate your suggestions into the final revision.\n\n---\n\n> Contribution of the $L_{\\text{Koopman}}$ loss: *Moreover, GET itself is not adversarially trained so the comparison between an adversarially trained KDM and a non-adversarially trained GET is not totally fair. The authors mention in Appendix D.3 that they struggled to train GET with an adversarial loss - it would be helpful to also see the results/loss curves for these attempts.*\n\nFollowing our original submission, we further explored how to incorporate adversarial training into GET and were able to successfully adapt it. On CIFAR-10, the adversarially trained GET achieves an FID of **6.13**, an improvement over the original GET’s 6.91, though still underperforming compared to our KDM’s **4.68**. Regarding the training dynamics, we observed a stable behavior in the loss curves during adversarial training:\n\n| Step           | 0%     | 10%    | 20%    | 30%    | 40%    | 50%    | 60%    | 70%    | 80%    | 90%    | 100%   |\n| :------------- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- | :----- |\n| KDM Gen. Loss  | 1.98   | 0.74   | 0.65   | 0.71   | 0.75   | 0.84   | 0.68   | 0.79   | 0.68   | 0.89   | 0.84   |\n| KDM Disc. Loss | 1.00   | 1.15   | 1.12   | 1.08   | 1.17   | 1.05   | 1.02   | 1.14   | 1.15   | 1.09   | 1.17   |\n| GET Gen. Loss  | 1.95   | 0.89   | 0.75   | 0.77   | 0.72   | 0.72   | 0.74   | 0.88   | 0.85   | 0.94   | 0.91   |\n| GET Disc. Loss | 1.01   | 1.09   | 0.99   | 1.05   | 1.15   | 1.06   | 1.17   | 1.12   | 1.07   | 1.13   | 1.19   |\n| CTM Gen. Loss  | -      | -      | -      | -      | -      | -      | -      | -      | -      | -      | -      |\n| CTM Disc. Loss | 0.025  | 0.009  | 0.008  | 0.009  | 0.010  | 0.008  | 0.008  | 0.011  | 0.010  | 0.012  | 0.012  |\n\n*Table: Convergence of training loss over uniformly sampled steps.*\n\nThese results suggest that while adversarial training does enhance GET’s generative quality, KDM provides additional benefits beyond what adversarial objectives alone can achieve.\n\nNote, the CTM framework computes the adversarial loss differently and on a different scale, as it employs a distinct discriminator architecture and adversarial setup. CTM released their adversarial loss values, which we also report in the table below for completeness. While the fidelity is less accurate compared to our method and GET since the loss is combined for both the generator and the discriminator, the reported loss exhibits similarly stable behavior, as expected, showcasing the robustness of their model.\n\n---\n\n> Weighting of the $L_{\\text{Koopman}}, L_{\\text{adv}}$ losses: *Following up from the previous comments, I wonder if the authors tried different weightings for the $L_{\\text{Koopman}}$ and $L_{\\text{adv}}$ losses? Currently, the weighting on the different components of $L_{\\text{Koopman}}$ is uniform, even though they may potentially lie on different scales during training. Moreover, there is no ablation for different values of $\\lambda_{\\text{adv}}$. It would be nice to see some ablations for this to better understand the effect of the adversarial loss and also the different components of $L_{\\text{Koopman}}$.*\n\nWe explore different weightings of the Koopman components $L_{\\text{rec}}, L_{\\text{lat}}$ and $L_{\\text{pred}}$. The results indicate that the model is relatively robust to variations in these weights, with FID scores remaining in a narrow range (4.94–5.03).\n\nIn addition, we evaluated the effect of the adversarial loss weight $\\lambda_{\\text{adv}}$, where we found that $\\lambda_{\\text{adv}} = 1.0$ yielded the best FID (4.57), compared to 4.98 for $\\lambda_{\\text{adv}} = 0.5$ and 4.67 for the default 0.25 setting reported in the main paper. We will include these results and clarify them further in the appendix to improve transparency.\n\n---\n\n>Practical relevance of the bound in Theorem 5.2: *Thm. 5.2 establishes that semantic proximity is preserved, showing that the distance between two generated images is bounded by the distance between their latent representations. However, clearly the quality of this bound depends on the norm $\\| C_\\eta \\|$ - can the authors show what kind of norms $\\| C_\\eta \\|$ actually achieves in practise?*\n\nIn practice, we observe an average Koopman operator spectral norm of 0.88 with std 0.18, which results in a reasonably tight bound and supports the practical utility of the theorem. Furthermore, our framework explicitly allows for controlling this norm by penalizing the eigenvalues of the Koopman operator. As discussed in Appendix C, the Koopman operator is factorized in a way that enables direct constraint of its spectrum. This provides a principled means to regularize $\\| C_\\eta \\|$, thereby improving the semantic consistency of generated samples while maintaining stability and interpretability."}, {"review_id": "6Vig6AKhUl", "reviewer": "Reviewer_zaaq", "summary": "This paper presents Koopman Distillation, an approach rooted in Koopman theory for nonlinear dynamics, in which a pretrained diffusion model is distilled by learning an autoencoder on the latent representation of the model and a linear operator inside that space in order to transfer data to noise.", "questions": "N/A", "limitations": "The authors do recognize the limitations of their performance, though I do think the overall poor performance is a bit understated and downplayed.", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The overall math an intuition behind the approach is well thought out, and exceedingly interesting and insightful. Unlike many modern distillation methods, this approach is novel and is rooted in both empirical observations and theoretical insights that make for a compelling distillation method.\n- The presented analysis in 6.1 is very interesting, and constitutes a thoughtful empirical analysis of the proposed KDMs.\n\n**Weaknesses**\n- I think the overall empirical results are a bit weak. The authors make clear emphasis the delineation between online and offline methods, but I think this emphasis is overstated and somewhat attempts to cover their weaker results.\n- Especially on unconditional CIFAR-10, KDMs barely beat *any* other distillation method, besides their chosen baseline of GET. The authors also incorrectly portray 2-rf++ as just a continuous flow method, when in fact **it is also a form of offline distillation** if I'm not mistaken.\n- Additionally, the relevant offline distillation methods that the authors *should* be comparing against, ADD and LADD, are generally done on much large Text2Image models rather than these small tasks (as GET is not particularly adopted if I'm not mistaken), which makes it hard to assess the overall future performance of KDMs, and whether such a framework could be competitive with these primarily adversarial methods, which are the most common forms of offline distillation. \n\n\n**Final Decision**: Overall, I do not think this paper is ready for publication yet. While I thoroughly enjoyed the methodological contribution and found these insights exciting and potentially useful for the wider acceleration community, it is hard to ignore the overall poor performance of the approach, and thus in all the paper seems like an interesting, but generally less performant distillation method. I would recommend the authors attempt what they can to either scale up their experiments, continue to explore the empirical design space of the method to improve performance, or recreate existing popular distillation methods at the scale feasible for their compute budget to demonstrate the strengths of KDMs.", "was_revised": true, "final_justification": "I am reasonably confident my score of weak rejection (and have adjusted my confidence accordingly):\n- The authors' main claims of success seem to hinge on what I see as an overly restrictive view of what constitutes \"offline\" distillation, as   they claim that access to *any* of the weights of the teacher model during distillation (including using part of the teacher as initialization for a discriminator as is done in LADD, Seaweed-APT, and UFO-Gen), even if there is no explicit supervision from the teacher as a diffusion model, would constitute as an online method. I have never seen this definition used before, and does not seem to be in line with the growing space of other offline methods. Additionally, upon further discussion, the author's stated that even initializing the student with the teacher would count as \"online\" distillation, despite this being the standard practice among nearly all modern distillation methods (both offline and online).\n- It is clear to me that 2-rf++ actually is an offline method, thus showing that KDMs are not even SOTA for offline methods.\n- Thus, despite the fascinating and concerted theoretical results throughout the paper, the experimental results are not strong enough to warrant acceptance.\n- I think there is a bit of disconnect between the current structure of the paper and the practical results of their experiments. Throughout the rebuttal period, the authors highlighted the benefits of their definition of \"offline\" distillation from a privacy perspective, which I think are valid points. However, the paper barely focuses on this aspect (besides passing mentions of offline distillation in a broad sense) and instead is formatted much like a standard diffusion distillation paper, which does not help their relatively moderate experimental results. While I don't think this paper is currently ready for publication, I do think that a future version of this paper, which could empirically motivate *how* KDMs can be better than their defined \"online\" methods from a privacy perspective and in general focuses on KDMs' privacy preserving abilities, would make for a strong contribution.", "rebuttal": "Thank you for your thoughtful and detailed review. We appreciate your recognition of the novelty and theoretical depth of our approach, as well as your engagement with our analysis in Section 6.1. Your feedback highlights both the strengths of our framework and important areas for further development. We address your points in detail below. Given the opportunity, we will incorporate our modifications into a final revision.\n\n---\n\n> W1: *I think the overall empirical results are a bit weak. The authors make clear emphasis the delineation between online and offline methods, but I think this emphasis is overstated and somewhat attempts to cover their weaker results.*\n\nWe respectfully disagree with the characterization that our emphasis on the offline setting is meant to obscure weaker results. Our goal is to highlight the distinctive general advantages of offline distillation compared to online distillation. Namely, reduced teacher NFEs, lower memory requirements, and improved privacy (as discussed in lines 36–42). While our method does not yet match the performance of leading online techniques, it currently achieves state-of-the-art results within the offline setting, and the gap is steadily narrowing. As shown in Tables 1–3 and our accompanying discussion, we are transparent about where our method stands relative to other paradigms. Part of our motivation is to encourage further attention to offline distillation, a promising yet underexplored direction where our results already demonstrate improved performance.\n\n---\n\n> W2: *Especially on unconditional CIFAR-10, KDMs barely beat any other distillation method, besides their chosen baseline of GET. The authors also incorrectly portray 2-rf++ as just a continuous flow method, when in fact it is also a form of offline distillation if I'm not mistaken.*\n\nWe would like to clarify a potential misunderstanding regarding 2-rf++. While the reviewer suggests it as an offline distillation method, 2-rf++ is an improvement over rectified flow, a simulation-free flow model conceptually closer to flow matching than to knowledge distillation. Unlike distillation methods, it does not learn from a pre-trained teacher model and does not use a distillation loss. Although 2-rf++ generates synthetic data (as we do), it also performs multiple forward passes through the model to fit rectified trajectories (see their Tab. 6), making it fundamentally different from offline distillation approaches, which rely on fixed teacher supervision. For these reasons, we did not include it among the offline distillation baselines.\n\nMore generally, there might be a misunderstanding regarding the differences between offline and online settings. In the offline distillation setting, the student model is trained solely on precomputed supervision (e.g., synthetic data) without access to the teacher during optimization. This contrasts with online distillation, where the student is continuously guided by the teacher through live queries or gradient-based feedback during training. Offline distillation offers several inherent advantages in terms of privacy and scalability (please see our discussion with Reviewer `mkU1` for further details regarding these benefits and trade-offs).\n\n---\n\n> W3: *Additionally, the relevant offline distillation methods that the authors should be comparing against, ADD and LADD, are generally done on much large Text2Image models rather than these small tasks (as GET is not particularly adopted if I'm not mistaken), which makes it hard to assess the overall future performance of KDMs, and whether such a framework could be competitive with these primarily adversarial methods, which are the most common forms of offline distillation.*\n\nWe would like to clarify that neither ADD nor LADD qualifies as an offline distillation method under its standard definition. ADD penalizes a distillation loss against the teacher during training (see their Fig. 2), requiring teacher access throughout. LADD, while using synthetic data, also supervises the student using the teacher’s latent features during training. In contrast, our method trains solely on precomputed synthetic samples without teacher interaction, adhering strictly to the offline setting. Therefore, these methods are not directly comparable baselines for our framework.\n\n---\n\n> Limitations: *The authors do recognize the limitations of their performance, though I do think the overall poor performance is a bit understated and downplayed.*\n\nWe appreciate the reviewer’s acknowledgment that we discuss our limitations. However, we would like to re-emphasize that our method achieves state-of-the-art results within the offline distillation setting which differs from the online setting. While it does not yet match the performance of online or simulation-free approaches, we are transparent about this gap throughout the paper and emphasize it to contextualize our contributions. Our intention is not to downplay limitations, but rather to highlight the strengths and practical trade-offs of offline distillation, and to encourage further exploration in this emerging area. We are open to revise our paper to reflect this better, following reviewer suggestions.\n\n---\n\n> Final Decision 1: *... overall poor performance of the approach, and thus in all the paper seems like an interesting, but generally less performant distillation method.*\n\nWe have addressed the performance concerns in detail (see W1 and Limitations), emphasizing that our method is state-of-the-art within the offline setting and closing the gap with other paradigms. We hope these clarifications help contextualize the results and encourage a reconsideration of the overall assessment. \n\n---\n\n> Final Decision 2: *... I would recommend the authors attempt what they can to either scale up their experiments, continue to explore the empirical design space of the method to improve performance, or recreate existing popular distillation methods at the scale feasible for their compute budget to demonstrate the strengths of KDMs.*\n\nWe agree that scaling and further exploration are promising directions and appreciate the constructive suggestions. As mentioned earlier, our current results already show strong promise in the offline setting. We hope our clarifications above will help demonstrate the strengths of KDMs within this context and support a more favorable evaluation."}]

Author rebuttal:

Meta-review: This paper leverages Koopman theory to propose a method for diffusion model offline single-function-evaluation distillation. The reviewers unanimously praised the theory, and the connection between Koopman theory and sampling in diffusion models. The main issue reviewers raised was the underperformance of the proposed method as compared to online distillation methods.

Although I agree with the reviewers that outperforming online distillation methods would strengthen the paper, I do not believe such results are necessary for acceptance, as the method performs competitively compared to existing offline distillation approaches, and the presented connections to Koopman theory are valuable to the community.
Decision: Accept (poster)