Purrception: Variational Flow Matching for Vector-Quantized Image Generation
Authors: Răzvan-Andrei Matișan
Vincent Tao Hu
Grigory Bartosh
Björn Ommer
Cees G. M. Snoek
Max Welling
Jan-Willem van de Meent
Mohammad Mahdi Derakhshani
Floor Eijkelboom
Venue: ICLR 2026 Poster (2026)
Keywords: generative models
flow matching
vector quantized
image generation
computer vision
variational flow matching
Primary area: generative models
TL;DR: We apply variational flow matching to VQ latent image generation through a hybrid discrete-continuous approach for improved image generation.

Abstract: We introduce Purrception, a variational flow matching approach for vector-quantized image generation that provides explicit categorical supervision while maintaining continuous transport dynamics. Our method adapts Variational Flow Matching to vector-quantized latents by learning categorical posteriors over codebook indices while computing velocity fields in the continuous embedding space. This combines the geometric awareness of continuous methods with the discrete supervision of categorical approaches, enabling uncertainty quantification over plausible codes and temperature-controlled generation. We evaluate Purrception on ImageNet-1k $256 \times 256$ generation. Training converges faster than both continuous flow matching and discrete flow matching baselines while achieving competitive FID scores with state-of-the-art models. This demonstrates that Variational Flow Matching can effectively bridge continuous transport and discrete supervision for improved training efficiency in image generation.

Published as a conference paper at ICLR 2026
PURRCEPTION: FOR
VARIATIONAL FLOW MATCHING IMAGE GENERATION
Räzvan-Andrei Matişan" "
Vincent Tao Hu
Grigory Bartosh
UvA-Bosch Delta Lab University of Amsterdam
Cimp LMU Munich Munich Center for Machine Learning
AMLab University of Amsterdam
Björn Ommer
Cees G. M. Snoek
Max Welling
CompVis Vis @ LMU Munich Munich Center for Machine Learning
Video & Image Sense Lab
UvA-Bosch Delta Lab University of Amsterdam
University of Amsterdam
Man-Willem van de Meent UvA -osshh Delta Lab University of Amsterdam
Mohammad Mahdi Derakhshani Video & Image Sense Lab University of Amsterdam
Floor Eijkelboom'
Delta Lab University of Amsterdam
ABSTRACT
We introduce Purrception, variational flow matching approach for vector- quantized image generation that provides explicit categorical supervision while maintaining continuous transport dynamics. Our method adapts Variational Flow Matching to vector-quantized latents by learning categorical posteriors over code- book indices while computing velocity fields in the continuous embedding space. This combines the geometric awareness of continuous methods with the discrete supervision of categorical approaches, enabling uncertainty quantification over plausible codes and lemperature-controllee generation. We evaluate Purrception on ImageNet- 1kk 256 x 256 generation. Training converges faster than both continuous flow matching and discrete flow matching baselines while achieving competitive FID scores with soate-of-the-ar models. This demonstrates that Variational Flow Matching effectively bridge continuous transport and discrete supervision for improved training efficiency in image generation.
INTRODUCTION
The task of generative modeling is to approximate a data distribution to enable sampling of new in- stances. Beyond high-fidelity synthesis in images, audio, and text, generative models increasingly used for augmentation, restoration, simulation, and in-silico design (e.g.. de novo molecules and proteins). Flow Matching (Lipman al. 2023 Allergo al., 2023 Liu al. (023) has emerged as an extremely effective approach for the generation of variety of data modalities. In Flow Matching, one first deñnes an interpolation between source (noise) and target (data) distribution, and then approximate the velocity field of continuous normalizing flow that transports samples between the two. As the target velocity can be understood the expected derivatiee of the interpolation, it can be learned in self-supervised manner by averaging over samples from the source and target distribution. The Flow Matching framework has been extended to general geometries (Chen &
Lipman 2024), discrete data (Gat et al. 2024), and has many applications (Wildberger et al. 2024
Dao et al. 2023 Hu et al. 2024 Kohler et al. 2023).
Variational Flow Matching (VFM) (Eijkelboom al. 2024) reframes Flow Matching as inference. Since the Flow Matching velocity field is the expectation of a conditional velocity, can be approxi- mated via variational posterior over endpoints (target samples) given the current interpolation point. Standard Flow Matching is recovered when this posterior is Gaussian, while other choices extend naturally different modalities. Applied to discrete data, VFM yields CatFlow previously used
Research done during an internship UvA-Bosch Delta Lab and visit LMU Munich. Equal contribution last authors.

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Published as a conference paper at ICLR 2026
Figure I: Purrception generates high-resolution images vector-quantized latent spaces, sampled as continuous transport learned through discrete supervision.
for graph generation and related to continuous diffusion for categorical data (Dieleman et al. 2022).
More broadly, VFM has been applied to mixed modalities (Guzmán-Cordero et al., 2025), ) molecula
generation (Eijkelboom et al., 2025 Sakalyan et al.), and general geometries (Zaghen et al. (025).
The variational view also enables problem-spese bounds like non-negative thickness are enforced through the loss (Finn et al. 2025).
constraints, e.g., for sea-ice forecasting, where
This paper leverages VFM in the context of image generation. We consider vector-quantized (VQ) latents, which map images into grids of discrete indices with associated embeddings, yielding compact representations that preserve perceptual fidelity at far lower dimensionality than pixels. However, their dual discrete-continuous nature poses modeling challenge not addressed by purely continuous discrete methods. Continuous methods (latent diffusion, flow matching) generate in the embedding space, enabling smooth transport and efficient high-resoluticn synthesis (Rombach et al.) 2022 Dao al. 2023). Yet they must discretize vectors back to indices: geometry is preserved, but categorical structure ignored - the model never learns which index to choose or how to represent uncertainty across codes. Conversely, fully discrete approaches (VQ-Diffusion (Gu et al. (2022), discrete flow matching (DFM) (Gat et al., 2024)) treat related embeddings unrelated tokens, discarding geometry. While DFM could use temperature-base indices each step commits to a single code whereas continuous flow matching (CFM) cannot use temperature at all, since it lacks logits.
sampling, this only produces stochastic "hops" between
To resolve this tradeoff, introduce Purrception, adaptation of VFM to vector- vector-quantized latents. By using a categorical posterior over indices while transporting probability bedding space, Purrception provides categorical learning signal while still leveraging geometry. This means the model can express uncertainty across plausible codes and translate it into smooth, geometry-aware transport rather than discrete jumps. Logits further enable temperature scaling: lowering temperature sharpens predictions, while raising temperature spreads probability across nearby embeddings. producing smoother generations and samples with more details. Empirically, this hybrid approach converges faster than both CFM and DFM on ImageNet- Ik, achieving competitive or superior FID while retaining the efficiency of flow matching.
the continuous em-
BACKGROUND
2.1 FLOW MATCHING
Flow matching (ipman et al. 2023 Liu et al. 2023 Albergo et al.) 2023) learns velocity field
of RDD [0, g parameterized by network with parameters 0 which induces transport
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of samples zo Po from a prior (e.g., standard noise) to D-dimensional points 21 that should approximate the data distribution. This is done by integrating the ordinary differential equation
dæ
dt o (x)
with Zg Po
(1)
which is equivalent to learning velocity field that satisfies the continuity equation, also known as continuous normalizing flow,
(ep((x) v (u)(=))((=)).
(2)
Flow matching starts from the observation that, given choice of interpolation between noise and data - e.g., linear, where 2, = to + -11-) 1) we can derive a conditional velocity field uz(x x1) that satisfies the continuity equation towards (i.e., conditional on) specific endpoint. A corresponding velocity field ue (z), which satisfies the continuity equation for the (marginal) probability path, can be expressed in terms of an (intractable) expectation with respect to the posterior
f ue(x x1) Pe(x1 I)) dæ = (x(())))) [ue( x1)]
Ue (x)
(3)
The goal of flow matching is therefore to learn a velocity field of (x) that approximates in (r). i.e., to minimize the flow matching objective
(FM( Et,- [1]x]()) (x)]]
(4)
which can be made tractable by optimizing
(CFM(0) E, (x))((x) ue (x xx)]1
(5)
ie.. Monte-Carlo estimate of the marginal objective through conditional objective. As shown in Lipman et al. (2023), indeed these two objectives have the same gradients w.r.t. 0. This can equivalently be understood trying to regress towards the expected time-derivative of the interpolant.
2.2 VARIATIONAL. FLOW MATCHING
Variational Flow Matching (VFM) (Eijkelboom et al. 2024) treats Flow Matching as a variational inference problem. By realizing (through Equation (3) the target marginal velocity field ue can be expressed as an expectation of the conditional field w.r.t the posterior distribution Pe(a1 æ)) the authors propose to learn this posterior directly, i.c., learn
LVFM (0) = R. [LI((p(21, x)) (x1,))) E [log of (x) [x)] const.,
(6)
where (21 x)) is the variational posterior approximating the posterior probability path pa (x wr). The resulting learning velocity field is thus given by
of (x) E (xtl)))) [x xx) OT (x)
where (x) := (=))))=E[([]11 and the conditional field is the linear (or optimal transport) interpolation. Though this objective initially looks intractable, we authors show that the task of learning the variational approximation only needs to be learned dimension-wise in the mean, as (x. [x]
only depends on the marginal of (xd æ) an approach called mean-field VFM.
VFM is flexible in choosing the variational distribution qf which makes a general framework for different data types. In Eijkelboom et al. (2024). the authors show significant improvement over CFM when the data discrete and the variational approximation is chosen to be categorical, a model coined CarFlow. VFM has also obtained strong performance in tabular data (Guzmán Cordero et al., 2025), molecular generation tasks (Fijkelboom et al. 2025 Sakalyan et al.), general geometries (Zaghen et al. 2025), and sea-ice modeling (Finn et al. 2025).
2.3 VECTOR- VECTOR-QUANTINN QUANTIZED AND LATENT GENERATIVE MODELS
High-resolution image modeling in pixel space is computationally prohibitive; a common remedy is to learn lower-dimensional latent space with an autoencoder. Vector-Quantized VAEs (Van Den Oord
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et al. 2017) and VQ-GANs (Esser et al. 2021) discrete codebook C [- C CR?, By
mapping images into a compact set of discrete tokens, vector-quantizee latents provide an efficient and stable representation: they alleviate posterior collapse and often yield sharper, higher-fidelity reconstructions than pixel-space models at comparable compute.
Given image I, the encoder output is quantized its nearest code:
z xxx) QQuanize(Euccodrrna)
arg min [Encoder(x) -exl) exEC
(8)
Equivalently, one can store the index
<(x) arg min [ncoder(()) ex 1 [K] = (1,..., K}.
| kE[K]
(9)
After training the encoder, decoder, and codebook, a generative model is learned in latent space and samples are decoded to pixels. For a grid of D discrete latents E [K] a common choice is an autoregressive model:
D II (ea cea). d=1
p(c)
(10)
While this formulation provides powerful and efficient representation, it also introduces funda- mental modeling tension: each latent at once discrete code index and a continuous embedding vector. Existing generative methods resolve this tension by making trade-off either operating in the continuous embedding space and ignoring the categorical structure, or modeling indices directly while discarding geometric information. This limitation motivates the hybrid perspective developed in Section 3
PURRCEPTION: VQ-VEN FOR IMAGES
3.1 MOTIVATION: HYBRID APPROACU TO VQ-LATENT FLOWS
Vector-quantized (VQ) latents encode data in ways simultaneously: as discrete indices drawn from a finite codebook and as continuous embeddings that capture geometric relations such as proximity and direction. Existing generative models are typically forced into one of two degenerate extremes, each of which breaks part of this dual structure:
Continuous flow models (e.g., latent diffusion and flow matching) operate in IR D. treating codebook vectors as continuous. From the perspective of Variational Flow Matching (VFM), this corresponds to Gaussian relaxation: endpoints are approximated continuous samples rather than categorical indices. Geometry is preserved, but discreteness is lost the model never receives a categorical learning signal, cannot express uncertainty over multiple plausible codes, and has no logits from which to derive controls such temperature scaling.
Fully discrete flow models instead predict categorical indices directly. While this aligns with the quantized structure, it collapses geometry: once reduced to raw indices, semantically related codes are treated as unrelated tokens. Predictions degenerate into discrete "teleports" between indices, eliminating interpolation and making both uncertainty modeling and temperature scaling meaningless.
An ideal solution should combine the strengths of both worlds: exploit the smooth geometry of embeddings and provide categorical supervision over indices. Our approach adapts VFM with categorical variational posterior, so that the velocity field evolves in continuous space while learning driven by cross-entropy over codebook entries. This hybridization allows the model to receive categorical learning signal, to reason over multiple plausible indices, and to convert that uncertainty into geometry-awee aware transport rather than discrete jumps. Crucially, working with logits also unlocks temperature knob: lowering enforees stronger commitments, which improves global fidelity but oversimplifies samples, while raising redistributes probability more broadly, adding detail and variety at the cost of overall quality.
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Codebook
Encoder
argaiin of + exll
RqD
STT
3
Quantization
Input
Output
t 21
Pc(c xe)
Cross Entropy
DIT
t)
Figure 2: Purrception approach. Purrception generates high-resolution images in a vector quantized latent space. For training. we pretrained encoder and codebook vector of size K to encode and quantize an image in latent space to obtain 21. Then, we train a diffusion transformer that predicts, given linear interpolant 2t, categorical distribution over the codebook vectors for each patch of the target 21 via a cross-entrope objective. For sampling, we generate quantized latent which we further pass through the decoder G to obtain the image pixel-space.
3.2 THE VQ- VFM OBJECTIVE
We begin from the key observation underlying VFM and CatFlow: the velocity at time f can be expressed an expectation over conditional continuations weighted by posterior over endpoints
(Eijkelboom et al. 2024):
(2)))) = Ep.(. =.) [u 1x1]].
(11)
This perspective reframes the learning problem: rather than predicting the vector field directly, we may approximate the posterior Pr(21 24) with variational distribution of (2) 22) and compute the
velocity as its expectation.
In the case of VQ-latents, this insight becomes particularly powerful. Each endpoint must be one of the finite codebook embeddings {c*)==1 f==1. so the posterior is categorical over the discrete latent codes. That is, variational posterior should be given by
df 24) Cat(c (2))).
(12)
where T (2t) is the probability distribution over the codebook vectors outputted by a neural network (e.g., Diffusion Transformer (Peebles Xie 2023)). Conditioning this posterior the noisy latent zi yields a distribution over discrete indices while still defining transport in continuous embedding
space, we can compute
(((2))
Σ
0. tt
(zt)
Ek Ze
(13)
where (e(())= == Σk 0.k (= and n. 0. (zz) is the probability to have as endpoint the codebook vector 6₁ given the deeeddeenddnt interpolant 24. This ensures that uncertainty over multiple plausible codes is translated into smooth, geometry-aware motion, rather than discrete "teleports" between unrelated indices.
Training follows from the VFM objective, which in this case reduces to the cross-entropy loss between the predicted posterior and the sround-truthe code indices:
Lpun (0) E.,..
[oog =x)]
(14)
where x ~ D is sampled from the data, 21 and c is the corresponding quantized image and latent code respectively, and 2, is obtained through 2 : 1.) (1 1)=0 for zn po and 2/ 1).

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Softmax temperature. ture T,
Because (2() is obtained from logits (2) via softmax with tempera-
(zt) 0.k Te (2x)
exp( STp (x)T+(
(15)
a 6.. ( (x1)/T) E exp(
our framework naturally inherits an inference-time degree of freedom that regulates how categorical uncertainty is expressed in the velocity field. When is small, the posterior distribution collapses toward the most likely index, enforcing early commitments and producing sharp, high-fidelity outputs that may, however, become overly simplistic as alternative hypotheses are ignored. Conversely, large r values flatten the distribution, assigning non-negligible weight to multiple neighboring codes. This broadening injects more detail and variability into the generated samples, but reduce overall fidelity as the barycenter drifts away from the most plausible embedding. Intermediate 7 values often strike the best balance, echoing the variance trade-off familiar from other generative frameworks. Such controllability is absent in continuous FM, where no categorical logits exist, and meaningless in fully discrete FM, where indices are collapsed immediately; it arises directly from the hybrid formulation, turning temperature into a principled knob for task-adaptive inference.
4 EXPERIMENTS AND RESULTS
We validate the performance of Purrception through series of experiments. In experiments, evaluate on ImageNet- Ik (Deng et al. 2009) on 256 x 256 resolution, using both the Stable Diffusion's vq- f8 (Esser et al. 2021) and Llama vq ds8 c2- (Sun et al. 2024) tokenizers, as well as the DiT-L/2 and DiT-XL/2 backbones (Peebles & Xie 2023). We provide a full description of the implementation details in Appendix C First, we perform comparative study between Purrception. continuous flow matching (Lipman et al. (023)), and discrete flow matching (DFM) (Gat et al. 2024). For continuous flow matching, we consider two objectives: the classical regression task of predicting the velocity field (denoted simply CFM) and the task of predicting the endpoint (denoted as CFM-endpoint), as seen in Ma( et al. (2024), allowing us to measure the effects both (1) switching to endpoint prediction, and (2) using discrete objective compared to the continuous baseline. We show that Purrception converges faster (i.e., in fewer training iterations) to a low FID, hence reducing computational resources. Then, show that Purrception generates high-fidelity and high-quality samples when trained at scale, achieving a competitive FID against a variety of state-of-the-ard autoregressive, diffusion, and masked generative baselines. Finally, we show that the softmax temperature parameter can be used to control the image sharpness and quality at inference time, property unique hybrid discrete-continuous models.
4.1 CONVERGENCE SPEED
A key requirement for practical generative modeling is the ability to reach high sample quality quickly, since faster convergence directly reduces training cost and compute requirements. To evaluate this, compare the convergence speed of Purrception against two strong baselines: the continuous flow model (CFM) (Lipman et al., (023) and the fully discrete flow model (DFM) (Gat et al., 2024). Given the great performance of Scalable Interpolant Transformers (SiT) Ma al. (2024). we include an additional bascline (denoted as CFM-endpoint) where the task is to predict via mean-squared error (similar to CFM) the endpoint 21 given the interpolant 2, (similar to Purrception). For fair comparison, we used the same training configurations, and we sample all images using Euler with 100 integration steps as ODE solver. We provide all implementation details in Appendix C
reports FID-10k scores for both DYT-1//2 and DÍT-X1/2 backbones. Across settings. Purrcep- tion not only achieves lower final FID but also reaches baseline performance substantially earlier. With DiT-L/2, Purrception checkpoint 2M iterations matches CFM's and CFM-endpoint's scores after ~ 1.2M iterations (1.65x faster), while Purrception checkpoint at IM iterations matches DFM's final score after 325k iterations (3.0x faster). With the larger DiT-XL//2 backbone, the gap grows further: Purrception converges 2.3x faster than both CFM baselines and 3.5x x faster than DFM.
These results underscore the advantage of Purrception's n's hybrid formulation. By receiving direct categorical supervision (unlike CFM), the model learns discrete structure more efficiently, while its use of continuous embedding space (unlike DFM) enables smooth geometry- transport

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50
Purreeption
50
Purreeption
CFM-endpoint
CFM-eendpoint
CFM
CFM
FID- 10k (1)
FIDD-10K (1)
DFM
DFM
30
30
3.0x faster
3.5x faster
20
20
1105x faster
2.3x faster
15
15
200k
600k
IM
1.4M
2M
200k
600k
1M
1,4M
2M
Training Iterations
Training Iterations
(a) Backbone: DIT-1//2
(b) Backhone: DÝT-X1/2
Figure 3: Convergence speed comparison on ImageNet-Ik, FID-10k scores are plotted against training iterations for Purrception, two CFM variants, and DFM. Results are shown for two DiT backbones: (a) DiT-L/2 and (b) DiT-XL/2. We train Purrception using the default 7 = 1.0 softmax temperature, while using т 0.9 during inference. The plots show that Purrception achieves lower final FID scores and converges significantly faster, matching the final performance of CFM and DFM in fewer training iterations. Here we used Stable Diffusion's fq tokenizer. Full training details are provided in Appendix C
rather than slow, discrete jumps. This combination accelerates optimization, leading to both faster convergence and stronger sample quality.
4.2
OPTIMIZING SAMPLE QUALITY VIA SOFTMAX TEMPERATURE SCALING
Temperature scaling is long-standing tech- nique in language modeling, used to balance coherence and diversity during sampling. In the context of VQ image synthesis. continuous flow methods (e.g., CFM) cannot exploit this mechanism at all, since they lack categorical logits. Fully discrete models (e.g.. DFM) can in principle apply temperature scaling to their logits, but because they commit to hard index selections at each step, adjusting has little prac- tical effect - the sampling collapses to discrete jumps regardless of the distribution' 's softness. In contrust, Purrception retains uncertainty in the logits while transporting through the continu- ous embedding space, which means temperature scaling can be naturally used.
50
Purrception
FID. S0k (1)
CFM
30
20
15
0.3
0.5
0.7
0.9
1.2
1.5
Softmax Temperature (7)
Figure 4: The effect of the softmax temperature on FID score. The plot depicts U-shape relation- ship between 7 parameter used in Purrception and the FID score. Both models have been trained for 1M iterations and under the same training condi- tions. Here we used the vg- f8 autoencoder.
To test the effect of the softmax temperature during inference, we conduct an ablation study with a DíT-XL/2 backbone trained for one mil- lion iterations. During training, we keep 7 the default 1.0, varying the temperature only at inference. Figures[aande show the FID-50k obtained and the effect on sample quality, respectively. We observe clear U-shaped curve: performance improves increases from very low values, reaches optimum around 0.8-0.9, and then degrades as T becomes larger. Qualitatively, low J values produce overly deterministic and simplistic images, while high values lead noisy and incoherent generations.
These findings highlight that: (1) even though Purrception has been trained with a constant r = 1.0. the data distribution is best approximated for lower softmax temperatures, and (2) adjusting is simple, training-fie free approach to improve the sample quality. Future work could consist developing principled softmax temperature schedules during inference varying during training.
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0.1
0.2
0.3
0.4
0.5
0.6
0.7 0.8
0.9
1.0
1.2
1.5
Figure Generated samples at different softmax temperatures. We can control the output of Purrception by changing the softmax temperature. A low temperature creates simpler, cleaner samples, while a high temperature adds more fine-grained details but can sometimes introduce flaws and reduce the image quality. Here we vary 7 from 0.1 to 1.5.
4.3 QUALITATIVE AND QUANTITATIVE RESULTS
To test how well Purrception performs against similar methods, we train Purrception at scale for 3.5M iterations with a DiT-XL/2 backbone, and report quantitative results on class-conditional ImageNet- 1 generation at 256 x 256 resolution.
Table 1 highlights comparison with popular image generation methods, similar to Purrception in model size and methodology, including autoregressive methods (Esser et al. [02 Yu et al. [021 Lee
ct al. 2022, Sun et al., 2024), discrete diffusion and masked generative models CChang et al. 2022]
Gu al. [222 Hu & Ommer 2024), , well continuous diffusion models (Dhariwal & Nichol
2021 Ho et al. 2022; Rombach et al., 2022 Peebles & Xie 2023 Ma et al. 2024). Purrception
competitive in FID score. Notably, Purrception outperforms all discrete diffusion and masked generative models. It also shows stronger performance against most autoregressive methods while having less parameters and/or benefiting from natively faster decoding than large-token autoregressive models (which often rely on inference optimizers such as vLLM Sun et al. (2024)). This firmly establishes Purrception as novel, state-of-the-art approach, among VQ-based latent generative models, demonstrating that our hybrid approaches in fidelity.
formulation can surpass traditional VQ
Against strong continuous diffusion baselines, Purrception falls short on important baselines like D/T-X1//2 and SiT-XI/2 baselines. We believe this is mainly due to two reasons: (1) the use of high-quality VAF autoencoders in those models, which are known to produce lower FID scores than VQ tokenizers at equivalent scales, and (2) their considcrably longer training schedules (twice as many iterations used for Purrception). Despite this gap. Purrception's 's strong results highlight that our hybrid design can approach the performance of top-tier diffusion models. This underscores that Purrception effectively bridges the fidelity of continuous diffusion with the categorical training objective suitable for VQ latent spaces, positioning it as a promising direction for future generative modeling.
5 RELATED WORK
Diffusion, flow matching, and latent spaces. Diffusion and score-based models synthesize data Via iterative denoising (Sohl-Dickstein et al., 2015 Ho et al. 2020 Song et al. (020)), while Flow Matching learns time-dependent velocity field that transports source distribution to the data distribution, yielding continuous normalizing flows with strong empirical results (Lipman et al. 2023
Liu et al. 2023 Albergo al.] 2023). Moreover, alternative parameterizations of flow matching exist, e.g. endpoint prediction shows improved performance in tasks like image generation and molecular generation (Ma et al. 2024 Bijkelboom al. 2024]. To reduce cost without sacrificing quality, many
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Table Class-conditional generation on ImageNet-1k 256 256. We compare Purrception against various autoregressive, diffusion, and masked generative models. We report the number of parameters in millions (M) or billions (B), as well as the FID scores for each model. Purrception achieves competitive FID of 3.88, showcasing the effectiveness of our hybrid discrete-continuous approach against strong baselines, particularly the VQ image generation methods. Here we use the LlamaGen's vq-ds8 c2i (Sun et al. 2024) tokenizer and Euler with 250 integration steps as ODE solver for FID computation.
Model
#Parameters
FID |
Autoregressive & Masked Generative Models
VQGAN [Esser
2021
1.4B
5.20
ViT-VQGAN Yu et al. 2021
1.7B
3.04
RQTransformer (Lee et al 2022
3.8B
3.80
LlamaGen-Xi XL (Sun et al. 2024
775M
3.39
MaskGIT (Chang al. 2022
227M
6.18
Open-MAGVIIZ-L Luo et al
2024
804M
2.51
Continuous Diffusion
ADM [Dhariwal & Nichol 2021
554M
10.94
CDM (o el al 2022
4.88
LDM- Rombach LDM-aRombach
2022)
400M
3.60
DiT-XLIZ (Pecbles & Xie
2023
675M
2.27
SiT-XL/2 (Ma et al. 2024
675M
2.06
Discrete Diffiusion Masked Generative Models
VQ-Diffusion (Gu et al. 2022]
5.84
Implicit Timestep Model Hu & Ommer 2024
546M
5.30
Hybrid Discrete-Continuort Models Purreeption (T 0.9, cfg 1.3)
750M
3.88
works apply these dynamics vector-quantized latent spaces, where autoencoders provide compact discrete indices with associated embeddings (Van Den Oord et 2017 latents underlic VQ-GAN and large-scale generative systems (Esser et al. 2021 Ramesh et al. 2021
Razavi et al. 2019). Such
2022). and running diffusion/flows on them enables efficient high-fidelity synthesis (Vahdat et al.
2021 Rombach et al. 2022 Dao et al. (202)), with recent work scaling to stronger backbones and
resolutions (Ma al. 2024 Esser al.).
Discrete dy dynamics and relaxations. Beyond continuous latents, discrete diffusion and flow models operate directly on tokens pixels (Hoogeboom et al.] 2021a Austin et al., 2021 Gat et al. 2024 Stark et al., 2024, Davis et al. 2024). Closer to our setting, discrete lutent diffusion denoises over VQ indices (Gu et al. 2022; Tang et al, 2022). making the indices explicit but typically discarding the geometry of their embeddings. A complementary approach is to embed categorical data into continuous space and run diffusion there, as in Continuous Diffusion for Categorical Data (CDCD) (Dieleman et al.] 2022), developed primarily for language modelling. CDCD preserves the continuous- time formulation by operating noisy embeddings while training with cross-entropy predictions, thereby capturing uncertainty and retaining guidance mechanisms. However, because the embeddings are learned jointly, the approach relies on continuous relaxations and may diverge from the true categorical structure. Our approach follows the same general spirit of combining categorical supervision with continuous transport.
token
6 CONCLUSIONS
We introduced Purrception, an adaptation of VFM to vector-quantized image generation. The method retains continuous transport in the embedding space while supervising with a categorical posterior over codebook indices. This coupling addresses the core trade-off of existing approaches: unlike CFM, Purrception benefits from categorical supervision, and unlike DFM, it avoids collapsing geometry into hard index jumps. The result is model that learns, broadly speaking, what choose and where to go, expressing uncertainty over plausible codes a geometry-awee -warr way. Empirically, Purrception outperforms both CFM and DFM on ImageNet- 1 256 x 256 benchmark, converging

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faster and achieving superior FID while preserving the efficiency of flow matching. Further ablations confirm that logits provide a controllable quality-divensy knob through temperature scaling.
Limitations and Future Work. Our approach is currently limited by its reliance on fixed, pretrained VQ autoencoder, which makes performance dependent on the initial tokenization quality. While the model is competitive on 256 256 ImageNet- Ik, its generalization higher resolutions needs validation, and it does not yet match the performance of top-tier continuous diffusion models. Future work could directly address these limitations by exploring different VQ models or jointly training the autoencoder with the flow model. Broader research directions include extending this hybrid perspective domains like audio, video, and 3D shapes, as well as developing principled temperature schedules and stronger theory for categorical objectives. Finally, because the model remains a continuous flow, it supports distillation into highly efficient, few-step samplers and can incorporate guidance, paving the way for practical generative pipelines.
other datasets or
Ethics Statement. All experiments this work rely exclusively on publicly available datasets (i.e., ImageNet) used under their original licenses. We do collect or annotate any new human data. As with other generative models, there exists a risk of misuse in privacy-invasive or unauthorized applications. We strongly caution against such uses and emphasize the importance of adhering to license terms, governance standards, and applicable legal requirements, though, as our approach is primarily methodological, we do not see immediate risks.
Reproducibility Statement. We provide pseudocode for training and sampling Purrception (Ap- pendix B as well as detailed implementation specifics in Appendix C which covers optimization settings and evaluation protocols. To facilitate replication, release the full codebase")
This project was supported by the Bosch Center for Artificial Intelligence. was also made possible in part by the ELLIS unit in Amsterdam, whose funding supported collaborative research visit to LMU Munich.
Acknowledgements.
httpsss//grthhbb.cm//rraaaaa http:/aaiiaaattaarrrrrttiiiioioi
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A USAGE OF LARGE LANGUAGE MODELS
During the preparation of this submission, Large Language Models (LLMs) were utilized as tool to enhance the quality and presentation of our work. Specifically, we employed LLMs for text refinement, including improving grammar, syntax, and clarity to ensure the readability of our research. Additionally, these models assisted in refining the aesthetic and structural layout of our data visualizations and plots, providing suggestions for more effective data presentation. It is important to note that the LLMs served solely as an assistive tool. The authors retained full responsibility for all intellectual content, including the underlying research, data analysis, interpretation of results, and the final articulation of all arguments and conclusions presented in this paper.
B ALGORITHMS
TRAINING
SAMPLING (EULER INTEGRATION)
for x D do
20 Pni
+ Quantize(Encoode (z)):
for {0..... ... , - do x- 8/T:
c - LatentCode(z1);
20 po:
Te + softmax( āe. (2.). T);
t~ ((, 1):
2. Σ ... Ck 2s 1-t 2s+1 t 20 + (1/T)us:
2 t2 + (1 zo:
[(0) CrossEntropy(c, m (z:));
Backprop and update 0:
end
end
Return x;;
Figure 6: Training and sampling algorithms for Purrception.
C IMPLEMENTATION DETAILS
Training specifications. We use DiT architectures of different sizes backbones for all models (i.e., Purrception, CFM, DFM). To train them, we mostly use the specifications from the original paper (Peebles & Xie 2023): we initialize the final linear layer of DiT with zeros and otherwise use the initialization techniques from the ViT (Dosovitskiy et al. (020). We optimize our models using AdamW (Kingma & Ba 2016, Loshchilov et al. 2017) with constant learning rate 1e 4, weight decay 0.01, (31, . :2) (0.9, 0.999). For Purrception, we use eps 1e - 6. We also use a global batch size 256. Based on the training details of prior image generation methods, we compute the exponential moving average (EMA) of the backbone parameters over training using a decay rate of 0.9999, and we do inference using solely the EMA model.
Additionally, we use two tokenizers: (1) the Stable Diffusion's 's tokenizer vo- f8 with downsampling factor f and a codebook C of shape 16, 384 (Esser et al., [021 and (2) the LlamaGen's tokenizer wwwdds---e c2222 with downsampling factor f 8 and codebook C of shape 16, 384 x 8 (Sun et al. 2024). This means that for given RGB image x 256 x 256 resolution, the shape of the latent {(x) is 32 x 32 x d (d = 4 for vq-f8 and d = 8 for ds- 2)), which is further quantized according to C. During sampling, we use the decoder G to map the generated latent back into pixel space. The encoder, decoder, and codebook are kept frozen during training.
Sampling and FID score computation. Flow models need to simulate an ODE to solve the generative modeling task. We use the torchdi library in PyTorch and (unless otherwise specified) the usual Euler method with 100 steps when generating samples.
For computing the FID scores, we first generate 10,000 samples for computing FID-10k scores and 50,000 samples for the FID-50k. Then, we the torch ffeliity Py PoToh library (Obukhov jet al.) 2020) to compute the FID score. For both FID-10k and FID-50k, we use 50k real samples (i.e., the entire validation with 256 256 resolution) to compute the statistics for the target dataset.
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Unless otherwise specified, we do nor use classifier-free guidance for the models trained conditionally on ImageNet-Ik.
Computational resources. All methods were trained using a distributed setup on the LUMI supercomputer, utilizing total of 16 AMD MI250x GPUs, each equipped with 128GB of HBM2e memory.
D TRAINING STABILITY
backhone to predict the codebook indices that compound the target datapoint. One of the biggest challenges encountered was maintaining stable training for larger DiT variants (i.e., DiT-L/2, DiT-XL/2), especially because such training instabilities occurred in the later stages of the training phase.
Architecturally,
diffusion transformer (DiT) Peebles & Xie (2023)
Since the major difference between training Flow Matching and Purrception is the training objective (i.e., mean-squan euanne error for the former one, cross- entropy for the latter one) and Flow Matching does not have such training instabilities, we hypothesize that the cause might be the final softmax operation. Indeed, this divergence the output logits from the log probabilities has been reported often bility issue by the research community when training large models scale (Chowdhery et al., 2023 Worts man et al. 2023). In their paper, Wortsman et al. (2023) name this issue the logit drift problem. To mit- igate this issue, they propose regularizing the train- ing using additional 2-loss which proved effective parameter
5.5
w/ z-loss
w/o z-loss
Training Loss
5
insta-
30k
60k
90k
120k
150k
Number of Training Iterations
Figure 7: Training loss curves with and without z-loss. An additional z-loss avoids training divergence. Raw data is shown in lighter colors, while exponentially smoothed curves (EMA) are shown bold. We used the same hyperparameters for both runs and DIT-X1// backbone. EMA smoothing factor is = 0.9.
in training recent ofute-of-the-arr billion- billion-parameter models such as Chameleon (Chameleon Team 2024).
Inspired by its success, we apply z-loss regulariza- tion as well. Similar to Chameleon Team (2024).
we add 10-5 10-" 7 to Purrception's 's loss function, where Z Σ
-i=1 em: and { the logits out- putted by the backbone. Ssoows Purrception achieves stability when z-loss is integrated. Thus, used the zoss by default when training Purrception.
E THE EFFECT OF VQ AUTOENCODERS
Table 2: Quantitative comparison of Purrception trained with different VQ tokenizers. Evalua- tion ImageNet-1k 1k 256 x 256 shows that the choice of tokenizer significantly influences generation quality, with vs-ds8 -2: outperforming vq- E across all FID thresholds.
vq-f8
vq-ds8-ca
rFID
1.19
0.59
FID (T 0.7)
11.80
7.44
FID (T 0.8)
10.85
6.46
FID (T 0.9)
10.82
7.03
FID (T 1.0)
12.33
9.60
When we train Purrception, we train exclusively the DiT backbone while keeping the encoder, decoder, and codebook vectors frozen. To test how much the performance of Purrception relies
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the VQ tokenizer, train two DiT-XL/2 models ImageNet- 1k 256 256 with the same training configurations, for 3.5M iterations each, using two different VQ autoencoders: (1) the Stable Diffusion's s vq-18 tokenizer (Rombach et al. 2022) and (2) the LlamaGen's vq-ds8-c2i
tokenizer (Sun et al. 2024).
The results in Table 2 show that Purrception's performance is tightly coupled to the quality and design of the underlying VQ autoencoder used to produce the latent tokens. We can observe that for LlamaGen's tokenizer (which has better rFID score compared Stable Diffusion's 's one), obtain better FID-50k scores in class-conditional ImageNet- Ik 256 x 256. This indicates that even when the DiT backbone is trained identically, the representational granularity and perceptual fidelity of the VQ tokenizer have decisive impact on downstream generation quality.
Overall, the experiment highlights that the performance of Purrception is not Eikenizer-agnosiie its effectiveness depends on the inductive biases and compression characteristics of the VQ autoencoder used. Future work could explore training custom VQ models co-optimized with the DiT backbone or investigating hybrid tokenizers that balance perceptual fidelity and codebook compactness.
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Reviews:
[{"review_id": "jdC8EawKXD", "reviewer": "Reviewer_i3Vq", "summary": "The authors propose a latent variational flow matching approach, Purrception, for vector quantized generative models. The motivation for the approach is to propose a hybrid between discrete and continuous latent flows for the latent generative models. They exploit the variational formulation of the marginal vector field and propose to model the discrete latent tokens as a categorical distribution (probabilities) from the continuous latent embeddings. With the experiments on the ImageNet-256 dataset, they show training efficiency over discrete and continuous latent generative models and improved final accuracy over discrete latent generative models.", "questions": "In line 294, what is e^{x_i}?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The idea of a continuous flow path and categorical token prediction combined under the variational flow matching framework is interesting. The application to VQ-VAEs does show improvement in generative performance compared to latent discrete flow-based approaches.", "weaknesses": "Major:\n\nImportant baseline missing - The main improvement compared to baselines is shown in the training efficiency. However, a comparison to the key baseline latent generative model SiT (Ma et al. 2024) is missing. I would request the authors to add this to their training efficiency analysis to improve thoroughness.\n\nOther:\n\nThe model doesn't reach competitive FID vs continuous models. This is not a major weakness, as it is known (empirically) that continuous latent models tend to perform better than the discrete ones in image generation. However, I would request the authors to add a discussion of why a 750M Purrception (discrete) will be more useful than a 675M DiT (continuous) model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ozn2clvMDO", "reviewer": "Reviewer_BoRK", "summary": "This paper introduces Purrception to address a dual nature of VQ-latents, which the authors argue that existing flow-based models make a poor trade-off: Continuous Flow Matching (CFM) respects the continuous geometry but ignores the discrete categorical structure, while Discrete Flow Matching (DFM) models the indices but discards the geometry.\n\nPurrception proposes a hybrid solution by adapting Variational Flow Matching (VFM). The model learns a categorical variational posterior over the discrete codebook indices, while the actual transport velocity is computed in the continuous embedding space.\n\nExperimental results on ImageNet-1k $256\\times256$ showing that Purrception converges significantly faster (1.7x-3.5x) and achieves a better final FID score than both CFM and DFM baselines when using the same DiT backbone88.", "questions": "1. As shown in Figure 5, the temperature does not affect the generated image too much. How is temperature useful in real application? And any tests on DFM baseline?\n1. Is z-loss just a specific patch that masks a deeper problem with scaling Purrception?\n1. Any reason why adapting VFM to a fixed, pretrained codebook is novel compared to the CDCD framework?\n1. Any evident that the final FID of 4.72 is not a property of Purrception, but just an inherent limitation of the vq-f8 latent space itself?\n1. Did you test how Purrception performs with any other VQ-VAE tokenizers?", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The motivation \"discrete-continuous tradeoff\" is a real problem for VQ-latents, and the paper's solution directly addresses it without compromise.\n1. The proposed solution is simple and elegant.\n1. Exprimental results show faster convergence than other DFM and CFM baselines.", "weaknesses": "1. Purrception failed to beat LlamaGen-XL, which has similar number of parameters, with respect to FID (Purrception 4.72 vs LlamaGen-XL 3.39), as well as other baselines.\n1. Purrception's encoder, decoder, and codebook are kept frozen during training, thus the entire method is fundamentally capped by the quality of the frozen vq-f8 tokenizer.\n1. Missing comparision with CDCD mentioned in related work, which also \"preserves the continuous-time formulation by operating on noisy embeddings while training with cross-entropy\".\n1. Table 1 does not include a CFM baseline. The paper only argues the converge curve \"strongly suggests\" Purrception would surpass CFM, but this is not convicing enough.\n1. Missing visual comparison with baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4Ho3rSL9By", "reviewer": "Reviewer_woJ8", "summary": "This paper proposes variational flow matching to train flow matching with discrete code from vq-vae. By jointly combining discrete and continuous, it shows that flow matching achieves better convergence than purely continuous or discrete.", "questions": "Please see the weakness above", "limitations": "", "rating": 4, "confidence": 5, "soundness": 1, "presentation": 3, "contribution": 2, "strengths": "1. The paper shows that under same discrete VQ-VAE, it outperforms the discrete flow matching training or the continuous flow matching training. The idea of uncertainty in sampling is quite interesting.\n\n2. The paper writing is easy to follow and clear.", "weaknesses": "1. The motivation is not well-convincing to me. It is not clear why receiving categorical signal is better for continuous flow matching. What is the geometry structure in continuous here ?\n\n2. The theory about VFM seems to be out of place for me. To my understanding, this method is based on VFM framework but instead of inputing the codebook index and let the flow matching learn internal codebook embedding, they utilize the codebook embedding from vq-vae to create soft embedding $z_1$ and input continuous signal. This leads to limited novelty.\n\n3. The model underperforms with DiT using continuous VAE. Furthermore, this technique heavily depends on a good VQ-VAE which is limited the model performance. It would be better to choose other discrete VQ-VAE, which has better rFID and some generative model training on that achieves very good FID like [1,2] to see how good the proposed technique can reached given different VQ-VAE. \n\n[1]: An Image is Worth 32 Tokens for Reconstruction and Generation\n[2]: Autoregressive Model Beats Diffusion: Llama for Scalable Image Generation", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9gvFQmoMx7", "reviewer": "Reviewer_8vtk", "summary": "The paper investigated flow-matching models for vector-quantized image generation. It pointed out the strengths and weaknesses of existing approaches: continuous flow models preserved geometric information but could not exploit categorical learning signals, while discrete models aligned with the quantized structure and allowed temperature-scaling control but failed to understand geometry. It then proposed Purrception, a variational flow-matching technique for vector-quantized image generation that combined the advantages of continuous and discrete flow models. Purrception learned categorical posteriors over codebook indices, which could be used for computing velocity fields in the continuous embedding space. Experiments showed that Purrception outperformed both continuous and discrete flow-matching baselines in both convergence speed and image generation quality on the ImageNet-1k 256 x 256 dataset.", "questions": "- The middle side of Equation (13) seems to be incorrect. \n- Section 4.1: Missing information on the averge NFE and the used temperature (I guess it is 0.9 based on Table 1).\n- Section 4.2: From my understanding, this experiment only varies the temperature at inference while keeping the training temperature (possibly 0.9) unchanged.\n  + As expected, using the inference temperature similar to the training one (the straight-forward configuration) provides the best performance. Deviating the inference temperature from the training one causes performance degradation, particularly in color saturation. Hence, the experiment cannot prove the benefits of the temperature-scaling control\n  + A more reasonable experiment is to use the same temperature for training & inference, and compare models trained w/ different temperatures\n  + There is no evidence showing that the temperature affects image diversity", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper discussed the strengths and weaknesses of existing flow-matching approaches for vector-quantized image generation. It then proposed to employ a variational flow-matching technique to combine the advantages of mentioned approaches.\n- The paper is technically sound.\n- Experiments showed that Purrception outperformed both continuous and discrete flow-matching baselines in both convergence speed and image generation quality on the ImageNet-1k 256 x 256 dataset.", "weaknesses": "- The middle side of Equation (13) seems to be incorrect. \n- Section 4.1: Missing information on the averge NFE and the used temperature (I guess it is 0.9 based on Table 1).\n- Section 4.2: From my understanding, this experiment only varies the temperature at inference while keeping the training temperature (possibly 0.9) unchanged.\n  + As expected, using the inference temperature similar to the training one (the straight-forward configuration) provides the best performance. Deviating the inference temperature from the training one causes performance degradation, particularly in color saturation. Hence, the experiment cannot prove the benefits of the temperature-scaling control\n  + A more reasonable experiment is to use the same temperature for training & inference, and compare models trained w/ different temperatures\n  + There is no evidence showing that the temperature affects image diversity\n- The results reported in Table 1 are far from state-of-the-art for Class-conditional generation on ImageNet-1k 256 x 256. The state-of-the-art ones, e.g., REPA, have FID less than 2. The reported result is only strong for vector-quantized image generation using standard DiT backbones. The authors should correct their claim.\n- Writting issues:\n  + Equation 12: $\\pi$ was used before being defined\n  + L265: The text should be in the same paragraph with the previous one\n  + Figure 5: Should add the temperature lable for each column", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)