Title: Transition Matching: Scalable and Flexible Generative Modeling
Venue: NeurIPS 2025
Accepted: yes
Keywords: generative models

flow matching

diffusion

large scale

multimodal
Abstract: Diffusion and flow matching models have significantly advanced media generation, yet their design space is well-explored, somewhat limiting further improvements. Concurrently, autoregressive (AR) models, particularly those generating continuous tokens, have emerged as a promising direction for unifying text and media generation, showing improved performance at scale. This paper introduces Transition Matching (TM), a novel discrete-time, continuous-state generative paradigm that unifies and advances both diffusion/flow models and continuous AR generation. TM decomposes complex generation tasks into simpler Markov transitions, allowing for expressive non-deterministic probability transition kernels and arbitrary non-continuous supervision processes, thereby unlocking new flexible design avenues. We explore these choices through three TM variants: (i) Difference Transition Matching (DTM), which generalizes flow matching to discrete-time  by directly learning transition probabilities, yielding state-of-the-art image quality and text adherence. (ii) Autoregressive Transition Matching (ARTM) and (iii) Full History Transition Matching (FHTM) are partially and fully causal models, respectively, that generalize continuous AR methods. They achieve continuous causal AR generation quality comparable to non-causal approaches and potentially enable seamless integration with existing AR text generation techniques. Notably, FHTM is the first fully causal model to match or surpass the performance of flow-based methods on text-to-image task in continuous domains. 
  We demonstrate these contributions through a rigorous large-scale comparison of TM variants and relevant baselines, maintaining a fixed architecture, training data, and hyperparameters.

Transition Matching: Scalable and Flexible Generative Modeling

Abstract

Diffusion and flow matching models have significantly advanced media generation, yet their design space is well-explored, somewhat limiting further improvements. Concurrently, autoregressive (AR) models, particularly those generating continuous tokens, have emerged as a promising direction for unifying text and media generation. This paper introduces Transition Matching (TM), a novel discrete-time, continuous-state generative paradigm that unifies and advances both diffusion/flow models and continuous AR generation. TM decomposes complex generation tasks into simpler Markov transitions, allowing for expressive non-deterministic probability transition kernels and arbitrary non-continuous supervision processes, thereby unlocking new flexible design avenues. We explore these choices through three TMvariants: (i) Difference Transition Matching (DTM), which generalizes flow matching to discrete-time by directly learning transition probabilities, yielding state-of-the-art image quality and text adherence as well as improved sampling efficiency. (ii) Autoregressive Transition Matching (ARTM) and (iii) Full History Transition Matching (FHTM) are partially and fully causal models, respectively, that generalize continuous AR methods. They achieve continuous causal AR generation quality comparable to non-causal approaches and potentially enable seamless integration with existing AR text generation techniques. Notably, FHTM is the first fully causal model to match or surpass the performance of flow-based methods on text-to-image task in continuous domains. We demonstrate these contributions through a rigorous large-scale comparison of TM variants and relevant baselines, maintaining a fixed architecture, training data, and hyperparameters.

1 Introduction

Recent progress in diffusion models and flow matching has significantly advanced media generation (images, video, audio), achieving state-of-the-art results [31, 24, 34, 4]. However, the design space of these methods has been extensively investigated [43, 20, 30, 40, 7], potentially limiting further significant improvements with current modeling approaches. An alternative direction focuses on autoregressive (AR) models to unify text and media generation. Earlier approaches generated media as sequences of discrete tokens either in raster order [37, 54, 6]; or in random order [3]. Further advancement was shown by switching to continuous token generation [25, 47], while also improving performance at scale [10].

This paper introduces Transition Matching (TM), a general discrete-time continuous-state generation paradigm that unifies diffusion/flow models and continuous AR generation. TM aims to advance both paradigms and create new state-of-the-art generative models. Similar to diffusion/flow models, TM breaks down complex generation tasks into a series of simpler Markov transitions. However, unlike diffusion/flow, TM allows for expressive non-deterministic probability transition kernels and arbitrary non-continuous supervision processes, offering new and flexible design choices.

FM

MAR

We explore these design choices and present three TM variants:

(i) Difference Transition Matching (DTM): A generalization of flow matching to discrete time, DTM directly learns the transition probabilities of consecutive states in the linear (Cond-OT) process instead of just its expectation. This straightforward approach yields a state-of-the-art generation model with improved image quality and text adherence, as well as significantly faster sampling.

(ii) Autoregressive Transition Matching (ARTM) and (iii) Full History Transition Matching (FHTM): These partially and fully causal models (respectively) generalize continuous AR models by incorporating a multi-step generation process guided by discontinuous supervising processes. ARTM and FHTM achieve continuous causal AR generation quality comparable to non-causal methods. Importantly, their causal nature allows for seamless integration with existing AR text generation methods. FHTM is the first fully causal model to match or surpass the performance of flow-based methods in continuous domains.

In summary, our contributions are:

Formulating transition matching: simplified and generalized discrete-time generative models based on matching transition kernels.

Identifying and exploring key design choices, specifically the supervision process, kernel parameterization, and modeling paradigm.

Introducing DTM, which improves upon state-of-the-art flow matching in image quality, prompt alignment, and sampling speed.

Introducing ARTM and FHTM: partially and fully causal AR models (resp.) that match non-AR generation quality and state-of-the-art prompt alignment.

Presenting a fair, large-scale comparison of the different TM variants and relevant baselines using a fixed architecture, data, and training hyper-parameters.

2 Transition Matching

We start by describing the framework of Transition Matching (TM), which can be seen as a simplified and general discrete time formulation for diffusion/flow models. Then, we focus on several, unexplored TM design choices and instantiations that goes beyond diffusion/flow models. In particular: we consider more powerful transition kernels and/or discontinuous noise-to-data processes. In the experiments section we show these choices lead to state-of-the-art image generation methods.

2.1 General framework

Notation We use capital letters X,Y,Z,A,B to denote random variables (RVs) and lower-case letter x, y, z, a, b to denote their particular states. One exception is time t where we abuse notation a bit and use it to denote both particular times and a RV . All our variables and states reside in euclidean spaces x ∈ R d . The probability density function (PDF) of a random variable Y is denoted p Y ( x ) . For RVs X t (and only for them) we use the simpler PDF notation p t ( x t ) . We use the standard notations for joints p X,Y ( x, y ) and conditional densities p X | Y ( x | y ) densities. We denote [ T ] = { 0 , 1 , . . . , T } .

Problem definition Given a training set of i.i.d. samples from an unknown target distribution p T , and some easy to sample source distribution p 0 . Our goal is to learn a Markov Process, defined by a probability transition kernel p θ t +1 | t ( x t +1 | x t ) , where t ∈ [ T -1] taking us from X 0 ∼ p 0 to X T ∼ p T . That is, we define a series of random variables ( X t ) t ∈ [ T ] such that X 0 ∼ p 0 and

Supervising process Training such a Markov process is done with the help of a supervising process , which is a stochastic process ( X 0 , X 1 , . . . , X T ) defined given data samples X T using a conditional process q 0 ,...,T -1 | T , i.e.,

X

T

X

0

, . . . , X

T

-

1

and q 0 ,...,T denotes the joint probability of the supervising process ( X t ) t ∈ T . The only constraint on the conditional process is that its marginal at time t = 0 is the easy to sample distribution p 0 , i.e.,

Note that this definition is very general and allows, for example, arbitrary non-continuous processes, and indeed we utilize such a process below. Transition matching engages with the supervising process ( X t ) t ∈ T by sampling pairs of consecutive states ( X t , X t +1 ) ∼ q t,t +1 , t ∈ [ T -1] .

Loss The model p θ t +1 | t is trained to transition between consecutive states X t → X t +1 in the sense of equation 1 by regressing q t +1 | t defined from the supervising process q . This motivates the loss utilizing a distance/divergence D between distributions

where t is sampled uniformly from [ T -1] . However, this loss requires evaluating q t +1 | t which is usually hard to compute. Therefore, to make the training tractable we require that the distance D has an empirical form , i.e., can be expressed as an expectation of an empirical one-sample loss ˆ D over target samples. We define the loss

where ( X t , X t +1 ) are sampled from the joint q t,t +1 with the help of equation 2, namely, first sample data X T ∼ p T and then ( X t , X t +1 ) ∼ q t,t +1 | T ( ·| X T ) . Notably, equation 5 can be used to learn arbitrary transition kernels, in contrast to e.g., Gaussian kernels used in discrete time diffusion models or deterministic kernels used in flow matching. The particular choice of the cost D depends on the modeling paradigm chosen for the transition kernel p θ t +1 | t , and discussed later.

Kernel parameterizations The first and natural option to parameterize p θ t +1 | t is to regress q t +1 | t directly as is done in equation 5. This turns out to be a good modeling choice in certain cases. However, in some cases one can use other parameterizations that turn out to be beneficial, as is also done for flow and diffusion models. To do that in the general case, we use the law of total probability applied to the conditional probabilities q t +1 | t with some latent RV Y :

where q Y | t is the posterior distribution of Y given X t = x t and q t +1 | t,Y is easy to sample (often a deterministic function of X t and Y ). Then the posterior of Y is set as the new target of the learning process instead of the transition kernel. That is, instead of the loss in equation 5 we consider

̸

Similarly, during training, sampling from the joint ( X t , Y ) ∼ q t,Y , is accomplished by first sampling data X T and then ( X t , Y ) ∼ q t,Y | T ( ·| X t ) . Once the posterior p θ Y | t is trained, sampling from the transition p θ t +1 | t during inference is done with the help of equation 6. To summarize, in cases we want to use non-trivial kernel parameterization, i.e., Y = X t +1 , we sample from q t +1 | t,Y (in sampling) and q t,Y | T (in training). See Algorithms 1 and 2 the training and sampling pseudocodes.

Kernel modeling Once a desirable Y is identified, the remaining part is to choose a generative model for the kernel p θ Y | t . Importantly, one of the key advantages in TM comes from choosing expressive kernels that result in more elaborate transition kernels than used previously. A kernel modeling is set by a choice of a probability model for p θ Y | t and a loss to learn it. We denote the probability model choice by B | A , where A denotes the condition and B the target . For example, Y | X t will denote a model that predicts a sample of Y given a sample of X t . We will also use more elaborate probability models, such as autoregressive models. To this end, consider the state Y reshaped into individual tokens Y = ( Y 1 , . . . , Y n ) , and then Y i | ( Y <i , X t ) means that our model samples the token Y i given previous tokens of Y , Y <i = ( Y 1 , . . . , Y i -1 ) , and X t .

All our models are learned with flow matching (FM) loss. For completeness, we provide the key components of flow matching formulated generically to learn to sample from B | A . Individual states of A and B are denoted a and b , respectively. Flow matching models p θ B | A via a velocity field u θ s ( b | a ) that is used to sample from p θ B | A ( ·| a ) by solving the Ordinary Differential Equation (ODE)

initializing with a sample B 0 ∼ N (0 , I ) (the standard normal distribution) and solving until s = 1 . In turn, B 1 is the desired sample, i.e., B 1 ∼ p θ B | A ( ·| a ) . The loss D , used to train FM, has an empirical form and minimizes the difference between q B | A and p θ B | A ,

where s is sampled uniformly in [0 , 1] , B 0 ∼ N (0 , I ) , B ∼ q B | A ( ·| a ) , and B s = (1 -s ) B 0 + sB .

We summarize the key design choices in Transition Matching:

2.2 Transition Matching made practical

The key contribution of this paper is identifying previously unexplored design choices in the TM framework that results in effective generative models. We focus on two TM variants: Difference Transition Matching (DTM), and Autoregressive Transition Matching (ARTM/FHTM).

Difference Transition Matching Our first instance of TM makes the following choices:

As the supervising process q we use the standard linear process (a.k.a., Conditional Optimal Transport), defined by

where X 0 ∼ p 0 = N (0 , I ) and X T ∼ p T . This is the same process used in [27, 28]. For the kernel parameterization Y we will use the difference latent (see Figure 3, left),

During training, sampling q t,Y | T ( ·| X T ) (i.e., given X T ) is done by sampling X 0 , and using 10 and 11 to compute X t , Y . Using the definition in 10 and rearranging gives

and this equation can be used to sample from q t +1 | t,Y ( ·| X t , Y ) during inference. See Figure 4 for an illustration of a sampled path from this supervising process. We learn to sample from the posterior p θ Y | t ≈ q Y | t using flow matching with A = ( t, X t ) and B = Y . This means we learn a velocity field u θ s ( y | t, x t ) and train it with Algorithm 1 and the CFM loss in equation 9.

Note that in this case one can also learn a continuous time t ∈ [0 , T ] which allows more flexible sampling.

The last remaining component is choosing the architecture for u θ s . Let x = ( x 1 , . . . , x n ) be a reshaped state to n tokens. For example, each x i can represent a patch in an image x . Next, note that in each transition step we need to sample Y ∼ p θ Y | t ( ·| X t ) by approximating the solution of the ODE in equation 8. Therefore, to keep the sampling process efficient, we follow [25] and use a small head g θ that generates all tokens in a batch and is fed with latents from a large backbone f θ . Our velocity model is defined as

where h i t is the i -th output token of the backbone, i.e., [ h 1 t , h 2 t , . . . , h n t ] = f θ t ( x t ) . See Figure 5 (DTM) for an illustration of this architecture. One limitation of this architecture worth mentioning is that in each transition step, each token y i is generated independently, which limits the power of this kernel. We discuss this in the experiments section but nevertheless demonstrate that DTM with this architecture still leads to state-of-the-art image generation model.

Connection to flow matching Although flow matching [27, 28, 1] is a deterministic process while DTM samples from a stochastic transition kernel in each step, a connection between the two is revealed by noting that the expectation of a DTM step coincides with Flow Matching Euler step, i.e.,

DTM

ARTM

FHTM

which is exactly the marginal velocity in flow matching, see Figure 3. In fact, as T → ∞ (or equivalently, steps are getting smaller), DTM is becoming more and more deterministic, converging to FM with Euler step, providing a novel and unexpected elementary proof (i.e., without the continuity equation) for FM marginal velocity. In Appendix C we prove

Theorem 1. (informal) As the number of steps increases, T →∞ , DTM converges to Euler step FM,

as k/T → 0 , where X ℓ , ∀ ℓ > t is defined by Algorithm 2 with a optimally trained p θ Y | t .

We attribute the empirical success of DTM over flow matching to its more elaborate kernel.

Autoregressive Transition Matching Our second instance of TM is geared towards incorporating state-of-the-art media generation in autoregressive models, and utilizes the following choices:

In this case we use a novel supervising process we call independent linear process , defined by

where X 0 ,t ∼ N (0 , I ) , t ∈ [ T ] are all i.i.d. samples. Sampling q t,t +1 | T ( ·| X T ) is done by sampling X 0 ,t and X 0 ,t +1 and using 15. Although the independent linear process has the same marginals q t as the linear process in equation 10, it enjoys better regularity of the conditional q t +1 | t ( ·| x t ) , see Figure 6 for an illustration, and as demonstrated later in experiments is key for building state-of-the-art Autoregressive image generation models.

For the transition kernel we use an Autoregressive (AR) model with the choice of Y = X t +1 . As before, we let a state written as series of tokens x = ( x 1 , . . . , x n ) and write the target kernel q t +1 | t using the probability chain rule (as usual in AR modeling),

where X < 1 t +1 is the empty state. We will learn to sample from q i t +1 | t using FM with A = ( t, X t , X <i t +1 ) and B = X i t +1 . That is, we learn a velocity field u θ s ( y i | t, x t , x <i t +1 ) trained with the CFM loss in equation 9. This method builds upon the initial idea [25] that uses such AR modeling to map in a single transition step from X 0 to X T using diffusion, and in that sense ARTM is a generalization of that method. Lastly the architecture for u θ s is based on a similar construction to DTM with a few, rather minor changes. Using the same notation for the head g θ and backbone f θ models we define

with h i t +1 = f θ t ( x t , x <i t +1 ) . Figure 5 (ARTM) shows an illustration of this architecture.

Full-History ARTM We consider a variant of ARTM that allows full "teacher-forcing" training and consequently provides a good candidate to be incorporated into multimodal AR model.

The idea is to use the full history of states, namely considering the kernel

and train an FM sampler from q i t +1 | 0 ,...,t with the choices A = ( X 0 , . . . , X t , X <i t +1 ) (no need to add time t due to the full state sequence) and B = X i t +1 . The architecture of the velocity u s is defined by

with h i t +1 = f θ ( x 0 , . . . , x t , x <i t +1 ) and we take f to be fully causal. See Figure 5 (FHTM).

3 Related work

Diffusion and flows We draw the connection to previous works from the perspective of transition matching. Diffusion models [41, 16, 42, 21] can be seen as an instance of TM by choosing D in the loss (5) to be the KL divergence, derived in diffusion literature as the variational lower bound [22]. The popular ϵ -prediction [16] in transition matching formulation is achieved by the design choices

where ( σ t , α t ) is the scheduler, and non-zero w t reproduces the sampling algorithm in [16], while taking the limit w t → 0 yields the sampling of [42]. Similarly, x -prediction [21] is achieved by the parametrization Y = X T . In contrast to these work, our TM instantiations use more expressive kernel modeling. Relation to flow matching[27, 28, 1] is discussed in Section 2.2. Generator matching [17] generalizes diffusion and flow models to general continuous time Markov process modeled with arbitrary generators, while we focus on discrete time Markov processes. Another line of works adopted supervision processes that transition between different resolutions; [19] used flow matching with a particular coupling between different resolution as the kernel modeling; [56] implemented a similar scheme but allowed the FM to be dependent on the previous states (frames) in an AR manner. Denoising Diffusion GANs [50] uses x -prediction parametrization Y = X T and utilize a GAN [12] model as the transition kernel. In a concurrent work, [57] proposes a similar parameterization to DTM ( Y = X T -X 0 ) however uses a backbone-only architecture consequently making transition sampling computationally very expensive and sub-par in generation quality compared to the backbone-head architecture.

Autoregressive image generation Early progress in text-to-image generation was achieved using autoregressive models over discrete latent spaces [37, 8, 54], with recent advances [46, 44, 14] claiming to surpass flow-based approaches. A complementary line of work explores autoregressive modeling directly in continuous space [25, 47], demonstrating some advantages over discrete methods. In [10] this direction is scaled further, achieving SOTA results. In our experiments we compare these models in controlled setting and show that our autoregressive transition matching variants improves upon these models and achieves SOTA text-to-image performance with a fully causal architecture. Lastly, DART-AR [13] uses a supervising process similar to the independent linear process 15 with an autoregressive backbone however utilizing a Gaussian transition kernel per patch in contrast to an FM head used in our case.

FM

MAR

FHTM (Ours)

DTM (Ours)

FM

MAR

FHTM (Ours)

DTM (Ours)

'A richly textured oil painting of a young badger delicately sniffing a yellow rose next to a tree trunk. A small

waterfall can be seen in the background. '

'a robot holding a sign with "Let's PAINT!" written on it'

'The Statue of Liberty with the face of an owl'

'a blue airplane taxiing on a runway with the sun behind it'

PLANIT!

LETS

PANT

LETSPANNT!

LeT's

PainT!

VERYDEEP

LEARNING

VEEREY

veriv

dep

loa Deep

LEARINIING

DEET

Leormg

Larning

'a racoon holding a shiny red apple over its head'

'A green sign that says "Very Deep Learning" and is at the edge of the Grand Canyon. '

4 Experiments

We evaluate the performance of our Transition Matching (TM) variants-Difference TM (DTM), with T = 32 TM steps, Autoregressive TM (ARTM-2,3) with T = 2 , 3 (resp.), and Full History TM(FHTM-2,3) with T = 2 , 3 (resp.) - on the text-to-image generation task. In Appendix B we provide training and sampling pseudocodes of the three variants in Algorithms 3-8, and python code for training in Figures 25,26, and 27. Our baselines include flow matching (FM) [9], continuoustoken autoregressive (AR) and masked AR (MAR) [25], and discrete-token AR [54] and MAR [3]. For continuous-token MAR we include two baselines: the original truncated Gaussian scheduler version [25], and the cosine scheduler used by Fluid (MAR-Fluid) [10].

Datasets and metrics Training dataset is a collection of 350M licensed Shutterstock image-caption pairs. Images are of 256 × 256 × 3 resolution and captions span 1-128 tokens embedded with the CLIP tokenizer [35]. Consistent with prior work [38], for continuous state space, the images are embedded using the SDXL-VAE [33] into a 32 × 32 × 4 latent space, and subsequently all model training are done within this latent space. For discrete state space, images are tokenized with Chameleon-VQVAE [2]. Evaluation datasets are PartiPrompts [54] and MS-COCO [26] text/image benchmarks. And the reported metrics are: CLIPScore [15], that emphasize prompt alignment; Aesthetics [39] and DeQA Score [53] that focus on image quality; PickScore [23], ImageReward [51], and UnifiedReward [49] which are human preference-based and consider both image quality and text adherence. Lastly, we report results on the GenEval [11] and T2I-CompBench [18] benchmarks.

Architecture and optimization All experiments are performed with the same 1.7B parameters DiT backbone ( f θ ) [32], excluding a single case in which we compare to a standard LLM architecture [48, 29]. Methods that require a small flow head ( g θ ), replace the final linear layer with a 40M parameters MLP [25]. Text conditioning is embedded through a Flan-UL2 encoder [45] and injected via cross attention layers, or as prefix in the single case of the LLM architecture. Finally, the models are trained for 500K iterations with a 2048 batch size. Precise details are in Appendix A.1. We aim to facilitate a fair and useful comparison between methods in large scale by fixing the training data, using the same size architectures with identical backbone (excluding the LLM architecture that use standard transformer backbone), and same optimization hyper-parameters. To this end, we restrict our comparison to baselines which we re-implemented.

4.1 Main results: Text-to-image generation

Our main evaluation results are reported in Tables 1 and 8 (in Appendix) on the DiT architecture. We find that DTM outperforms all baselines, and yields the best results across all metrics except the CLIPScore, where on the PartiPrompts benchmark it is a runner-up to MAR and our ARTM-3 and FHTM-3. On the MS-COCO benchmark, the discrete-state space models achieve the highest CLIPScore but lag behind on all other metrics, as well as on the GenEval benchmark. DTM

shows a considerable gain in text adherence over the baseline FM and sets a new SOTA on the text-to-image task. Next, our AR kernels with 3 TM steps: ARTM-3 and FHTM-3, demonstrate a significant improvement compared to the AR baseline, see comparison of samples in Figure 15 in the Appendix. When compared to MAR, ARTM-3 and FHTM-3 have comparable CLIPScore, but improve considerably on all other image quality metrics, where this is also noticeable qualitatively in Figure 7 and Figures 11-14 in the Appendix. GenEval and T2I-Compbench results are reported in Tables 2 and 9 (resp.) showing that overall DTM is leading with FHTM-3/ARTM-3 and MAR closely follows. To our knowledge, FHTM is the first fully causal model to match FM performance on text-to-image task in continuous domain, with improved text alignment.

Image generation with causal model Beyond improving prompt alignment and image quality in text-to-image task, a central goal of recent research [58, 55] is to develop multimodal models also capable of reasoning about images. This direction aligns naturally with our approach, as the fully causal FHTM variant enables seamless integration with large-language models (LLM) standard architecture, training, and inference algorithms. As a first step toward this goal, we demonstrate in Table 1 and 8 that FHTM, implemented with an LLM architecture replacing 2D with 1D positional encoding and input the text condition only at the first layer, can match

and even surpass the performance of approximately the same size DiT architecture. Furthermore, it matches or improve upon all baselines across all metrics. Further implementation details are in Appendix A.1.

Global ranking As part of our main effort to empirically validate the Transition Matching framework, we trained 6 variants: DTM, ARTM-2/3, FHTM-2/3, and FHTM-3 (LLM) and 6 baselines: FM, MAR, MAR-Fluid, MAR-discrete, AR, AR-discrete. All models were evaluated on four benchmarks: PartiPrompts, GenEval, MS-COCO, and T2I-CompBench (Tables 1-9). In total, we considered 12 models and 27 metrics per model. To derive a single measure of overall performance, we assigned each model a rank from 1 (best) to 12 (worst) per metric and summed them across all benchmarks. Table 3 reports the global rank, where DTM substantially outperforms all other TM variants and baselines, followed by FHTM, ARTM, and MAR.

4.2 Evaluations

Sampling efficiency One important benefit in the DTM variant is its sampling efficiency compared to flow matching. In Table 10 we report CLIPScore and PickScore for DTM and FM for different numbers of backbone and head steps while in Table 11 we log the corresponding forward times. Notably, the number of backbone forwards in DTM sampling can be reduced con-

Table 4: FM and DTM sampling times.

siderably without sacrificing generation quality. Table 4 presents the superior sampling efficiency of DTM over FM: DTM achieves state-of-the-art results with only 16 backbone forwards, leading to an almost 7-fold speedup compared to FM, which requires 128 backbone forwards for optimal quality in this case. In contrast to DTM, ARTM/FHTM do not offer any speed-up, in fact they require backbone forwards equal to the number of transition steps times the number of image tokens, as specified in Tables 1,2,8; Figure 8 reports CLIPScore and PickScore for different number of head forwards which demonstrates that this number can be reduced up to 4 with some limited reduction in performance for ARTM/FHTM sampling.

Dependent vs. independent linear process To highlight the impact of the supervising process, we compare the linear process (10), where X 0 is sampled once for all t ∈ [ T ] , with the independent linear process (15), where X 0 ,t is sampled for each t ∈ [ T ] independently, on our autoregressive kernels: ARTM-3 and FHTM-3. The models are trained for 100K iterations and CLIPScore and PickScore are evaluated every 10K iterations. As shown in Figure 9, the independent linear process is far superior to the linear process on these kernels, see further discussion in Appendix A.4.

DTMKernel expressiveness The DTM kernel (see equation 13) generates each token y i of dimension 2 × 2 × 4 , corresponding to an image patch, independently in each transition step. This architecture choice is done mainly for performance reasons to allow efficient transitions. In Figure 10 we compare performance using a higher dimension y i , corresponding to a 2 × 8 × 4 patches. As can be seen in these graphs, performance improves for this larger patch kernel for low number of transition steps (1-4 steps) and surprisingly stays almost constant for very low number of head step, up to even a single step. The fact that performance does not improve for larger number of transition steps can be partially explained with Theorem 1 that shows that larger number of steps result in a simpler transition kernel (which in the limit coincides with flow matching).

5 Conclusions

We introduce Transition Matching (TM), a novel generative paradigm that unifies and generalizes diffusion, flow and continuous autoregressive models. We investigate three instances of TM: DTM, which surpasses state-of-the-art flow matching in image quality and text alignment; and the causal ARTM and fully causal FHTM that achieve generation quality comparable to non-causal methods. The improved performance of ARTM/FHTM comes at the price of a higher sampling cost, i.e., NFE counts are proportional to the number of transition steps, see e.g., in Table 1. DTM, in contrast, requires less backbone forwards and leads to significant speed-up over flow matching sampling, see e.g., Table 4. Future research directions include improving the training and/or sampling via different time schedulers and distillation, as well as incorporating FHTM in a multimodal system. Our work does not introduce additional societal risks beyond those related to existing image generative models.

References

Michael S Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants. arXiv preprint arXiv:2209.15571 , 2022.

Chameleon-Team. Chameleon: Mixed-modal early-fusion foundation models, 2025.

Huiwen Chang, Han Zhang, Lu Jiang, Ce Liu, and gledhhnddinerbdcilnulnfjWilliam T. Freeman. Maskgit: Masked generative image transformer, 2022.

Yushen Chen, Zhikang Niu, Ziyang Ma, Keqi Deng, Chunhui Wang, Jian Zhao, Kai Yu, and Xie Chen. F5-tts: A fairytaler that fakes fluent and faithful speech with flow matching. arXiv preprint arXiv:2410.06885 , 2024.

Mostafa Dehghani, Basil Mustafa, Josip Djolonga, Jonathan Heek, Matthias Minderer, Mathilde Caron, Andreas Steiner, Joan Puigcerver, Robert Geirhos, Ibrahim Alabdulmohsin, Avital Oliver, Piotr Padlewski, Alexey Gritsenko, Mario Luˇ ci´ c, and Neil Houlsby. Patch n' pack: Navit, a vision transformer for any aspect ratio and resolution, 2023.

Prafulla Dhariwal, Heewoo Jun, Christine Payne, Jong Wook Kim, Alec Radford, and Ilya Sutskever. Jukebox: A generative model for music, 2020.

Prafulla Dhariwal and Alex Nichol. Diffusion models beat gans on image synthesis, 2021.

Ming Ding, Zhuoyi Yang, Wenyi Hong, Wendi Zheng, Chang Zhou, Da Yin, Junyang Lin, Xu Zou, Zhou Shao, Hongxia Yang, and Jie Tang. Cogview: Mastering text-to-image generation via transformers, 2021.

Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, Dustin Podell, Tim Dockhorn, Zion English, Kyle Lacey, Alex Goodwin, Yannik Marek, and Robin Rombach. Scaling rectified flow transformers for high-resolution image synthesis, 2024.

Lijie Fan, Tianhong Li, Siyang Qin, Yuanzhen Li, Chen Sun, Michael Rubinstein, Deqing Sun, Kaiming He, and Yonglong Tian. Fluid: Scaling autoregressive text-to-image generative models with continuous tokens, 2024.

Dhruba Ghosh, Hanna Hajishirzi, and Ludwig Schmidt. Geneval: An object-focused framework for evaluating text-to-image alignment, 2023.

Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks, 2014.

Jiatao Gu, Yuyang Wang, Yizhe Zhang, Qihang Zhang, Dinghuai Zhang, Navdeep Jaitly, Josh Susskind, and Shuangfei Zhai. Dart: Denoising autoregressive transformer for scalable text-to-image generation, 2025.

Jian Han, Jinlai Liu, Yi Jiang, Bin Yan, Yuqi Zhang, Zehuan Yuan, Bingyue Peng, and Xiaobing Liu. Infinity: Scaling bitwise autoregressive modeling for high-resolution image synthesis, 2024.

Jack Hessel, Ari Holtzman, Maxwell Forbes, Ronan Le Bras, and Yejin Choi. Clipscore: A reference-free evaluation metric for image captioning, 2022.

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models, 2020.

Peter Holderrieth, Marton Havasi, Jason Yim, Neta Shaul, Itai Gat, Tommi Jaakkola, Brian Karrer, Ricky T. Q. Chen, and Yaron Lipman. Generator matching: Generative modeling with arbitrary markov processes, 2025.

Kaiyi Huang, Chengqi Duan, Kaiyue Sun, Enze Xie, Zhenguo Li, and Xihui Liu. T2icompbench++: An enhanced and comprehensive benchmark for compositional text-to-image generation, 2025.

Yang Jin, Zhicheng Sun, Ningyuan Li, Kun Xu, Kun Xu, Hao Jiang, Nan Zhuang, Quzhe Huang, Yang Song, Yadong Mu, and Zhouchen Lin. Pyramidal flow matching for efficient video generative modeling, 2025.

Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-based generative models. Advances in neural information processing systems , 35:26565-26577, 2022.

Diederik P. Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models, 2023.

Diederik P Kingma, Max Welling, et al. Auto-encoding variational bayes, 2013.

Yuval Kirstain, Adam Polyak, Uriel Singer, Shahbuland Matiana, Joe Penna, and Omer Levy. Pick-a-pic: An open dataset of user preferences for text-to-image generation, 2023.

Black Forest Labs. Flux. https://github.com/black-forest-labs/flux , 2024.

Tianhong Li, Yonglong Tian, He Li, Mingyang Deng, and Kaiming He. Autoregressive image generation without vector quantization. Advances in Neural Information Processing Systems , 37:56424-56445, 2024.

Tsung-Yi Lin, Michael Maire, Serge Belongie, Lubomir Bourdev, Ross Girshick, James Hays, Pietro Perona, Deva Ramanan, C. Lawrence Zitnick, and Piotr Dollár. Microsoft coco: Common objects in context, 2015.

Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747 , 2022.

Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.03003 , 2022.

Llama 3 Team Meta. The llama 3 herd of models, 2024.

Alex Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models, 2021.

Patrick, Robin Rombach, and Björn Ommer. Taming transformers for high-resolution image synthesis, 2021.

William Peebles and Saining Xie. Scalable diffusion models with transformers, 2023.

Dustin Podell, Zion English, Kyle Lacey, Andreas Blattmann, Tim Dockhorn, Jonas Müller, Joe Penna, and Robin Rombach. Sdxl: Improving latent diffusion models for high-resolution image synthesis, 2023.

A Polyak, A Zohar, A Brown, A Tjandra, A Sinha, A Lee, A Vyas, B Shi, CY Ma, CY Chuang, et al. Movie gen: A cast of media foundation models, 2025. URL https://arxiv. org/abs/2410.13720 , page 51.

Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. Learning transferable visual models from natural language supervision, 2021.

Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. Exploring the limits of transfer learning with a unified text-to-text transformer, 2023.

Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation, 2021.

Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models, 2022.

Christoph Schuhmann, Romain Beaumont, Richard Vencu, Cade Gordon, Ross Wightman, Mehdi Cherti, Theo Coombes, Aarush Katta, Clayton Mullis, Mitchell Wortsman, Patrick Schramowski, Srivatsa Kundurthy, Katherine Crowson, Ludwig Schmidt, Robert Kaczmarczyk, and Jenia Jitsev. Laion-5b: An open large-scale dataset for training next generation image-text models, 2022.

Neta Shaul, Ricky T. Q. Chen, Maximilian Nickel, Matt Le, and Yaron Lipman. On kinetic optimal probability paths for generative models, 2023.

Jascha Sohl-Dickstein, Eric A. Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics, 2015.

Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models, 2022.

Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations, 2021.

Peize Sun, Yi Jiang, Shoufa Chen, Shilong Zhang, Bingyue Peng, Ping Luo, and Zehuan Yuan. Autoregressive model beats diffusion: Llama for scalable image generation, 2024.

Yi Tay, Mostafa Dehghani, Vinh Q Tran, Xavier Garcia, Jason Wei, Xuezhi Wang, Hyung Won Chung, Siamak Shakeri, Dara Bahri, Tal Schuster, et al. Ul2: Unifying language learning paradigms. arXiv preprint arXiv:2205.05131 , 2022.

Keyu Tian, Yi Jiang, Zehuan Yuan, Bingyue Peng, and Liwei Wang. Visual autoregressive modeling: Scalable image generation via next-scale prediction, 2024.

Michael Tschannen, Cian Eastwood, and Fabian Mentzer. Givt: Generative infinite-vocabulary transformers, 2024.

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need, 2023.

Yibin Wang, Yuhang Zang, Hao Li, Cheng Jin, and Jiaqi Wang. Unified reward model for multimodal understanding and generation, 2025.

Zhisheng Xiao, Karsten Kreis, and Arash Vahdat. Tackling the generative learning trilemma with denoising diffusion gans, 2022.

Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li, Ming Ding, Jie Tang, and Yuxiao Dong. Imagereward: Learning and evaluating human preferences for text-to-image generation, 2023.

Yilun Xu, Mingyang Deng, Xiang Cheng, Yonglong Tian, Ziming Liu, and Tommi Jaakkola. Restart sampling for improving generative processes. Advances in Neural Information Processing Systems , 36:76806-76838, 2023.

Zhiyuan You, Xin Cai, Jinjin Gu, Tianfan Xue, and Chao Dong. Teaching large language models to regress accurate image quality scores using score distribution, 2025.

Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, Ben Hutchinson, Wei Han, Zarana Parekh, Xin Li, Han Zhang, Jason Baldridge, and Yonghui Wu. Scaling autoregressive models for content-rich text-to-image generation, 2022.

Lili Yu, Bowen Shi, Ramakanth Pasunuru, Benjamin Muller, Olga Golovneva, Tianlu Wang, Arun Babu, Binh Tang, Brian Karrer, Shelly Sheynin, et al. Scaling autoregressive multi-modal models: Pretraining and instruction tuning. arXiv preprint arXiv:2309.02591 , 2023.

Zhihang Yuan, Yuzhang Shang, Hanling Zhang, Tongcheng Fang, Rui Xie, Bingxin Xu, Yan Yan, Shengen Yan, Guohao Dai, and Yu Wang. E-car: Efficient continuous autoregressive image generation via multistage modeling, 2024.

Yichi Zhang, Yici Yan, Alex Schwing, and Zhizhen Zhao. Towards hierarchical rectified flow, 2025.

Chunting Zhou, Lili Yu, Arun Babu, Kushal Tirumala, Michihiro Yasunaga, Leonid Shamis, Jacob Kahn, Xuezhe Ma, Luke Zettlemoyer, and Omer Levy. Transfusion: Predict the next token and diffuse images with one multi-modal model, 2024.

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A Experiments

A.1 Implementation details

DiT architecture The DiT architecture [32] uses 24 blocks of a self-attention layer followed by cross attention layer with the text embedding [36], with a 2048 hidden dimension, 16 attention heads, and utilize a 3D positional embedding [5]. Embedded image [33] size is 32 × 32 × 4 and input to the DiT trough a patchify layer with patch size of 2 × 2 × 4 . The total number of parameters is 1.7B. Since ARTM gets as input both X t and X t +1 , which results in a longer sequence length, for a fair comparison across TM variations and baselines, for all other models we pad the input sequence to double its length.

LLM architecture The LLM architecture [29] is similar to the DiT with the following differences: (i) time injection is removed, (ii) cross attention layer is removed and text embedding is input as a prefix (iii) it uses a simple 1D instead of 3D positional embedding. To compensate for reduction in number of parameters, we increase the number of self-attention layers to 34, reaching 1.7B total number of parameters (comparable to the DiT). FHTM-DiT gets X 0 as input but does not take a loss on it, while for FHTM-LLM we remove the X 0 all together and instead use a single boi (begin of image) token to save sequence length.

Flow head architecture Following [25] we use an MLP with 6 layers and a hidden dimension of 1024. to convert from the backbone hidden dimension (2048) to the MLP hidden dimension (1024) we use a simple linear layer. Finally, we replace the time input with AdaLN[32] time injection.

Optimization The models are trained for 500K iterations, with a 2048 total batch size, 1 ∗ e -4 constant learning rate and 2K iterations warmup.

Classifier free guidance To support classifier free guidance (CFG), during training, with probability of 0 . 15 , we drop the text prompt and replace it with empty prompt. Following [25], during sampling, we apply CFG to the velocity of the flow head ( g θ ) with a guidance scale of 6 . 5 .

A.2 Main results: Text-to-image generation

Additional Kernels and Baselines Similar to the extension of the AR kernel to ARTM, We extend the MAR-Fluid kernel to 2 and 3 transition steps, resulting with the MARTM -2 and MARTM -3 kernels. Furthermore, we investigate the performance of the Restart sampling algorithm [52] on the FM kernel, were noise is added during the sampling process. We follow the authors' suggestion and perform 1 restart from t = 0 . 6 to t = 0 . 4 , 3 restarts from t = 0 . 8 to t = 0 . 6 , and an additional 3 restarts from t = 1 to t = 0 . 8 . The sampling is performed on a base of 1000 steps, resulting with a total of 2400 NFE. As an additional baseline, we sample the FM kernel with 2400 NFE. Results can be found in Tables 5,6,7.

Flow head NFE We ablate the number of NFE required by the flow head ( g θ ) to reach best performance for each model. As shown in Figure 8, we observe the models reach saturation with relatively low NFE, and decide to report results on Tables 1, 8 and 2 with 64 NFE for the flow head.

TMsteps vs Flow head NFE for DTM We test the performance of the DTM variant as function of TM steps and Flow head NFE. As shown in Table 10, our DTM model achieve reach saturation about

16 TM steps and 4 Flow head steps, according to CLIPScore and PickScore. Generation time for a single image on a single H100 GPU is provided in Table 11.

A.3 Sampling efficiency

(a) DTM CLIPScore

TMsteps

1

2

4

8

16

32

64

128

Head NFE

1

15.8

17.0

20.4

22.8

23.2

23.2

23.0

22.8

2

16.1

18.6

24.2

26.2

26.4

26.4

26.2

26.2

4

17.9

21.1

25.4

26.7

26.8

26.7

26.5

26.5

8

18.8

21.2

25.5

26.6

26.8

26.6

26.5

26.5

16

18.9

21.3

25.5

26.7

26.8

26.6

26.5

26.4

32

19.0

21.2

25.5

26.7

26.7

26.7

26.6

26.5

64

19.0

21.3

25.4

26.7

26.8

26.6

26.4

26.5

128

18.9

21.3

25.4

26.7

26.9

26.7

26.5

26.4

(b) DTM PickScore

TMsteps

1

2

4

8

16

32

64

128

Head NFE

1

17.6

17.8

18.6

19.4

19.6

19.7

19.6

19.6

2

17.7

18.3

19.7

20.6

20.9

21.0

21.0

21.0

4

18.1

18.8

20.0

20.8

21.1

21.1

21.1

21.1

8

18.3

18.8

20.0

20.8

21.1

21.1

21.1

21.2

16

18.3

18.8

20.0

20.9

21.1

21.1

21.1

21.1

32

18.3

18.8

20.0

20.9

21.1

21.1

21.1

21.1

64

18.3

18.8

20.0

20.9

21.1

21.1

21.1

21.1

128

18.3

18.8

20.0

20.8

21.1

21.1

21.1

21.1

(c) FM CLIPScore

Euler steps

1

2

4

8

16

32

64

128

0

15.8

16.6

19.7

23.8

25.6

25.9

25.9

26.0

(d) FM PickScore

Euler steps

1

2

4

8

16

32

64

128

0

17.9

18.0

18.7

20.0

20.8

21.0

21.0

21.0

1

2

4

8

16

32

64

128

19

20

21

22

23

24

25

26

27

CLIPScore

ARTM 3-steps (DiT)

FHTM 3-steps (DiT)

FHTM 3-steps (LLM)

1

2

4

8

16

32

64

128

18.5

19.0

19.5

20.0

20.5

21.0

PickScore

ARTM 3-steps (DiT)

FHTM 3-steps (DiT)

FHTM 3-steps (LLM)

Head NFE

Head NFE

A.4 Dependent vs. independent linear process

Further analysis of the generated images reveals that the AR kernels are unable to learn the linear process, resulting in low quality image generation. We hypothesize that the AR kernels exploit the linear relationship between X t and X t +1 during training, which leads the model to learn a degenerate function and causes it to fail in inference.

MS-COCO

PartiPrompts

0

20000

40000

60000

80000

Training iteration

10

12

14

16

18

20

22

24

26

CLIPScore

ARTM:Dependent

ARTM:Independent

FHTM:Dependent

FHTM:Independent

0

20000

40000

60000

80000

Training iteration

12

14

16

18

20

22

24

26

CLIPScore

ARTM:Dependent

ARTM:Independent

FHTM:Dependent

FHTM:Independent

0

20000

40000

60000

80000

Training iteration

17

18

19

20

PickScore

ARTM:Dependent

ARTM:Independent

FHTM:Dependent

FHTM:Independent

0

20000

40000

60000

80000

Training iteration

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

PickScore

ARTM:Dependent

ARTM:Independent

FHTM:Dependent

FHTM:Independent

A.5 DTMKernel expressiveness

1

2

4

8

16

32

64

128

TM steps

19

20

21

22

23

24

25

26

27

CLIPScore

Flow Head 2x2x4

Flow Head 2x8x4

1

2

4

8

16

32

64

128

Head NFE

23.5

24.0

24.5

25.0

25.5

26.0

26.5

CLIPScore

Flow Head 2x2x4

Flow Head 2x8x4

1

2

4

8

16

32

64

128

18.5

19.0

19.5

20.0

20.5

21.0

PickScore

Flow Head 2x2x4

Flow Head 2x8x4

1

2

4

8

16

32

64

128

19.6

19.8

20.0

20.2

20.4

20.6

20.8

21.0

21.2

PickScore

Flow Head 2x2x4

Flow Head 2x8x4

TM steps

Head NFE

A.6 Scheduler ablation for independent linear process

We have experimented with two transition scheduler options: uniform (as described in 15) and "exponential", i.e., t T ∈ { 0 , 0 . 5 , 0 . 75 , 1 } . The results for ARTM and FHTM are reported in Table 12 and show almost the same performance with a slight benefit towards exponential in DiT architecture and these are used in our main implementations.

A.7 Additional generated images comparison

FM

MAR

FHTM

DTM

'a coffee mug floating in the sky'

'

'The Alamo with bright white clouds above it'

'

'a tornado passing over a corn field'

'

'a harp with a carved eagle figure at the top'

FM

MAR

FHTM

'a living room with a large Egyptian statue in the corner'

FM

MAR

FHTM

DTM

'a blue wooden pyramid on top of a red plastic box'

'A bowl of soup that looks like a monster knitted out of woo'

'a dolphin in an astronaut suit'

g

'a moose standing over a fox'

FM

MAR

FHTM

DTM

'panda mad scientist'

2

'a futuristic city in synthwave style'

AR

ARTM-2

ARTM-3

'a half-peeled banana'

'beer'

n

0

START

SDR

T

SIARI

'the word 'START' '

A.8 Generation process visualization

'a comic about an owl family in the forest'

'A photo of a maple leaf made of water. '

STARTSTARTSTARTSTARSTARTSTARPTARTTARPTARTSTARPTAR

ESARSTARTSTARTSTARTSTARTTSTARTTSTARTSTARTSTARTTSTART

TART

TART

'the word 'START' written in chalk on a sidewalk'

'a cloud in the shape of a teacup'

3

SS

SS

sS

79.

'A giant cobra snake made from sushi'

'a stained glass window of a panda eating bamboo'

'a cross-section view of a walnut'

'A photo of a lotus flower made of water. '

'A map of the United States made out sushi. It is on a table next to a glass of red wine. '

'a green pepper cut in half on a plate'

'a cloud in the shape of a elephant'

'A photo of a teddy bear made of water. '

'a robot painted as graffiti on a brick wall. a sidewalk is in front of the wall, and grass is growing out of cracks in

the concrete. '

'a capybara'

'A television made of water that displays an image of a cityscape at night. '

A.9 Classifier free guidance sensitivity

4

6

8

10

12

14

Guidance scale

22

23

24

25

26

27

CLIPScore

DTM (DiT)

FHTM 3-step (LLM)

FM (DiT)

AR (DiT)

AR-discrete (DiT)

MAR (DiT)

MAR-fluid (DiT)

MAR-fluid-discrete (DiT)

4

6

8

10

12

14

Guidance scale

19.50

19.75

20.00

20.25

20.50

20.75

21.00

21.25

PickScore

3

4

5

6

7

8

9

10

11

12

13

14

'A raccoon wearing formal clothes, wearing a tophat and holding a cane. The raccoon is holding a garbage bag.

Oil painting in the style of Hokusai. '

'a stop sign knocked over on a sidewalk'

'a beach with a cruise ship passing by'

'a spaceship hovering over The Alamo'

STOP

STOP

STOP

STOP

STOP

[STOP]

[STOP]

[STOP]

STOPH

[STOP]

[STOP)

[STOP]

DS

STeG

Sep

STOP

SOP

[SOP]

[STOP)

[STOP)

[STOP)

STOP

[STOP!

STOP!

STOP

STOP

STOP

STOP

STOP

STOP

STOP

THEALCALINO

e1 e1 Pe 28 2g 2ge 28e 38e 09

'five red balls on a table'

B Training and sampling algorithms

Algorithms 1 and 2 describe and training and sampling (resp.) of transition matching for a general supervision process, kernel parametrization, and kernel modeling. In this section, we provide training and sampling algorithms tailored to the specific desgin choices of our three variants: (i) DTM is described in Figure 22, (ii) ARTM is described in Figure 23, and (iii) FHTM is described in Figure 24. Additionally, we provide Python code of a training step for each variant: (i) DTM in Figure 25, (ii) ARTM in Figure 26, and (iii) FHTM in Figure 27.

Algorithm 3 DTM Training

Algorithm 4 DTM Sampling

Algorithm 5 ARTM Training

Algorithm 6 ARTM Sampling

Algorithm 7 FHTM Training Require: p T ▷ Data Require: T ▷ Number of TM steps 1: while not converged do 2: Sample X T ∼ p T 3: parallel for t = 0 , ..., T do 4: Sample X 0 ,t ∼ N (0 , I d ) 5: X t ← ( 1 -t T ) X 0 ,t + t T X T 6: end for 7: parallel for t = 0 , ..., T -1 , i = 1 , ..., n do 8: h i t +1 ← f θ t ( X 0 , ..., X t , X <i t +1 ) 9: Sample Y i 0 ∼ N (0 , I d/n ) 10: Sample s ∼ U ([0 , 1]) 11: Y i s ← (1 -s ) Y i 0 + sX i t +1 12: L i t ( θ ) ← ∥ ∥ g θ s ( Y i s , h i t +1 ) -( X i t +1 -Y i 0 )∥ ∥ 2 13: end for 14: L ( θ ) ← 1 nT ∑ i,t L i t ( θ ) 15: θ ← θ -γ ∇ θ L ▷ Optimization step 16: end while 17: return θ Sample ( X t , Y ) ∼ q t,Y | T ( ·| X T ) L ( θ ) ← ˆ D ( Y, p θ Y | t ( ·| X t ))

Algorithm 8

Figure 24: n is the effective sequence length after patchify layer. The parallel for operations run simultaneously across the "sequence length" dimension of the tensor; ode_solve is any generic ODE solver for solving equation 8.

C Convergence of DTM to flow matching

Here we want to prove the following fact: Assume we have a sequence of Markov chains { X 0 , X h , X 2 h , . . . , X 1 } , with an initial state X 0 = x , where h = 1 T and T → ∞ . For convenience note that we index the Markov states with fractions ℓh , ℓ ∈ [ T ] , and we denote the RV

Assume the Markov chains satisfy:

The function f t ( x ) = E [ Y t | X t = x ] is Lipshcitz continuous. By Lipschitz we mean that ∥ f s ( y ) -f t ( x ) ∥ ≤ c L ( | s -t | + ∥ x -y ∥ ) .

Let k = k ( h ) ∈ N be an integer-valued function of h such that k →∞ and 1 2 ≥ kh → 0 as h → 0 . We will prove that the random variable

converges in mean to f 0 ( x ) . That is, we want to show

Theorem 2. Considering a sequence of Markov processes { X 0 , X h , X 2 h , . . . , X 1 } satisfying the assumptions above, then

Proof. First,

and if we open the squared norm we get three terms:

̸

We will later show that E [ Y ℓh | X 0 = x ] = f 0 ( x ) + O ( kh ) and for ℓ = m we have E [ Y ℓh · Y mh | X 0 = x ] = ∥ f 0 ( x ) ∥ 2 + O ( kh ) . Plugging these we get that equation 22 equals

as h → 0 , where we used assumption 2 above to bound E [ ∥ Y ℓh ∥ 2 | X 0 = x ] ≤ c ( x ) . Now to conclude we show

Now for m<ℓ we have

Therefore,

where we used equation 34 and equation 45, and the proof is done since kh → 0 as h → 0 .

C.1 The DTM case

We note show that the DTM process satisfies the two assumptions above. We recall that the DTM process is defined by Y t ∼ q Y | t ( ·| X t ) where Y = X 1 -X 0 .

First we check the Lipchitz property.

which is Lipschitz for t < 1 as long as p t | 1 ( x | x 1 ) > 0 for all x , and is continuously differentiable in t and x , both hold for the Gaussian kernel p t | 1 ( x | x 1 ) = N ( x | tx 1 , (1 -t ) I ) .

Let us check the second property. For this end we make the realistic assumption that our data is bounded, i.e., ∥ X 1 ∥ ≤ r for some constant r > 0 . Then, consider some RV X ′ 1 -X ′ 0 = Y t ∼ p Y | t ( ·| X t ) . Then by definition we have that X t + h = X t + h ( X ′ 1 -X ′ 0 ) and X t = (1 -t ) X ′ 0 + tX ′ 1 . Therefore,

We apply this to t + h = ℓh where ℓ ∈ [ k ] and therefore

where we used kh ≤ 1 2 . Finally,

where we used again X t = (1 -t ) X ′ 0 + tX ′ 1 . Lastly, applying this to t = ℓh ≤ kh ≤ 1 2 and using equation 60 we get

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a general framework called Transition Matching (TM), which unifies continuous-state autoregressive models. The framework is built upon three core design choices: the supervision process, kernel parameterization, and the modeling paradigm. By making specific selections within these dimensions, the authors propose three powerful new variants: DTM, ARTM, and FHTM. The empirical results show that the proposed method FHTM can match the quality of full-sequence flow model in image generation.\n\nSTRENGTHS AND WEAKNESSES:\n**[Distinctions to previous works]**\n\nThe paper proposes a general framework of continuous state autoregressive models by defining the three major design spaces. \n\nHowever, the overall writing was not easy to follow with unconventional notations and missing justifications and explanations. \nFirst, design space analysis has been widely explored in previous works [1] (diffusion), [2-3] (diffusion and flow), [4] (diffusion, flow, and \nCTMC). For me Sec. 2.1 seems as a preliminary, not a core component of the paper, and the author's claims of unifying flow, diffusion under TM is too bold. \n\nIn Sec. 2.2, the paper specifically focuses on continuous state autoregressive models, which I could not spot the differences of DTM from MAR [5]. While the authors propose modeling the latent difference, rather than the conventional $\\epsilon$-prediction in diffusion models or velocity prediction in flow models, I am not convinced why this is needed and such a design choice should give better results. Similarly, the use of independent linear process is not justified. What does it exactly mean by a better regularity of the conditional reverse kernel? \n\n\n**[DTM results]**\n\nThe authors show that DTM achieves the best performances to full sequence denoising such as FM. \nHowever, I find the results contradictory since the DTM also outperforms AR-based methods that adopt causal attention. \nAdditionally, how does DTM compete against FM in inference time or training stability? \n\n\nMinor typo: well-explored (L2), mixed use of discrete, continuous time (L125-126). \n\n\n\n[1] Denoising Diffusion Implicit Models\n\n[2] Flow Matching for Generative Modeling\n\n[3] Stochastic Interpolants: A Unifying Framework for Flows and Diffusions\n\n[4] Generator Matching: Generative modeling with arbitrary Markov processes\n\n[5] Autoregressive Image Generation without Vector Quantization\n\nQUESTIONS:\nPlease address the weaknesses above.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a novel auto-regressive generative model based on \"Transition Matching\". Here, the idea is to form a Markov model that takes some easily sampled data to the target distribution gradually based on a fixed process called \"supervising process\". The idea is then to use various transition modeling strategies and train them to generate data at the test time. This results in a generic and rich set of generative models, for which diffusion models and flow matching could be considered as its special cases. Because of the generality of the model, one would expect a greater flexibility (e.g. could be made auto-regressive as a result of the modeling choices) and potentially improved generative capabilities compared to previous models whose design choices are saturated for the most part. Three variants of the proposed model was trained on massive datasets \"350M licensed Shutterstock\" and tested based on PartiPrompts and MS-COCO benchmarks, based on various metrics such as CLIPScore, PickScore, etc.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper overall looks promising and clear. I have few concerns:\n\n- The problem of text-to-image generation is multi-facet problem and may previous work focused on improving specific aspects, e.g. compositionality, quality, bias, style transfer, etc. It is not clear which of these aspects is expected to be improved by the proposed method. With regard to compositionaliity, one would expect benchmarks such as T2I-CompBench and other related ones to be tested against. Furthermore, many of the proposed metrics lack alignment with human perception, making results less reliable. \n- The fact that DTM (Difference Transition Matching) gave better results compared to more complex ARTM and FHTM sounds puzzling and was not discussed in details in the paper. My question to the authors is why DTM outperforms such models. \n- As the authors mentioned in line 286 \"The improved performance of DTM/ARTM/FHTM comes at the price of a higher sampling cost\". But we know certain test-time scaling methods, such as noise optimization, that improve the image generation quality by large margins when given higher test-time compute budget (e.g. see ReNO). The authors should elaborate how their method compare to such schemes.\n\nQUESTIONS:\nPlease see above.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents Transition Matching (TM), a new class of generative models that merges the strengths of diffusion and flow-based approaches. TM broadens the generative design space by introducing non-deterministic probabilistic transitions and flexible, non-consecutive supervision. The authors develop three variants: Difference (DTM), Autoregressive (ARTM), and Full History (FHTM) Transition Matching. DTM enhances image quality and text consistency while speeding up sampling. As partially and fully causal models, ARTM and FHTM achieve generation quality on par with or exceeding non-causal methods and integrate easily with existing text generation technologies. In particular, FHTM is the first fully causal model in the continuous domain to outperform flow-based methods, excelling in text-to-image synthesis. The paper validates the advantages of the TM framework through extensive comparisons and outlines future work on improving efficiency and integrating FHTM into larger multimodal systems.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths:**\n\n1. The paper presents a novel generative model called Transition Matching (TM) that combines the advantages of diffusion and flow models. The authors introduce non-deterministic probability transfer kernels and arbitrary non-continuous supervised processes, significantly expanding the design space. They propose three different TM variants - Difference Transition Matching (DTM), Autoregressive Transition Matching (ARTM), and Full History Transition Matching (FHTM) - each achieving performance optimization in specific aspects.\n\n2. The proposed TM framework offers a powerful tool for generating high-quality samples while maintaining computational efficiency. It addresses limitations of existing approaches and provides a new direction for future research in the field of generative modeling. The authors highlight the significance of their work in terms of its potential impact on various applications such as image generation, text-to-image synthesis, and multi-modal systems.\n\n### **Weaknesses:**\n\n1. As authors stated, the improved performance of DTM/ARTM/FHTM comes at the price of a higher sampling cost.\n\n2. The analysis of the model's performance is simplistic.\n\nQUESTIONS:\n1. Has this method been validated on large-scale generative models?\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-05"}

comments: SUMMARY:
This paper introduces Transition Matching (TM), a general framework for generating discrete-time continuous-state sequences. TM unifies both flow-based methods and continuous autoregressive generation. The framework explores three novel variants: Difference Transition Matching, Autoregressive Transition Matching, and Full History Transition Matching. Experiments demonstrate that FHTM outperforms flow-based methods in the text-to-image task for continuous domains.

STRENGTHS AND WEAKNESSES:
**Strengths**

1. ARTM and FHTM generate continuous causal AR with quality comparable to non-causal approaches.

2. The proposed methods achieve state-of-the-art image quality and text adherence in text-to-image generation. They significantly improve text adherence compared to Flow matching and set a new state-of-the-art on this task.

**Weaknesses**

Experiments indicate that the Non-Formal Error (NFE) for ARTM/FHTM is high. Further analysis or experiments specifically focusing on the trade-off between quality and NFE for these causal models would be beneficial.

QUESTIONS:
Please refer to the weaknesses.

LIMITATIONS:
I’m unable to find the discussion about the limitation except line 157.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-08