Title: Exploring the Design Space of Diffusion Bridge Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion bridge models

stochastic interpolants

image-to-image translation
Abstract: Diffusion bridge models and stochastic interpolants enable high-quality image-to-image (I2I) translation by creating paths between distributions in pixel space. However, recent diffusion bridge models excel in image translation but suffer from restricted design flexibility and complicated hyperparameter tuning, whereas Stochastic Interpolants offer greater flexibility but lack essential refinements. We show that these complementary strengths can be unified by interpreting all existing methods within a single SI-based framework. In this work, we unify and expand the space of bridge models by extending Stochastic Interpolants (SIs) with preconditioning, endpoint conditioning, and an optimized sampling algorithm.
These enhancements expand the design space of diffusion bridge models, leading to state-of-the-art performance in both image quality and sampling efficiency across diverse I2I tasks. Furthermore, we identify and address a previously overlooked issue of low sample diversity under fixed conditions. We introduce a quantitative analysis for output diversity and demonstrate how we can modify the base distribution for further improvements.

Exploring the Design Space of Diffusion Bridge Models

Abstract

Diffusion bridge models and stochastic interpolants enable high-quality imageto-image (I2I) translation by creating paths between distributions in pixel space. However, recent diffusion bridge models excel in image translation but suffer from restricted design flexibility and complicated hyperparameter tuning, whereas Stochastic Interpolants offer greater flexibility but lack essential refinements. We show that these complementary strengths can be unified by interpreting all existing methods within a single SI-based framework. In this work, we unify and expand the space of bridge models by extending Stochastic Interpolants (SIs) with preconditioning, endpoint conditioning, and an optimized sampling algorithm. These enhancements expand the design space of diffusion bridge models, leading to state-of-the-art performance in both image quality and sampling efficiency across diverse I2I tasks. Furthermore, we identify and address a previously overlooked issue of low sample diversity under fixed conditions. We introduce a quantitative analysis for output diversity and demonstrate how we can modify the base distribution for further improvements. Code is available at https://github.com/szhan311/ECSI .

1 Introduction

Denoising Diffusion Models (DDMs) and flow matching create a stochastic process to transition Gaussian noise into a target distribution [33, 14, 34, 19]. Building upon this, diffusion bridge-based models (DBMs) have been developed to transport between two arbitrary distributions, π T and π 0 , including I2SB [21], DSBM [39], DDBM [18], DBIM [42], Bridge Matching [28]. DBMs achieve superior image quality in I2I translation compared to DDMs [18, 21, 2], primarily because the distance between source and target image distributions is typically smaller than that between Gaussian and target distributions.

While DBMs like DDBM [39], DBIM [42], and I2SB [21] achieve state-of-the-art FID scores in image-to-image translation, they suffer from limited design flexibility, constrained bridge path formulations, and complex parameter tuning. In contrast, Stochastic Interpolants (SIs) [1, 2] offer a simpler and more flexible framework, but they have yet to integrate practical advances from recent diffusion bridge models, such as preconditioning. Besides, SIs require training two separate models, unlike the more efficient single-model setup in DDBM. Table 1 summarizes the key characteristics of these methods, highlighting that their complementary strengths had not yet been unified.

Another overlooked issue stemming from restrictive design choices in previous bridge models is the lack of diversity in outputs. While some image translation tasks are one-to-one, we find that in one-to-many translation tasks, like black and white edges to color images, previous methods produce limited variation in colors and textures. We refer to this as the conditional diversity problem and show that our approach leads to significant improvements.

ECSI(Ours)

ECSI(Ours)

DBIM

DDBM

I2SB

DBIM

I2SB

DDBM

(a)

Bridge paths

(b)

Samplers

Deblurring

Edges to Hangbags

Our main contributions are as follows:

We propose Endpoint-Conditioned Stochastic Interpolants (ECSI), which extend stochastic interpolants by incorporating endpoint conditioning and preconditioning. Previous bridge methods artificially coupled unrelated aspects of the transition kernel. ECSI introduces a decoupled parametrization that expands and simplifies the design space for bridge paths and samplers. To further improve sampling quality and efficiency, we develop a novel noise control scheme and an efficient sampling algorithm.

We identify a previously overlooked issue: the low diversity of outputs conditioned on fixed source images. To address this, we propose modifying the base distribution. Furthermore, to quantitatively evaluate conditional output diversity, we introduce a new metric-Average Feature Diversity (AFD).

Experimental results demonstrate our model's state-of-the-art performance in both image quality and sampling speed across various I2I tasks, including deblurring, edges-to-handbags translation, and depth-to-RGB conversion. Notably, for handbag generation, our approach yields significantly more diverse outputs with varied colors and textures.

2 Background

Notations Let π T , π 0 , and π 0 T represent the base distribution, the target distribution, and the joint distribution of them respectively. π cond and π data represent the distributions of the input and output data. Let p be the distribution of a diffusion process; we denote its marginal distribution at time t by p t , the conditional distribution at time t given the state at time s by p t | s , and the distribution at time t given the states at times 0 and T by p t | 0 ,T , i.e., the transition kernel of a bridge.

2.1 Denoising Diffusion Bridge Models

DDBMs [18] extend diffusion models to translate between two arbitrary distributions π 0 and π T given samples from them. Consider a reference process given by:

whose transition kernel is given by q t | 0 ( x t | x 0 ) = N ( x t ; a t x 0 , σ 2 t I ) . This process can be conditioned (or "pinned") at both an initial point x 0 and a terminal point x T to construct a diffusion bridge. Under mild assumptions, the pinned process is given by Doob's h -transform [29]:

where ∇ X t log p T | t ( x T | X t ) = ( a t /a T ) x T -X t σ 2 t (SNR t / SNR T -1) and SNR t := a 2 t /σ 2 t [18]. Eq. (2) is a stochastic process that transport from p 0 = π 0 and p t = π t , which is a valid bridge process. To sample from the conditional distribution p ( x 0 | x T ) , we can solve the reverse SDE or probability flow ODE from t = T to t = 0 [18]:

where X T = x T , s = ∇ X t log p T | t ( x T | X t ) , h = ∇ X t log p t | T ( X t | x T ) . Generally, the score ∇ x t log p t | T ( x t | x T ) in Eqs. (3) and (4) is intractable. However, it can be effectively estimated by denoising bridge score matching. Let ( x 0 , x T ) ∼ π 0 ,T ( x 0 , x T ) , x t ∼ p t | 0 ,T ( x t | x 0 , x T ) , t ∼ U (0 , T ) , and ω ( t ) be non-zero loss weighting term of any choice, then the score ∇ x t log p T | t ( x T | x t ) can be approximated by a neural network s θ ( x t , x T , t ) with denoising bridge score matching objective [18]:

where E dentotes expectation over x t ∼ p t | 0 ,T ( x t , x 0 ) , ( x 0 , x T ) ∼ π 0 ,T , t ∼ U (0 , T ) .

2.2 Diffusion Bridge Implicit Models

The transition kernel of the bridge process in Eq. (2) is given by [18, 42]:

where α t = a t (1 -SNR T SNR t ) , β t = a t a T SNR T SNR t , γ 2 t = σ 2 t (1 -SNR T SNR t ) . Suppose we sample in reverse time on the discretized timesteps 0 = t 0 < t 1 < · · · t N -1 < t N = T . Then we can sample x 0 by the initial value x T and the updating rule:

where ˆ x θ 0 ( x t , x T , t ) has the relation with the score function:

3 Have the bridge paths been fully explored?

Given the forward process defined in Eq. (1), diffusion bridge models [18, 42, 12, 8] utilize Doob's h -transform to construct a corresponding bridge process (Eq. (2)). While the resulting process effectively bridges the initial distribution π T and the target distribution π 0 , such diffusion bridge approaches exhibit several limitations.

Parameter coupling. Notice that the parameters a t and σ t are convolved in the transition kernel (Eq. (6)). Such coupling is unnecessary and decoupling those parameters is helpful for searching the 'best' bridge path.

Limited design space. Despite Eq. (2) provides an infinite number of bridge paths by tuning a t and σ t , but the space of bridge paths is still artificially restricted.

In contrast, the stochastic interpolants [1] framework allows a larger design space of bridge path with more decoupled parameters. Specifically, stochastic interpolants build a bridge path directly via the flow map:

where z ∼ N (0 , I ) . Eq. (9) builds a transport with π 0 and π T as boundary conditions if the kernel parameters satisfy [1]:

α 0 = β T = 1 and α T = β 0 = γ 0 = γ 1 = 0 ;

α t , β t , γ t > 0 for t ∈ (0 , T ) .

The transition kernel of the stochastic interpolants in Eq. (9) is a Gaussian distribution: N ( x t ; α t x 0 + β t x T , γ 2 t I ) . Unlike DDBM, which is parameterized by only two variables a t and σ t , stochastic interpolants introduce decoupled parameters α t , β t , and γ t , offering a more flexible and expressive design space for constructing bridge paths.

A detailed discussion on the rationale behind the choices of α t , β t , and and an ablation study on the shape of γ t is provided in App. E. Notably, the DDBM-VP and DDBM-VE models presented in [18] can be considered as special cases by choosing different α t , β t , and γ t , see App. D for more details. In the experiments, we limit the scope to linear transition kernels and set T = 1 , i.e., p t | 0 ,T ( x t | x 0 , x T ) = N ( x t ; (1 -t ) x 0 + tx 1 , 4 γ 2 max t (1 -t ) I ) .

Stochastic interpolants expands the space of bridge paths and leads to decoupled parameters compared to DDBM and DBIM.

4 Has the sampler space been fully explored?

For diffusion models, EDM [17] demonstrated that the design of training and sampling schemes could be decoupled to significantly improve results. We now explore whether a similar decoupling is possible for bridge models, and what freedom we have to improve sampling quality with a given trained model.

4.1 Endpoint-Conditioning for Stochastic Interpolants (ECSI)

Given transition kernel p t | 0 ,T ( x t | x 0 , x T ) = N ( x t ; α t x 0 + β t x T ; γ t I ) , we can identify the training objective 11, reverse sampling SDEs (Eq. (10)), as demonstrated in Proposition 4.1, see App. C for the proof.

Proposition 4.1 (Endpoint-conditioned Stochastic Interpolants) . Suppose the transition kernel of a diffusion bridge process is given by p t | 0 ,T ( x t | x 0 , x T ) = N ( x t ; α t x 0 + β t x T , γ 2 t I ) , then the evolution of conditional probability q t ( X t | x T ) is given by the SDE:

where b ( t, x t , x T ) = ˙ α t ˆ x 0 + ˙ β t x T +(˙ γ t + ϵ t γ t )ˆ z t , ˆ x 0 = E [ x 0 | x t , x T ] , ˆ z t =: ( x t -α t ˆ x 0 -β t x T ) /γ t . Besides, ˆ x 0 can be approximated by neural networks ˆ x θ 0 by minimizing a regression objective with the observed x 0 , x T as targets,

where E denotes an expectation over ( x 0 , x T ) ∼ π ( x 0 , x T ) and x t ∼ p t ( x t | x 0 , x T ) .

Relation to Stochastic Interpolants (SI) . Both SI and ECSI in Prop. 4.1 can be seen as special cases of Conditioned SI. A key advantage of ECSI is its efficiency: while SI need to estimate two terms: E [ x 0 | x 0 ] and E [ x 1 | x t ] , ECSI only estimate E [ x 0 | x t , x 1 ] . A detailed comparison was demonstrated in App. B.

For training, we found that we could define an expanded space of bridge paths in terms of α t , β t , γ t , where γ t apparently controlled the stochasticity of the path. For sampling, we see from the proposition above that the sampling design space is expanded even further, as the sampling dynamics depend on α t , β t , γ t and ϵ t , where ϵ t appears as an additional degree of freedom to control stochasticity.

Training. Eq. (11) provides the training objective of the denoiser ˆ x θ 0 ( t, x t , x T ) . In the implementation, we include additional preconditioning as DDBM [18] and DBIM [42], see App. G for more details.

Sampling. We can generate samples from the conditional distribution q 0 | T ( x 0 | x T ) by solving the stochastic differential equation in Eq. (10) from t = T to t = 0 .

4.2 Existing samplers are a strict subset of ECSI samplers

We now show that existing samplers implement a strict subset of the ECSI samplers, see Figure 1.

DDBMsampler. When ϵ t = 0 , Eq. (10) reduces to a deterministic ODE. Setting ϵ t = γ t ˙ γ t -˙ α t α t γ 2 t recovers the sampling SDE used in DDBM [18]. However, DDBM only provides a single reverse SDE and a single corresponding reverse ODE; it does not explore alternative choices of ϵ t .

DBIMsampler. For small enough ∆ t and γ 2 t -∆ t -2 ϵ t ∆ t > 0 , the sampling SDE can be discretized as:

where ¯ z t ∼ N (0 , I ) , ˜ z = √ γ 2 t -∆ t -2 ϵ t ∆ t ˆ z t + √ 2 ϵ t ∆ t ¯ z t . Eq. (12) recover the DBIM sampler. Note that the condition γ 2 t -∆ t -2 ϵ t ∆ t > 0 limits the design space of samplers. For example, our best result in the experiments is achieved by setting α t = 1 -t , γ t = γ 2 max 4 t (1 -t ) and ϵ t = γ t ˙ γ t -˙ α t α t γ 2 t , DBIM sampler fails under this setting since γ 2 t -∆ t -2 ϵ t ∆ t > 0 cannot be guaranteed all the time.

I 2 SB sampler. When 2 ϵ t ∆ t = γ 2 t -∆ t -β 2 t -∆ t γ 2 t /β 2 t , the coefficient of x T in Eq. (12) vanishes. This special case corresponds to the Markovian bridge introduced in [42], and notably allows us to recover the sampling procedure of I2SB [21]. We provide a detailed derivation of this connection in Appendix D. The design space of the I 2 SB sampler is also limited, as it can be interpreted as a special case of the DBIM sampler.

Endpoint-Conditioned Stochastic Interpolants (Prop. 4.1) identify a class of sampling SDEs that share the same marginal distribution, but offer greater flexibility and a broader design space for sampler construction compared to DDBM, DBIM, and I2SB.

4.3 Our implementation

Our sampler based on Euler's discretization of the sampling SDE in Eq. (10):

We set ϵ t = η ( γ t ˙ γ t -˙ α t α t γ 2 t ) , where η ∈ (0 , 1) is an interpolation parameter. This formulation provides continuous control over the sampling process, ranging from purely deterministic ODE sampling ( η = 0 ) to fully stochastic SDE sampling ( η = 1 ). In our implementation, we let ϵ t = 0 for the last two steps, Eq. (12) gets reduced to: x t -∆ t ≈ α t -∆ t ˆ x 0 + β t -∆ t x T + γ t -∆ t ˆ z t . For other steps, we apply Eq. (13) and let ϵ t = η ( γ t ˙ γ t -˙ α t α t γ 2 t ) , where η is a constant. Putting all ingredients together leads to our sampler outlined in Algorithm 1.

Algorithm 1 ECSI Sampler

Diffusion path

N(0,b²I)

N(0, b²m)

Our bridge path

Previous bridge path

Tcond

Tcond

* N(O, b²I)

Tcond * N(0, b²I)

Tcond

Tdata

(a) Bridge paths in state space

(b) Bridge paths in density space

5 Is there any benefit to modifying the starting point of a bridge?

We expanded the paths in distribution space connecting a base and target distribution, but so far left the endpoints fixed. While the target distribution should remain fixed, we could, in principle, modify the base distribution. At first glance this seems counter-intuitive - because of the data processing inequality we can only lose information about the target by modifying the base distribution. Hence, this angle has not been explored in the bridge literature. However, we found a surprising result modifying the base distribution can help significantly. The situation is analogous to the benefits of lossy compression in VAEs [6]. Information in the base distribution is not necessarily helpful, so by modifying the base distribution (which destroys some information) the model can align better with natural factors of variation.

5.1 Low conditional diversity in one-to-many translations

In our experiments (see Sec. 6), we observe that existing diffusion bridge models tend to produce low-diversity outputs under fixed conditioning. For instance, when generating handbags from a single edge map, the model is expected to produce varied outputs in terms of color, texture, and fine details. However, we find that current bridge models generate visually similar images across different sampling runs, despite the injection of different noise realizations during the diffusion process.

To address the issue of low output diversity, we propose modifying the base distribution used in the bridge model. Prior works [18, 1] typically treat the base distribution π T as equivalent to the input data distribution, denoted π cond . In contrast, our approach introduces a controlled perturbation by redefining the base distribution as π T = π cond ∗ N (0 , b 2 I ) , where b is a constant that governs the magnitude of noise added to the input distribution. This modification enables greater diversity in the generated outputs while maintaining conditional alignment.

Intuitively, this modification can be interpreted as a trade-off between standard diffusion models and traditional diffusion bridge models. As illustrated in Fig. 3, diffusion models typically generate samples starting from pure Gaussian noise, while diffusion bridge models begin sampling from fully conditioned inputs, such as edge maps. Our approach introduces an intermediate regime by

sampling from noisy conditioned inputs, thereby blending the benefits of both paradigms-preserving conditional guidance while enhancing output diversity.

Modifying the base distribution with lossy compression can significantly improve the conditional diversity of the generated images.

5.2 How to measure the conditional diversity?

While existing metrics like FID implicitly capture the unconditional diversity of generated images, we need to capture the diversity of outputs (e.g. color images) for a single input image (a black and white edge map). To measure the conditional diversity, we will adopt Vendi Score (VS) [11] as a metric. Besides, We propose the Average Feature Distance (AFD) metric to quantify the conditional diversity among generated images. Initially, we select a group of source images { x ( i ) T } i M =1 . For each x ( i ) T , we then generate L distinct target samples. The j -th generated sample corresponding to the i -th source image is denoted by y ij . Then the AFD is calculated as follows:

̸

where F ( · ) is a function that extracts the features of images, and ∥ · ∥ represents Euclidean norm. Intuitively, a larger AFD indicates the better conditional diversity. Here, F ( x ) can be x to evaluate the diversity directly in the pixel space. Alternatively, F ( · ) can be defined using the Inception-V3 model to assess the diversity in the latent space. In our experiments, we use AFD in latent space. Furthermore, we provide additional justification for the validity of our proposed metric in App. A.

A comparison between AFD and VS . Both AFD and the VS quantify diversity in the feature space of images, using features extracted from the Inception-V3 model. AFD measures the average pairwise Euclidean distance between feature vectors, making it sensitive to outliers. In contrast, the Vendi Score evaluates diversity by computing the effective number of unique feature patterns, based on the eigenvalues of the similarity matrix, emphasizing the overall structural diversity of the feature set. These metrics are complementary, capturing different aspects of diversity.

6 Experiments

In this section, we demonstrate how greatly expanding the space of bridge paths with ECSI leads to significantly improved performance for I2I translation tasks, in terms of sample efficiency, image quality and conditional diversity. We evaluate on I2I translation tasks on Edges → Handbags [16] scaled to 64 × 64 pixels and DIODE-Outdoor scaled to 256 × 256 [37], and Deblurring on ImageNet dataset [9]. For evaluation metrics, we use Fréchet Inception Distance (FID) [13] for all experiments, and additionally measure Inception Scores (IS) [3], Learned Perceptual Image Patch Similarity (LPIPS) [41], Mean Square Error (MSE), following previous works [42, 18]. In addition, we use VS and AFD, Eq. 14, to measure conditional diversity. Further details of the experiments and design guidelines are provided in Appendix G and E.

Sampler . We evaluate different sampling algorithms in Fig. 4 (a), the results demonstrate that setting ϵ t = 0 and using Eq. (12) for the last 2 steps can significantly improve sampled image

300

DDBM sampler

50

Ymax= 0.025

15.0

Ymax = 0.125,b = 0

250

A+B

A

40

Ymax = 0.25

Ymax = 0.125

12.5

Ymax=0.25,b =0

Ymax = 1.0, b = 0

200

A+C

Ymax = 0.5

10.0

Ymax = 0.125, b = 0.5

150

A+B+C

30

Ymax= 1

FID

7.5

100

20

5.0

50

10

2.5

0

10

20

30

40

50

1

2

3

4

5

6

7

0.0

0.2

0.4

0.6

0.8

1.0

1.2

NFE

Iterations (x105)

n

(b). Ablation study on max for

(c).Ablation study on Et for

(a). Ablation study on discretization.

SDBmodelwithLinearpath.

SDBmodelwithLinearpath.

quality compared with simple Euler discretization and DDBM sampler. Furtheremore, By specifically designing noise control during sampling, our sampler surpasses the sampling results by DDBM and DBIM with the same pretrained model. The results are demonstrated in Table 2. We set the number of function evaluations (NFEs) from the set [5 , 10 , 20] .

Bridge paths . We introduced an extensive bridge design space and begin by focusing on linear transition paths with different strength of maximum stochasticity, i.e., p t | 0 ,T ( x t | x 0 , x T ) = N ( x t ; (1 -t ) x 0 + tx T , 1 4 γ 2 max t (1 -t ) I ) . We conducted detailed ablation studies on γ max and η for the Linear path on DIODE ( 64 × 64 ) dataset, as shown in Fig. 4 (b) and (c). The optimal values for γ max were found to be 0 . 125 and 0 . 25 , while the best performance for η was achieved with η = 0 . 8 and η = 1 . 0 . Performance deteriorates when either parameter is too small or too large. Based on the results of these ablation studies, we further trained ECSI models on the Edges2handbags ( 64 × 64 ) and DIODE ( 256 × 256 ) datasets by taking γ max ∈ { 0 . 125 , 0 . 5 } and setting η = 1 . 0 . The results are presented in Table 3. Our models establish a new benchmark for image quality, as evaluated by FID, IS and LPIPS. Despite our models having slightly higher MSEs compared to the baseline DDBM and DBIM, we believe that a larger MSE indicates that the generated images are distinct from their references, suggesting a richer diversity.

Modifying base distribution . Through controlling noise in the base distribution, we achieved a more diverse set of sample images, while this diversity comes at the cost of slightly higher FID scores and slower sampling speed. We show generated images in Fig. 5. More visualization can be found in Appendix I, which shows that by introducing booting noise to the input data distribution, the model can generate samples with more diverse colors and textures. Further quantitative results are presented

DBIM

NFE=10,FID=2.46,AFD=5.20

DDBM

NFE=118,FID=1.83,AFD=6.99

AFD

FID

Linear (b = 0.5)

NFE=10,FID=2.07,AFD=9.35

in Table 4, confirming that our model surpasses the vanilla DDBM in terms of image quality, sample efficiency, and conditional diversity.

Deblurring on ImageNet Dataset . We evaluate our models for Gaussian deblurring applying a Gaussian kernel with σ = 10 and Uniform deblurring, shown in Table 5. The results demonstrates that our ECSI models achieve much lower FID score.

7 Related Work

Diffusion Bridge Models . Diffusion bridges are faster diffusion processes that could learn the mapping between two random target distributions [39, 35], demonstrating significant potential in various areas, such as protein docking [32], mean-field game [20], I2I translation [21, 18]. According to different design philosophies, DBMs can be divided into two groups: bridge matching and stochastic interpolants. The idea of bridge matching was first proposed by Peluchetti et al. [28], and can be viewed as a generalization of score matching [34]. Based on this, diffusion Schrödinger bridge matching (DSBM) has been developed for solving Schrödinger bridge problems [35, 39]. In addition, Liu et al. [21] utilize bridge matching to perform image restoration tasks and noted benefits of noise empirically, the experiments shows the new model is more efficient and interpretable than score-based generative models [21]. Furthermore, our benchmark DDBM [18] achieve significant improvement for various I2I translation tasks, DBIM [42] improved the sampling algorithm for DDBM, significantly reducing sampling time while maintaining the same image quality.

Image-to-Image Translations . While diffusion models are strong at generating images, applying them to image-to-image (I2I) translation is more difficult due to artifacts in the output. DiffI2I improves quality and alignment with fewer diffusion steps [5]. In latent space, S2ST speeds up translation and reduces memory use [27]. Other methods improve guidance using features like frequency control [26, 15, 38]. A common challenge is that many models require joint training on both source and target domains, raising privacy concerns. Injecting-Diffusion tackles this by isolating shared content for unpaired translation [24]. SDDM improves interpretability by breaking down the score function across diffusion steps [30].

8 Conclusion

We introduced Endpoint-Conditioned Stochastic Interpolants (ECSI)-an improved version of stochastic interpolants that adds endpoint conditioning, modifies the base distribution, and uses discretization to explore the design space of Diffusion Bridge Models (DBMs). We highlighted a key issue often overlooked: one-to-many image translation tasks lack conditional diversity. Our findings show that resolving this requires adjusting the starting distribution, not the path or sampler. ECSI sets new benchmarks in image quality, sampling efficiency, and conditional diversity on tasks like 64 × 64 edges2handbags, 256 × 256 DIODE-outdoor, and ImageNet deblurring.

Limitations . (i) We note that optimal path design may vary by task, leaving room for future refinement. (ii) Incorporating guidance techniques may further enhance model performance.

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Narek Tumanyan, Michal Geyer, Shai Bagon, and Tali Dekel. Plug-and-Play Diffusion Features for Text-Driven Image-to-Image Translation. Computer Vision and Pattern Recognition , 2023.

Or Greenberg, Eran Kishon, and Dani Lischinski. S2ST: Image-to-Image Translation in the Seed Space of Latent Diffusion. arXiv.org , 2023.

Stefano Peluchetti. Non-denoising forward-time diffusions. arXiv preprint arXiv:2312.14589 , 2023.

L Chris G Rogers and David Williams. Diffusions, Markov processes, and martingales: Itô calculus , volume 2. Cambridge university press, 2000.

Shurong Sun, Longhui Wei, Junliang Xing, Jia Jia, and Qi Tian. SDDM: Score-Decomposed Diffusion Models on Manifolds for Unpaired Image-to-Image Translation. International Conference on Machine Learning , 2023.

Kihyuk Sohn, David Berthelot, Nicholas Carlini, Zizhao Zhang, Han Zhang, Colin A Raffel, Ekin Dogus Cubuk, Alexey Kurakin, and Chun-Liang Li. Fixmatch: Simplifying semisupervised learning with consistency and confidence. Advances in neural information processing systems , 33:596-608, 2020.

Vignesh Ram Somnath, Matteo Pariset, Ya-Ping Hsieh, Maria Rodriguez Martinez, Andreas Krause, and Charlotte Bunne. Aligned diffusion schrödinger bridges. In Uncertainty in Artificial Intelligence , pages 1985-1995. PMLR, 2023.

Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems , 32, 2019.

Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456 , 2020.

Stefano Peluchetti. Diffusion Bridge Mixture Transports, Schr\"odinger Bridge Problems and Generative Modeling. arXiv.org , 2023.

Xuan Su, Jiaming Song, Chenlin Meng, and Stefano Ermon. Dual diffusion implicit bridges for image-to-image translation. arXiv preprint arXiv:2203.08382 , 2022.

Igor Vasiljevic, Nick Kolkin, Shanyi Zhang, Ruotian Luo, Haochen Wang, Falcon Z Dai, Andrea F Daniele, Mohammadreza Mostajabi, Steven Basart, Matthew R Walter, et al. Diode: A dense indoor and outdoor depth dataset. arXiv preprint arXiv:1908.00463 , 2019.

Xiang Gao, Zhengbo Xu, Junhan Zhao, and Jiaying Liu. Frequency-Controlled Diffusion Model for Versatile Text-Guided Image-to-Image Translation. AAAI Conference on Artificial Intelligence , 2024.

Yifeng Shi, Valentin De Bortoli, Andrew T. Campbell, and Arnaud Doucet. Diffusion Schr\"odinger Bridge Matching. Neural Information Processing Systems , 2023.

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A AFD validation

In this section, we thoroughly validate the effectiveness of our proposed metric, AFD, for measuring conditional diversity and demonstrate its role as a complementary metric to FID. In unconditional generation scenarios, the FID is widely used to evaluate the diversity of generated images. While low FID scores generally indicate high diversity across the entire dataset, they do not necessarily imply high conditional diversity. For instance, we observed that samples generated by the DDBM model often lack diversity when conditioned on edge images, despite achieving very low FID scores. To address this limitation, we introduce the concept of conditional diversity and propose a corresponding metric to quantify it.

The first question is why FID failed to measure the conditional diversity. To illustrate the limitations of FID in capturing conditional diversity, consider an extreme case: if the images generated by a generative model are identical to a set of baseline images, the FID score can be very low since the two distributions are indistinguishable. However, this scenario does not reflect diversity within the conditional outputs.

To further support our point, we designed two classes of pseudo-generative models capable of controlling the diversity of the generated images, which are further validated by FID and AFD. The experiments are evaluated on Imagenet dataset [9].

A.1 Pseudo-generative models by random selection

We designed four pseudo-generative models: ImageNet-1-mode, ImageNet-2-modes, ImageNet-5modes, and ImageNet-10-modes. The experimental setup is as follows:

We selected 11,000 samples from the ImageNet validation dataset, randomly choosing 11 images per class.

From these, we designated 1,000 images as the "real" set, while the remaining images served as the source pool for the generative models.

Each ImageNet-k-modes model simulates a generative process by randomly sampling images from a pool of k distinct images within a given class.

We present sampled images in Fig. 6, where it is evident that the ImageNet-10-modes model generates images with the highest conditional diversity. To quantify this, we conducted experiments to calculate both FID and AFD for the four generative models. The results are summarized in Table 6. While the FID scores are nearly identical across all models, the AFD values increase as the conditional diversity of the generative models improves. This highlights that AFD is a more effective metric for capturing conditional diversity than FID.

A.2 Pseudo-generative models by strong augmentation

Strong augmentation has been widely used in computer vision to generate synthetic data while preserving its underlying semantics [7, 40, 31, 4]. The intensity of augmentation can be adjusted, with higher intensities producing more diverse images. To further validate our proposed metric, AFD, as a measure of diversity, we construct pseudo-generative models using strong augmentation.

We selected 1,000 images from the ImageNet-1k dataset, one from each category. These images were subjected to data augmentation, specifically using ColorJitter, with varying magnitudes to enhance diversity. For each image, the augmentation was applied 16 times, creating an augmented dataset for

each magnitude setting. We then calculated the AFD for these augmented datasets to evaluate the relationship between dataset diversity (as influenced by augmentation magnitude) and the AFD value.

Table 7 summarizes the AFD results across various augmentation magnitude settings. The results show that as diversity increases, AFD values also rise, further confirming that the proposed AFD metric is a reliable indicator of image diversity.

B Relation to Stochastic Interpolants

Conditioned Stochastic Interpolants build a marginal probability path p t | y using a mixture of interpolating densities: p t | y ( x ) = ∫ p t ( x t | x 0 , x 1 ) π ( x 0 , x 1 | y ) dx 0 dx 1 , where π ( x 0 , x 1 | y ) is a joint distribution with marginals π 0 | y ( x 0 | y ) and π 1 | y ( x 1 | y ) . For linear interpolants given by: X t = α t X 0 + β t X 1 + γ t z . The conditional kernel p t ( x t | x 0 , x 1 ) is given by a Gaussian distribution:

p t ( x t | x 0 , x 1 ) = N ( α t x 0 + β t x 1 , σ 2 t I ) , ∀ t ∈ [0 , 1] . Then we can sample from the conditional distribution p 0 | y ( x 0 | y ) by running a stochastic process p t | y ( x t | y ) from time t = 1 to t = 0 , which is given by the following SDE:

where the drift term b ( t, x, y ) is:

As y represents null conditioning, Eq. (15) recover the original sampler of Stochastic Interpolants. In the drift term, E [ x 0 | x, y ] , E [ x 1 | x, y ] and E [ z | x, y ] are unknown, but we only need to estimate two of them, since

We can further reduce the number of unknown term by endpoint-conditioning. Here as we replace condition y to be endpoint x 1 , the term E [ x 1 | x, x 1 ] = x 1 . So we have:

This is exactly the sampler for ECSI in Prop. 4.1. Therefore, both SI and ECSI can be seen as special cases of Conditioned SI. A key advantage of ECSI is its efficiency: while SI need to estimate two terms: E [ x 0 | x t ] and E [ x 1 | x t ] , ECSI only estimates E [ x 0 | x t , y ] .

C Proofs

There are infinitely many pinned processes characterized by the Gaussian transition kernel p t | 0 ,T ( x t | x 0 , x T ) = N ( x t ; α t x 0 + β t x T , γ 2 t I ) . Specifically, we formalize the pinned process as a linear Itô SDE, as presented in Lemma C.1.

Lemma C.1. There exist a linear Itô SDE

where f t = ˙ α t α t , s t = ˙ β t -˙ α t α t β t , g t = √ 2( γ t ˙ γ t -˙ α t α t γ 2 t ) , that has a Gaussian marginal distribution N ( x t ; α t x 0 + β t x T , γ 2 t I ) .

Proof. Let m t denote the mean function of the given Itô SDE, then we have d m t dt = f t m t + s t x T . Given the transition kernel, the mean function m t = α t x 0 + β t x T , therefore,

Matching the above equation:

Further, For the variance γ 2 t of the process, the dynamics are given by:

Solving for g 2 t , we substitute f t = ˙ α t α t :

Given the pinned process (18), we can sample from the conditional distribution p 0 | T ( x 0 | x T ) by solving the reverse SDE or ODE from t = T to t = 0 :

where the score ∇ X t log p t ( X t | x T ) can be estimated by score matching objective (5).

For dynamics described by ODE d X t = u t dt , we can identify the entire class of SDEs that maintain the same marginal distributions, as detailed in Lemma C.2. This enables us to control the noise during sampling by appropriately designing ϵ t .

Lemma C.2. Consider a continuous dynamics given by ODE of the form: d X t = u t dt , with the density evolution p t ( X t ) . Then there exists forward SDEs and backward SDEs that match the marginal distribution p t . The forward SDEs are given by: d X t = ( u t + ϵ t ∇ log p t ) dt + √ 2 ϵ t d W t , ϵ t > 0 . The backward SDEs are given by: d X t = ( u t -ϵ t ∇ log p t ) dt + √ 2 ϵ t d W t , ϵ t > 0 .

Proof. For the forward SDEs, the Fokker-Planck equations are given by:

This is exactly the Fokker-Planck equation for the original deterministic ODE d X t = u t dt . Therefore, the forward SDE maintains the same marginal distribution p t ( X t ) as the original ODE.

Now consider the backward SDEs, the Fokker-Planck equations become:

This is again the Fokker-Planck equation corresponding to the original deterministic ODE d X t = u t dt . Therefore, the backward SDE also maintains the same marginal distribution p t ( X t ) .

Therefore,

Lemma C.3. Let ( x 0 , x T ) ∼ π 0 ( x 0 , x T ) , x t ∼ p t ( x | x 0 , x T ) , Given the transition kernel: p ( x t | x 0 , x T ) = N ( x t ; α t x 0 + β t x T , γ 2 t I ) , if ˆ x 0 ( x t , x T , t ) is a denoiser function that minimizes the expected L 2 denoising error for samples drawn from π 0 ( x 0 , x T ) :

then the score has the following relationship with ˆ x 0 ( x t , x T , t ) :

we can minimize L ( D ) by minimizing L ( D ; x t , x T ) independently for each { x t , x T } pair.

Thus we conclude the proof.

Proof of Prop. 4.1.

Proof.

̸

Proof. Recall Eqs. (24) (25) and Lemma C.2,

Next we take the reparameterized score in Eq. (34) into Eq. (48):

D Reframing previous methods in our framework

We draw a link between our framework and the diffusion bridge models used in DDBM.

D.1 DDBM-VE

DDBM-VE can be reformulated in our framework as we set :

Proof. In the origin DDBM paper, the evolution of conditional probability q ( x t | x T ) has a time reversed SDE of the form:

and an associated probability flow ODE

Compare Eqs. (54) and 55 with Lemma C.1. We only need to prove:

In the original paper,

Therefore,

In our framework, f t , s t , g 2 t can be calculated:

Therefore,

which matches the formulation in DDBM.

D.2 DDBM-VP

DDBM-VP can be reformulated in our framework as we set :

Proof. In the original DDBM-VP setting,

Therefore,

In our framework, f t , s t , g 2 t can be calculated:

which matches the formulation in DDBM.

D.3 EDM

ODE formulation . The ODE formulation in EDM can be formlated in our framework as we set α t = 1 , β t = 0 , γ t = σ t .

Proof. Recall 25, the ODE formulation is given by:

Therefore,

where f t = ˙ α t α t , s t = ˙ β t -˙ α t α t β t , g t = √ 2( γ t ˙ γ t -˙ α t α t γ 2 t ) . As α t = 1 , β t = 0 , γ t = σ t , The sampling ODE is given by:

Sampling SDEs with noise added . Recall Proposition 4.1, as α t = 1 , β t = 0 , γ t = σ t , then the SDE has the form:

Now we recover the stochastic sampling SDE in original EDM paper.

D.4 I2SB

I2SB can be reformulated in our framework as we let:

where σ 2 t := ∫ t 0 β τ dτ .

When 2 ϵ t ∆ t = γ 2 t -∆ t -β 2 t -∆ t γ 2 t /β 2 t , the coefficient of x T in Eq. (12) vanishes. Thus, Eq. (12) can be simplified as:

Using discretization in Eq. (85):

In the I2SB paper, define a 2 n := ∫ t n +1 t n β τ d τ , σ 2 n := ∫ t n 0 β τ d τ . Therefore,

Thus, we reproduce the sampler of I2SB.

0.25

25

k=0.5

k=0.7

0.20

k=1.0

20

0.15

k=2.0

k=4.0

0.10

10

0.05

0.00

5

0.0

0.2

0.4

0.6

0.8

1.0

1

2

3

4

5

6

7

T

8

Iterations (x105)

(b) Ablation study on the shape of Yt.

Ymax

= 0.025

Ymax

= 0.125

Ymax

= 0.25

Ymax

=0.5

Ymax

=1

E Additional design guideline

α t and β t . Theoretically, α t and β t can be freely designed, and future work may explore alternative design choices. However, in this paper, we focus on the simple case where α t = 1 -t and β t = t . The rationale is as follows: consider the scenario where α t = 1 -β t , which represents an interpolation along the line segment between x 0 and x 1 . For the path p (1) t ( x ) = N ((1 -β t ) x 0 + β t x 1 , γ 2 t I ) , where β t is invertible, it is straightforward to construct another path p (2) t ( x ) = N ((1 -t ) x 0 + tx 1 , γ 2 β -1 t I ) , which achieves the same objective function but uses a different distribution of t during training. Based on this equivalence, setting α t = 1 -t and β t = t is a reasonable choice.

The shape of γ t . We conducted an ablation study on γ t with different shapes. Specifically, we assumed γ t has the form γ t = 2 γ max √ t k (1 -t k ) , as shown in Fig. 7, γ t will have different shape as we set different k . The results indicate that the best performance is achieved when k = 1 , which is the exact setting used in this paper.

γ max . Our ablation studies on γ max demonstrate that the optimal values of γ max are approximately 0 . 125 or 0 . 25 . Furthermore, the sampling paths corresponding to different choices of γ t are shown in Fig. 8. Adding an appropriate amount of noise to the transition kernel helps in constructing finer details.

ϵ t . We use the setting ϵ t = η ( γ t ˙ γ t -˙ α t α t γ 2 t ) . The ablation studies on ϵ t demonstrate that the optimal choice of η for the DDBM-VP model is approximately 0 . 3 , while the best choice for the ECSI model with a Linear Path is around 1 . 0 . Additionally, we present sample paths and generated images under different η settings to illustrate heuristic parameter tuning techniques. The results are shown in Figures 10, 11, and 12. Too small a value of η results in the loss of high-frequency information, while too large a value of η produces over-sharpened and potentially noisy sampled images.

(a) Yt.

Linear (Ymax = 0.5)

Linear (Ymax = 1.0)

DDBM-VP

DDBM-VE

XT

Linear (Ymax = 0.5)

Linear (Ymax = 1.0)

DDBM-VP

DDBM-VE

1.0

1.0

1.0

40

αt

0.5

0.5

0.5

20

Vt

0.0

0.0

0.0

0

0

1

0

0

0

50

time (t)

time (t)

time (t)

time (t)

n=0

= 0.3

n = 0.5

n = 0.8

= 1.0

=1.5

n=0

= 0.3

= 0.8

= 1.0

=1.2

F Impact Statement

Our method can improve image translation and solving inverse problem, which may benefit applications in medical imaging. However, it is important to note that as with many generative and restoration models, our method could be misused for malicious image manipulation.

= 0

= 0.3

=0.5

= 0.8

= 1.0

n = 1.5

G Experiment Details

Architecture . We maintain the architecture and parameter settings consistent with [18], utilizing the ADM model [10] for 64 × 64 resolution, modifying the channel dimensions from 192 to 256 and reducing the number of residual blocks from three to two. Apart from these changes, all other settings remain identical to those used for 64 × 64 resolution.

Training . We include additional pre- and post-processing steps: scaling functions and loss weighting, the same ingredient as [17]. Let D θ ( x t , x T , t ) = c skip ( t ) x t + c out( t ) ( t ) F θ ( c in ( t ) x t , c noise ( t )) ,

where F θ is a neural network with parameter θ , the effective training target with respect to the raw network F θ is: E x t , x 0 , x T ,t [ λ ∥ c skip ( x t + c out F θ ( c in x t , c noise ) -x 0 ∥ 2 ] . Scaling scheme are chosen by requiring network inputs and training targets to have unit variance ( c in , c out ) , and amplifying errors in F θ as little as possible. Following reasoning in [18],

where σ 2 0 , σ 2 T , and σ 0 T denote the variance of x 0 , variance of x T and the covariance of the two, respectively.

Wenote that TrigFlow [23], adopts the same score reparameterization and pre-conditioning techniques. It can be considered a special case of our framework by setting α t = cos( t ) , β t = 0 , γ t = σ 0 sin( t ) , t ∈ [0 , π 2 ] . In this case, σ T = 0 , σ 0 T = 0 ,

Then we recover TrigFlow.

In our implementation, we set σ 0 = σ T = 0 . 5 , σ 0 T = σ 2 0 / 2 for all training sessions. Other setting are shown in Table 9.

Sampling . We use the same timesteps distributed according to EDM [17]: ( t 1 /ρ max + i N ( t 1 /ρ min -t 1 /ρ max )) ρ , where t min = 0 . 001 and t max = 1 -10 -4 . The best performance achieved by setting ρ = 0 . 6 for Edges2handbags and ρ = 0 . 8 for DIODE datasets.

H Licenses

Edges → Handbags [16]: BSD license.

DIODE-Outdoor [37]: MIT license.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed a flexible framework for diffusion bridge models used in image-to-image translation. It extends existing stochastic interpolant methods by incorporating endpoint conditioning, preconditioning, and a tunable noise-based sampling strategy, thereby enlarging the design space of both bridge paths and samplers. \n\nAdditionally, the authors identify a key limitation in prior models, low conditional diversity, and address it by modifying the base distribution. To quantify this aspect, they introduce a new metric, Average Feature Diversity.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1.  This method performs well on conditional sampling tasks, generating high-quality outputs that stay aligned with the input. Its design helps capture important details between the source and target. \n\n2. The method outperforms existing baselines across various conditional generation tasks. Also, there are enough ablation and experiments.\n\nWeaknesses\n\nThe writing style of this paper is too overclaimed. The main contributions of this paper are the introduction of noise to the conditioning input to improve output diversity, and the proposal of a new metric to quantify conditional diversity. Other claims are mostly reformulations or combinations of existing ideas.\n\n1. In Section 3, \\emph{Have the bridge paths been fully explored?}, here the author an important question, whether the space of bridge paths has been fully explored, I was expected this work could provide some extension viewpoints, but the paper does not actually address it. The boundary conditions for alpha and beta are standard, and the decoupled parameterization is already known in stochastic interpolant literature. The authors merely test a specific gamma schedule and present limited ablations, falling short of a meaningful exploration, even no theoretical validation, why the new gamma schedule works? \n\n2. Similar to Section 3, Section 4 \\emph{Has the sampler space been fully explored?}, also poses a strong question—whether the sampler space has been fully explored—but offers a limited answer. \n\nI personally find the questions posed in Section 3 and Section 4 are very interesting and important. However, in reading the paper, I feel that the proposed method only touches a small part of these spaces and does not fully engage with the broader implications of those questions. From my perspective, the work presents useful contributions, but they seem more modest than the framing suggests.\nOverall, I believe the paper does introduce some new ideas. However, it falls significantly short of delivering on the promises implied by its title and section headings. The actual contributions are relatively minor compared to what is claimed. \n\nIn summary, while the technical contribution may merit acceptance, the overly exaggerated presentation makes me reluctant to recommend acceptance in its current form.\n\nQUESTIONS:\nsee above Strengths and Weaknesses.\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes Endpoint-Conditioned Stochastic Interpolants (ECSI), an extension of Stochastic Interpolants to the endpoint-conditioned setting. The conditional dynamics of a stochastic process starting at $T=1$ and transforming a given sample ​into a sample from an unknown distribution at $T=0$ are derived while preserving the same marginals as a spatially linear stochastic interpolant. Additionally, a loss function is proposed to approximate these dynamics, enabling conditional sampling from the unknown distribution via SDE or ODE dynamics. To improve the diversity of generated samples, the paper proposes modifying the distribution at $T=1$ using a Gaussian convolution and introduces the Average Feature Distance (AFD), which sums Inception-V3 feature distances, to measure conditional diversity. Lastly, ECSI is evaluated empirically on Edges2handbags ( $64 \\times 64$ ). DIODE-Outdoor datasets ( $256 \\times 256$ ), deblurring on ImageNet ( $256 \\times 256$ ) where ECSI is compared to diffusion bridge models and flow based methods.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- To the best of the reviewer’s knowledge, the theoretical contribution presented in Proposition 4.1 constitutes an extension of the IS framework of Albergo et al. [1] to the endpoint-conditioned setting, and is therefore a valuable contribution in its own right.\n- The paper tackles the important issue of diversity in conditional image generation and proposes a simple approach to improve diversity by modifying the base distribution.\n- The empirical results suggest that the proposed ECSI method yields superior performance compared to Denoising Diffusion Bridge Models and Diffusion Bridge Implicit Models.\n\n### Weaknesses\n- The presented method, which extends the IS framework to the endpoint-conditioned setting, is not compared to the unconditional Stochastic Interpolants (IS) framework of Albergo et al. [1], neither at a conceptual level nor in terms of performance. An unconditional version of Proposition 4.1 has already been derived in [1], as [Corollary 2.18, 1] gives\n\n\\begin{equation}\ndX_t = b(t,X_t)dt + \\sqrt{2\\epsilon}d W_t, \\quad X_T \\sim \\pi_T\n\\end{equation}\n\nand [eq. (4.4), 1] yields\n\n\\begin{equation}\nb(t,x) = \\dot{\\alpha}(t)\\mathbb{E}[X_0|X_t=x] + \\dot{\\beta(t)}\\mathbb{E}[X_1|X_t=x] + (\\gamma(t)+\\frac{\\epsilon(t)}{\\gamma(t)})\\mathbb{E}[Z|]X_t=x].\n\\end{equation}\n\nThe reviewer acknowledges that Proposition 4.1 is stated in the conditional setting $X_1=x_1$, and, in the reviewer’s opinion, it is not straightforward to derive Proposition 4.1 from Albergo et al. [1]. Moreover, the training objective proposed in this work differs from the loss functions presented in Albergo et al. [Eq. (4.6), 1], and hence, to the best of the reviewer’s knowledge, ECSI constitutes a novel method. However, the strong connection of Proposition 4.1 to the above derivations should be explicitly acknowledged, and a detailed discussion of the conceptual differences between ECSI and SI should be provided. Most importantly, the paper lacks a performance comparison between ECSI and the unconditional SI framework as described in Albergo et al. [1]. This comparison is especially relevant since Albergo et al. [1] already provides, for linear interpolants given by \n\n\\begin{equation}\nX_t = \\alpha(t)X_0 + \\beta(t)X_1 + \\gamma(t)z\n\\end{equation}\n\na detailed discussion of the choice of $\\alpha(t),\\beta(t)$ and $\\gamma(t)$ in [Section 4, 1].\n\n- The proposed metric, Average Feature Distance (AFD), to assess diversity in conditional image generation, is only compared to the Fréchet Inception Distance (FID) [2] and is justified solely by stating that assessing diversity of generated data via FID alone is insufficient. While the reviewer agrees with this point, it is a well-known issue, and several works have already addressed it. The paper lacks a detailed discussion of the differences and advantages of AFD compared to more recent methods for diversity quantification in generative models, such as Improved Recall [3], $\\beta$-Recall [4], Vendi Score (VS) [5] and Feature Likelihood Divergence [6]. The reviewer suggests that the authors provide a thorough discussion of why AFD is preferable to the metrics proposed in [3,4,5,6] and/or include these metrics in their evaluation to support the claim that ECSI produces more diverse data. In particular, VS [5] is explicitly designed to measure the diversity of generated data and should be considered in both the conceptual discussion and the quantitative evaluation.\n\n[1] Albergo et al. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint arXiv:2303.08797, 2023.\n\n[2] Heuse et al. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. NIPS 2017.\n\n[3] Kynkäänniemi et al. Improved Precision and Recall Metric for Assessing Generative Models. NeurIPS 2019.\n\n[4] Alaa et al. How Faithful is your Synthetic Data? Sample-level Metrics for Evaluating and Auditing Generative Models. ICML 2022.\n\n[5] Friedman et al. The Vendi Score: A Diversity Evaluation Metric for Machine Learning. Transactions on Machine Learning Research, 2023.\n\n[6] Jiralerspong et al. Feature Likelihood Divergence: Evaluating the Generalization of Generative Models Using Samples. NeurIPS 2023.\n\nQUESTIONS:\n- The authors write in the abstract that “the proliferation of techniques based on incompatible mathematical assumptions have impeded progress.” What are these assumptions, and why do they impede the progress of diffusion bridge models for image-to-image translation? \n- Could the authors clarify the advantages and conceptual differences of ECSI compared to the spatially linear interpolants presented in Albergo et al. [Section 4, 1]?\n- How does ECSI perform compared to the unconditional setting of SI?\n- What are the advantages of the proposed Average Feature Distance (AFD) compared to the Vendi Score (VS) [5] and other established metrics (see Weaknesses above)?\n- Does the quantitative evaluation of ECSI using the Vendi Score (VS) support the authors’ claim that modifying the distribution at $T=1$ via a Gaussian convolution yields more diverse samples?\n- What is the computational cost of calculating the Average Feature Distance (AFD)? \n\n[5] Friedman et al. The Vendi Score: A Diversity Evaluation Metric for Machine Learning. Transactions on Machine Learning Research, 2023.\n\n### Minor comments:\n- Is it a typo that the authors write $\\bar{f},\\bar{g}$ in eq. (1) instead of $f,g$? The reviewer would assume with this notation $\\bar{f}(t):=f(T-t)$ \n- There is a reference missing for the derivation of the reverse model in eq. (3) and eq. (4)\n- There is a reference missing for the derivation of the denoising bridge score matching objective in eq. (5)\n- $\\log p_{t|0,T}$ is missing it's input in eq. (5)\n- (line 105): $tx_1$ instead of $tx_0$ \n- (line 113): \"by the SDE\" instead of \"by SDE\"  \n- There is no reference in the main paper to the proof of Proposition 4.1 provided in the Appendix.\n- (line 118): \"we found that we could define [...] controlled the stochasticity\" should be \"we find that we can define [...] controls the stochasticity\"\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents Endpoint-Conditioned Stochastic Interpolants (ECSI), a novel and unified framework that significantly expands the design space of diffusion bridge models for image-to-image (I2I) translation tasks. The authors systematically investigate the role of stochasticity across three key components: the transition kernel, the sampling SDE, and the base distribution. By extending stochastic interpolants with endpoint conditioning, preconditioning, and a more flexible sampling algorithm, the proposed method achieves new state-of-the-art results. A key contribution is the identification of the low conditional diversity problem in one-to-many translation tasks, which the authors address by modifying the base distribution and quantify using a newly proposed metric, Average Feature Distance (AFD). Experimental results on tasks like edges-to-handbags, depth-to-RGB, and deblurring demonstrate that ECSI surpasses previous methods in terms of image quality, sampling efficiency, and conditional output diversity.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths (Pros)\n\n**Unified Framework and Expanded Design Space**: The paper presents a compelling and well-motivated framework (ECSI) that successfully unifies and expands upon previous diffusion bridge models like DDBM, DBIM, and Stochastic Interpolants. By systematically decoupling and exploring stochasticity in the transition kernel, the sampling process, and the base distribution, the authors provide a much broader and more flexible design space for I2I translation tasks. This is a strong conceptual contribution to the field.\n\n**Significant and Comprehensive Empirical Gains**: The proposed method achieves impressive empirical results, establishing new state-of-the-art performance on multiple benchmark datasets. The improvements are not limited to a single dimension but are demonstrated across three critical axes:\n\n*Sampling Quality and Speed*: The ECSI model and its sampler show a remarkable improvement in efficiency, achieving superior image quality (FID) with significantly fewer function evaluations (NFE) than prior art.\n\n*Conditional Diversity*: A major strength is the identification and resolution of the often-overlooked issue of low conditional diversity. The proposed modification of the base distribution, coupled with the new Average Feature Distance (AFD) metric, provides a novel and effective way to generate more varied and realistic outputs for one-to-many translation tasks.\n\n*Clarity and Rigor*: The paper is well-written and clearly structured. The theoretical connections between ECSI and prior models are well-explained, and the experimental section is thorough and convincing.\n\n\nWeaknesses (Cons)\n\n**Heuristic Design Principles and Lack of Theoretical Guidance**: While the paper successfully expands the design space, the exploration within this new space remains largely empirical and heuristic. The paper provides limited theoretical guidance or practical principles on how to optimally choose the bridge path parameters (αt, βt, γt) or the sampler noise schedule (ϵt) for a new, unseen task. This suggests that applying the framework to different problems might still require extensive and costly hyperparameter tuning, which somewhat tempers the practical benefits of the expanded flexibility.\n\n**Limited Demonstration of Scalability**: The experiments, while thorough, are conducted on relatively small-scale datasets and resolutions (e.g., 64x64 handbags, 256x256 DIODE). It remains an open question whether the demonstrated gains in efficiency and quality will hold when the framework is scaled up to higher-resolution images or more complex, large-scale industrial applications. Demonstrating the method's effectiveness on a more challenging, large-scale benchmark would significantly strengthen the paper's claims about its general applicability.\n\n**Insufficient Analysis of Interacting Stochasticity Sources**: The paper introduces two key sources of randomness: the path stochasticity (controlled by γt, which determines the noise z_t in the forward process) and the sampler stochasticity (controlled by ϵt during reverse sampling). However, the relationship and potential interplay between these two noise sources are not deeply analyzed. For instance, could a wider, more stochastic path (larger γt) be compensated for by a more deterministic sampler (smaller ϵt), or vice-versa? A more in-depth analysis or ablation study on how these two forms of stochasticity interact would provide deeper insights into the model's behavior and could lead to even better design principles.\n\nQUESTIONS:\nPlease see the weakness.\n\nLIMITATIONS:\nPlease see the weakness.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a unified approach to expand the design space of diffusion bridge models by incorporating preconditioning, endpoint conditioning, and an optimized sampling algorithm into Stochastic Interpolants (SIs). The proposed framework advances the state of image-to-image translation by improving both image quality and sampling efficiency across diverse tasks. Furthermore, it addresses the overlooked issue of low sample diversity under fixed conditions through a quantitative analysis of output diversity, demonstrating how base distribution modifications can enhance results.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. The proposed bridge model is unified approach and surpasses pervious models (e.g. DDBM, DDIM) in handling varying  characteristics.\n1. The experimental results are highly illustrative, providing clear empirical support.\n\n**Weaknesses**\n\n1. The novelty of ECSI primarily resides in introducing additional noise along the sampling path compared to the original bridge models' sampling process. While this design enhances diversity, the underlying intuition is not entirely novel, as similar concepts of noise augmentation in diffusion sampling have been discussed in prior work (e.g., EDM).\n2. It remains unclear whether the training scheme of ECSI differs substantially from that of DDBM. If there are differences, the approach resembles applying noised stochastic sampling to DDBM models, analogous to endpoint conditioning techniques that utilize $x_0$-prediction for original DDBM frameworks.\n3. The paper lacks theoretical analysis on the convergence behavior of the newly proposed method, which is a notable gap in the methodological justification.\n\nQUESTIONS:\n1. What is the difference between the training scheme of the ECSI to the previous bridge models? \n2. Will the proposed ECSI sampler mitigate the sampling discretization error? \n3. Does the marginal distribution of the ECSI sampler coincide with the underlying true marginal distribution from a rigorous Fokker-Planck equation analysis?\n\nLIMITATIONS:\nClarification of the paper is limited.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-02"}