Exploring the Design Space of Diffusion Bridge Models
Authors: Shaorong Zhang
Yuanbin Cheng
Greg Ver Steeg
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion bridge models
stochastic interpolants
image-to-image translation
Primary area: deep_learning

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
Shaorong Zhang, Yuanbin Cheng, Greg Ver Steeg Unversity of California Riverside (szhan311, ychen871, gregoryv) @ ucr.edu
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 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 ex- isting methods within 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 algo- rithm. These enhancements expand the design space of diffusion bridge mod- els, leading to state-of-the-art performance in both image quality and sampling efficiency across diverse I2I tasks. Furthermore, we identify and address pre- viously overlooked issue of low sample diversity under fixed conditions. We introduce quantitative analysis for output diversity and demonstrate how can modify the base distribution for further improvements. Code is available at https: htt///gttttt giggttt..b..... com htmm//shan311/Ee
Introduction
Denoising Diffusion Models (DDMs) and flow matching create a stochastic process to transition
Gaussian noise into a target distribution [33 14 19]. Building upon this, diffusion bridge-based models (DBMs) have been developed to transport between two arbitrary distributions, TT and To- including 12SB (1T, DSBM [39] DDBM [8. DBIM 42 Bridge Matching (28). DBMs achieve superior image quality in 12I translation compared to DDMs 18 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 12SB 21 achieve state-of-the-an 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 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 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, 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 this the conditional diversity problem and show that our approach leads to significant improvements.
39th Conference on Neural Information Processing Systems (NeurIPS 2025).

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ECSI (Ours)
(Ours)
DDBM
DBIM
DSBM
ECSI (ours)
DBIM
Endpoinl conditioning
X
DDBM DBIM
12SB
Uncoupled parameters
DDBM
Extensive hridge paths
Extensive samplers Preconditioning
X
X
Bridge paths
(b) Samplers
Modified base density
X
X
Figure 1: The design space of bridge paths and samplers.
Table I: Characteristics of different bridge models.
Deblurring
Depth to RGB
Edges to Hangbags
Figure 2: Samples for I2I translation with our ECSI models: Deblurring, Depth-RGBB and Edges to Handbags. For each pair of images, we show the input image (upper) and the output image (bottom).
Our main contributions are as follows:
Stochastic Interpolants (ECSI), which extend stochas- tic interpolants by incorporating endpoint conditioning and preconditioning. Previous bridge methods artificially coupled unrelated aspects of the transition kernel. ECSI introduces decoupled parametrization that expands and simplifies the design space for bridge paths and samplers. To further improve sampling quality and efficiency, develop novel noise control scheme and an efficient sampling algorithm.
We propose
We identify 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, introduce metric Average Feature Diversity (AFD).
Experimental results demonstrate model's state-of-ther oh-tte--rtttttttoo performance in both image quality and sampling speed across various 121 tasks, including deblurring, edges-to- to-handbags translation, and depth-to-RGB conversion. Notably, for handhag generation, approach yields significantly more diverse outputs with varied colors and textures.
Background
Notations Let my, To, and Tor represent the base distribution, the target distribution, and the joint distribution of them respectively. Weond and Taata represent the distributions of the input and output diffusion process; we denote its marginal distribution at time / by Pt, the conditional distribution at time & given the state at time by Prja. and the distribution at time t given the states at times and T by Poo.r, i.c., the transition kernel a bridge.
data. Let p the distribution
2.1 Denoising Diffusion Bridge Models
DDBMs 18 extend diffusion models to translate between two arbitrary distributions wo and given samples from them. Consider reference process given by:
dX, fixyet g.dWe,
(I)

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whose transition kernel is given by 4ej0 (2) (o)) (2 a,20, of This process can be conditioned (or "pinned") at both an initial point zn and terminal point 27 to construct diffusion bridge. Under mild assumptions, the pinned process is given by Doob's h-transform 29
dX, = {&X, + of Vx, ( dt g.dWW
(2)
where V X, log (or X1) lar/ar)rr -X,
(SNN, (SNR--1 and SNR, /o 18] Eq. is stochastic
process that transport from Po = To and pt = Ne, which is a valid bridge process. To sample from the conditional distribution p(ro) (TT), we can solve the reverse SDE or probability flow ODE from
t to [8]
dx, = (J.X, + 98 h)jdt + gddW,,
dx,- = Uix, + 5 - h))dd,
where XT = IT, V log PTy (xr X2). h = Vx, log (X) Generally, the score V., log P: ((()))) in Eqs. and (4) is intractable. However, it can be effectively estimated by denoising bridge score matching. Let (In, ~ To.r (Fo, FT). 22 ~ Proor(z; (oo xT), t ~ (0, 7), and (()) be non-zero loss weighting term of any choice, then the score Va, log Pry ( can be approximated by neural network sø s0(2 TT. t) with denoising bridge score matching objective 18
(()) = E [wo(t)sss( FT, ze log | To, xT) 2T)]
(5)
where I dentotes expectation over H, (Ft, 20), (T), ~ To, t ((.,TT)
2.2
Diffusion Bridge Implicit Models
The transition kernel of the bridge process in Eq. (2) is given by
p0 IT) a 72211
(6)
where a1
SNR, ), 3, SNRT 72 = of(1 SNR.
SNR, ). Suppose we sample in reverse time
SNR.
SNH.
on the discretized timesteps 0 = to t < ... fN-1 < IN = T. Then we can sample by the initial value Fy and the updating rule:
Ft,, = 0t, IT + Br, dQ + vi.
Pr.
2tnt
Pints
O4t PT Beat 20 + pe,ss e e~,( (7)
where ag (2,, XT, t) has the relation with the score function:
so(2,, 1) Tt. QxTT 3r08 (2t,
(8)
7t
3
Have the bridge paths been fully explored?
Given the forward process defined in Eq. T.. diffusion bridge models 18 42 12
utilize Doob's
h-transform construct a corresponding bridge process (Eq. (2). While the resulting process effectively bridges the initial distribution गт and the target distribution To. such diffusion bridge approaches exhibit several limitations.
Parameter coupling. Notice that the parameters and are convolved in the transition kernel (Eq. (6)) Such coupling unnecessary and decoupling those parameters helpful for searching the 'best' bridge path.
3

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Limited design space. Despite Eq. provides an infinite number of bridge paths by tuning a, and os but the space of bridge paths still artificially restricted.
In contrast, the stochastic interpolants [ framework allows a larger design space bridge path with more decoupled parameters. Specifically, stochastic interpolants build bridge path directly via the flow map:
Φ: 0x400 + B 7
where 2 ~ (O. Eq. (9) builds a transport with Ty and my as boundary conditions the kernel parameters satisfy .T:
on = Br and or = =B0- = 70 = 71 O:
042 Bc, > O for & E (),
The transition kernel of the stochastic interpolants in Eq. (9) is Gaussian distribution: A (zr; ay 3elt, Unlike DDBM, which is parameterized by only two variables a, and ot stochastic interpolants introduce decoupled parameters az, 33, and yt, offering more flexible and expressive design space for constructing bridge paths.
A detailed discussion on the rationale behind the choices of a, 3r, and and an ablation study on the shape of ") is provided in App. E Notably, the DDBM-VP and DDBM- VE models presented in I8 can be considered as special cases by choosing different Br. and see App. D for more details. In the experiments, we limit the scope linear transition kernels and T 1, i.e. Pejo, (Ze (o, FT) = (xri (I t(zo 4- (1 t)ll..
Stochastic interpolants expands the space bridge paths and leads to decoupled parameters compared to DDBM and DBIM.
4 Has the sampler space been fully explored?
For diffusion models, EDM demonstrated that the design of training and sampling schemes could be decoupled to significantly improve results. We now explore whether similar decoupling possible for bridge models, and what freedom we have improve sampling quality with given trained model.
4.1 Cadpoint-Conddttionngg for Stochastic Interpolants (ECSI)
Given transition kernel (Z) 2o, FT) -N(x;; ayzo + B,zr; 7/ll), we can identify the training
objective II reverse sampling SDEs (Eq. (00 ), as demonstrated in Proposition 4.1 see App. App.
the proof.
Proposition 4.1 (Eadipint-conddttoond Stochastic Interpolants). Suppose the transition kernel of a diffusion bridge process is given by ( Fo. = (N 0430 + Berr,,??1 then the evolution of conditional probability qe (Xx(((T) given by the SDE:
dX, b(t, X1, dd + V2e,dW,,
(10)
where b(t, xz, x1, dy B, + (fr $) 2n = E(zo Fy, æT), 2 =: (x, a, 3,TT) //n- Besides, to can be approximated by neural networks 2 by minimizing a regression objective with the observed 2o, xr targets,
Loldg]
E||l E Fe, xT) xoll3]dt
(11)
where E denotes expectation over (oo, ØT) x (xo, and xs Pe 2O,

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Relation to Stochastic Interpolants (SI), Both SI and ECSI in Prop. 4.1 can be seen as special cases of Conditioned SI. key advantage of ECSI is its efficiency: while SI need estimate two terms: Riro | [o] and E [1 sie), ECSI only estimate Blzo Ut, Z1, 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 at, 5t, Tt. where ") apparently controlled the stochasticity of the path. For sampling, see from the proposition above that the sampling design space is expanded even further, as the sampling dynamics depend on aie: 3e, Yt and Cg. where E appears as an additional degree of freedom control stochasticity.
Training. Eq. (11 provides the training objective of the denoiser a Te. 2r). In the implementation,
include additional preconditioning DDBM [8] and DBIM 42 see App. G for more details.
Sampling. We can generate samples from the conditional distribution (0/r(20 (T)) by solving the stochastic differential equation in Eq. (TO) from = 7 to & = 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 |
DDBM sampler. When e, = 0, Eq. 100 reduces to a deterministic ODE. Setting 6, = -xxx 2/
recovers the sampling SDE used in DDBM [8]. However, DDBM only provides a single reverse SDE and single corresponding reverse ODE; it does not explore alternative choices of c2-
DBIM sampler. For small enough A/ and - -2, > O, the sampling SDE can be discretized as:
It At = a4 04-A000 + BI-ACTT +
(12)
where &n (0, V 1 2e, At + Ae, Eq. 12) recover the DBIM sampler.
Note that the condition 7²- 22,At limits the design space of samplers. For example, best result the experiments is achieved by setting a t, Yt - t(1 t) and c, = 7e de DBIM sampler fails under this setting since 72 At > O cannot be guaranteed all the time.
ot Yt
ISB sampler. When 2G, -: 8 At 7o /3, the coefficient of ær 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 ISB sampler also limited, as it can be interpreted as special case of the DBIM sampler.
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. (00)
Ft RJ 8. b(1, XT A + 2e,
(13)
We 6, "(x de>:)) where 7 (0,1) is interpolation parameter. This formulation
de
provides continuous control the sampling process, ranging from purely deterministic ODE sampling (1) 0) to fully stochastic SDE sampling (n = 1). our implementation, we let 6 = 0
for the last two steps, Eq. 12) gets reduced to: Iı-At 01-Arã 3i-ACTT -- For other
steps, apply Eq. (13) and let e, n/(2/ din?), where is constant. Putting all ingredients together leads our sampler outlined in Algorithm
5

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Algorithm EESS Sampler
1: Input: 2T, timesteps (t, 1/ distribution Teond, schedule ot St, Yt,
2: Sample IT Teond, no ~ (N(0,
3: av 2T nh
4: for
ttol do in DD IT, Lt), & (xx cu,ã0 -0te-0u,
Br, x 7/7
6: i 2 2 then
7:
Sample & (0,1))
8:
de = u0 2o + 31, NN + (fh; + 24,/3,
2:-1 a; dt: 4-11) + /2er, 1 (li 1--1)
10:
else
11: Ri-1 = at,-1 20 + Be, 2
NN + 74-11
12:
end if
13:
end for
Ditfusion path
NO 12
(0,b
Our bridge path
Previous bridge path
cond
Aeond ( 1
Wondd -N,
#coond
data
(a) Bridge paths in state space
(b) Bridge paths in density space
Figure 3: Modifying the base distribution corresponds to a lossy compression of the input that leads to 'trade-off' between unconditional diffusion and diffusion bridge models.
Is there any benefit to modifying the starting point of bridge?
We expanded the paths in distribution space connecting base and target distribution, but 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 the bridge literature. However, we found a surprising result modifying the base distribution 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.
help significantly. The situation is analogous to the benefits of
5.1 Low conditional diversity in one-to-many translations
In experiments (see Sec. G). 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 typically treat the base distribution कт equivalent to the input data distribution, denoted Teond In contrast, our approach introduces a controlled perturbation by redefining the base distribution magnitude of noise added to the input distribution. This modification enables greater diversity in the generated outputs while maintaining conditional alignment.
गe Teond * (O, 621), where 6 is a constant that governs the
Intuitively, this modification can be interpreted as a trade-off between standard diffusion models and traditional diffusion bridge models. As illustrated in 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 intermediate regime by

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Table 2: Validation of our sampler via DDBM pretrained VP model (Evaluated by FID), where
= 0.3( de yt)
Edges- Handdags (64 64)
DIODE-Outdoor (256
256)
Sampler
NFE-5
NFE=10
NFE=20
NFE=5
NFE=10
NFE=20
DDBM [8
317.22
137.15
46.74
328.33
151.93
41.03
DBIM 42
3.60
2.46
1.74
14.25
7.98
4.99
RCST (Ours)
2.36
2.25
1.53
10.87
6.83
4.12
sampling from noisy conditioned inputs, thereby blending the benefits of both paradigms- pressevving 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 single input image (a black and white edge map). To measure
conditional diversity, we will adopt Vendi Score (VS) [[]] as metric. Besides, We propose the Average Feature Distance (AFD) metric to quantify the conditional diversity among generated images. Initially, select group of source images {-1
{2(9) LM For each
fi=1
(4) JT we then generate distinct target samples. The j-th generated sample corresponding to the i-th source image is denoted by Vig. Then the AFD calculated as follows:
AFD
Σ k11=1,
[F(y)) F(wa)l
(14)
2
where F()) is function that extracts the features of images, and || represents Euclidean norm. Intuitively, a larger AFD indicates the better conditional diversity. Here, F(x) can 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. our experiments, we use AFD in latent space. Furthermore, we provide additional justification for the validity of our proposed metric in App. 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 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 121 translation tasks, in terms of sample efficiency, image quality and conditional diversity. We evaluate I2I translation tasks on Edges-Handbags [16] scaled to 64 x 64 pixels and DIODE-Outdoor scaled to 256 x 256 37], and Deblurring on ImageNet dataset [9] For evaluation metrics, use Fréchet Inception Distance (FID) [3] for all experiments, and additionally measure Inception Scores (IS) B. Learned Perceptual Image Patch Similarity VS and AFD, Eq. [4] to measure conditional diversity. Further details of the experiments and design guidelines provided in Appendix [add]
(LPIPS) [1]. Mean Square Error (MSE), following previous works 442 [8]. In addition,
Sampler. We evaluate different sampling algorithms in Fig. 4 (a), the results demonstrate that setting 0 and using Eq. (12) for the last 2 steps significantly improve sampled image
7

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300
DOBM sampler
# 02025
Pow 00122,
0.125
- 0025,
A+B
0.25
110,b=0
A+C
Wru
10.0
0.125,b-00
130
A+B+C
Yrax
74-
100
5.0
23-
0.4
1.0
10
NFB
Iterations (x10%)
(a). Ablation study on discretization.
(b). Ablation study on Ymax for SDB model with Lincar path.
(c). Ablation study
SDB model with Lincar path.
Figure 4: Ablation studies on discretization, "mmx and 6t- (a). We evaluate different discretization
schemes on Edges2handbags (64 x 64) dataset using DDBM-VP pretrained model, A represents
simple Fuler discretization in Eq. (33). B reprents setting e for the last 2 steps, C represents using Eq. (12) for E, 0. (b). Ablation study on Tmax evaluated by DIODE (64 64) dataset. (c).
Ablation study on et through our ECSI model with Linear path on Edges2handbags (64 x 64) dataset, di ax 2
where G = (7
Table 3: Quantitative results in the 12I translation task Edges2handbags (64 x 64) and DIODE (256 x 256) datasets. Our results were achieved by Linear transition kernel and setting 1j 1.
hands-hhandbags (64
64)
DIODE-Outdon (256 256)
Model
NFE
FID 4
f
LPIPS
MSE
FID
IS t
LPIPS
MSE
Pix2Pix [16
74.8
3.24
0.356
0.209
82.4
4.22
0.556
0.133
DDIB 36
2 40
186.84 84
2.04
0.869
1.05
242.3
4.22
0.798
0.794
SDEdit 25
40
26.5
3.58
0.271
0.510
31.14
5,70
0.714
0.534
Rectified Flow
22
40
25.3
2.80
0.241
0.088
77.18
5.87
0.534
0.157
PSB 21
40
7.43
3,40
0.244
0.191
9.34
5.77
0.373
0.145
DDRM
118
1.83
3.73
0.142
0.040
4.43
0.244
0.084
DBIM 42
20
1.74
3.64
0.095
0.005
4.99
6.10
0.201
0.017
0.89
4.10
0.049
0.024
12.97
5.49
0.269
0.074
ECST ("mmxs 0.125)
0.67
4.11
0.045
0.024
10.12
5.56
0.255
0.076
20
0.56
4.11
0.044
0.024
8.62
5.62
0.248
0.078
1.46
4.21
0.040
0.016
4.16
5.83
0.104
0.029
ECSI (Ymax 0.25)
1.38
4.22
0.038
0.017
3.44
5.86
0.098
0.029
20
1.40
4.20
0.038
0.017
3,27
5.85
0.094
0.029
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 [ 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., Pro, (10 = -Nxx; (1- tizo+ 2 t(1 - -)) We conducted detailed ablation studies on Ymax and for the Linear path on DIODE (64 x 64) dataset, as shown in Fig. (b) and (c). The optimal values for Ymax were found to be 0.125 and 0.25, while the best performance for was achieved with 17 = 0.8 and // 1.0. Performance deteriorates when either parameter is too small or too large. Based on the results of these ablation studies, we further trained FCSI models on the Fdges2handbags (64 x 64) and DIODE (256 x 256) datasets by taking Tmax E {00.15, 0.5} and setting 1) = 1.0. The results are presented in Table 3 Our models establish new benchmark for image quality, evaluated by FID, IS and LPIPS. Despite our models having slightly higher MSEs compared to the bascline DDBM and DBIM, we believe that 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 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 [ 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

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AFD
FID
DBIM
DDBM
Linear (b 0.5)
NFE-10, FID-2.46, AFD-5.20
NFB-118, FID-1.83 AFD-6.99
NFE=10, FID-2.07, AFD-9.35
Figure 5: Visualization of conditional diversity via sampled images one-to-many translation task.
While FID measures diversity within columns, AFD evaluates diversity across rows. The visualization further proved the effectiveness of AFD. More sampled images can be found in Appendix
Table 4: Quantitative results for Different denoisers and samplers on Edges2handbags (64 x 64). Our
baseline is achieved by DDBM pretrained checkpoint and DBIM sampler.
Method
FID J
AFD
VS
NFE=5
NFE=10
NFE=20
NFE=5
NFE-10
NFE-20
NFE=5
NFE=10
NFE=20
(pre) DBIM sampler
3.60
A: DDRM (pre) ECSI sampler
2.36
2.25
1.53
5.11
5.70
6.04
1.15
1.20
1.23
B: ECSI (pre) ECSI sampler
0.89
0.67
0.56
6.00
6.05
6.25
1.22
1.25
1.28
Modified base density
3.31
2.07
1.74
8.53
9.35
9.65
1.48
1.63
1.69
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 10 and Uniform deblurring, shown in Table The results demonstrates that our ECSI models achieve much lower FID score.
Related Work
Diffusson 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 [8]] 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 al. (2
and can be viewed as generalization of score matching 34. Based on this, diffusion Schrödinger bridge matching (DSBM) has been developed for solving Schrödinger bridge problems [5] 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 more efficient and interpretable than score-based generative models (2 Furthermore, our benchmark DDBM [18) achieve significant improvement for various 12I 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 strong generating images, applying them to image-to-image (12I) translation is more difficult due to artifacts in the output. Diff121 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 common challenge that many models require joint training on both source and target domains, raising privacy concerns. Injecting-Diffusii tackles this by isolating shared content for unpaired translation (24). SDDM improves interpretability by breaking down the
frequency control 26 15 38.
score function across diffusion steps [30].

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Table Deblurring results with respect different kernels, evaluated by FID on the 10k ImageNet (256 x 256) validation subset. Our results are achieved by 20 NFEs.
Kernel
DDRM
DDNM
Pallette
CDSB
ISBB
ECSI (ours)
Uniform
9.9
3.0
4.1
15.5
3.9
1.11
Gaussian
6.1
3.1
3.0
0.41
8 Conclusion
introduced Cadpoint-Conditionnel 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 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 tasks like 64 x 64 edges2handbags, 256 x 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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12

---

Table 6: Evaluation for generative models: BongeNet-l-mode ImageNet-22mmods, ImageNet-5- modes, and ImageNet- 10-modes.
Model
ImageNet-I-men -mode
ImageNet-2-modes
ImageNett-Smmods
ImageNet-10-modde
FID 58.30 0
57.34 57.78
57.26
AFD
8. 14
12.84
14.47
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 complementary metric to FID. In unconditional generation scenarios, the FID widely used to evaluate the diversity 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 edge images, despite achieving very low FID scores. To address this limitation, we introduce the concept conditional diversity and propose a corresponding metric to quantify it.
The first question why FID failed to measure the conditional diversity. To illustrate the limitations of FID in capturing conditional diversity, consider extreme case: if the images generated by 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 useudo-generative models capable of controlling the diversity 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 pscudo-generative models: MangeNet-I-mode. ImageNet-2-modes, ImageNet-5- modes, and ImageNet- 10-modes. The experimental setup is as follows:
We selected 11,000 samples from the ImageNet validation dataset, randomly choosing images per class.
From these, we designated 1,000 images as the "real" set, while the remaining images served the source pool for the generative models.
Each ImageNet-k-modes model simulates a generative process by randomly sampling images from pool of k distinct images within given class.
We present sampled images in Fig. where 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 While the FII 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 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, models using strong augmentation.
as a measure of diversity, we construct
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 applied 16 times, creating augmented dataset for
13

---

ImageNet-I-mode: FID-58.30, AFD-0
ImageNet-2-modee. FID-57.34, AHD-8.14 14
SmageNet-SSmodees FID-57.78, 12.812.84
ImageNet-10-mon FID=57.26, AFD-14.47
Figure 6: Sampled images from 4 generative models: ImageNet- BrpeNet---moode -mode, ImageNet-2-moods, ImageNet-5-modee SrageNet-10-modee
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 diversity increases, AFD values also rise, further confirming that the proposed AFD metric a reliable indicator of image diversity.
B Relation to Stochastic Interpolants
Conditioned Stochastic Interpolants build marginal probability path Ptly using mixture of in- terpolating densities: (ty(2) p. (o, x1 ( æ1 yddrddr1, where π (<0, æ1 y) is joint distribution with marginals ( y) and T1yy(1 | y). For linear interpolants given by:
X; axXo B, + n. The conditional kernel (x, Io. Ø1) is given by Gaussian distribution:
14

---

Table AFD results across different augmentation magnitudes
Augmentation magnitude
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
AFD FID
2.16
3.77
5.13
6.16
6.98
7.63
8.22
9.01
0.20
2.95
7.02
11.62
16.33
20.84
25.12
28.89
Pr(et | x1) (a,20 + o [0,1].
Then we can sample from the conditional distribution Puj (2ro y) by running stochastic process Prly(re y) from time to which is given by the following SDE:
da = 1,11,,,uddL + /2 in Piles
(15)
where the drift term b(t, x. y) is:
b(t, x, y) = á, B/2 2, vl + 3, vl + (% + St BB22 v.
(16)
As y represents null conditioning, Eq. 15 recover the original sampler of Stochastic Interpolants. In the drift term, B[zo| Iz, y/. B[x1 x, and E[z x, y] are unknown, but we only need to estimate two of them, since
E[ra Te, oz B[zo 2e, yl 3,E[[1 22,
24, y]
2t
We can further reduce the number of unknown term by endpoint--oondttoonngg Here as we replace condition y to be endpoint æ1, the term E[x1 I x1] = .1. So we have:
6(t, v. v) E
xr 01 + 3,0 + (3r +
E "
B[
xr x
(17)
This is exactly the sampler for ECSI in Prop. 4.1
Therefore, both SI and ECSI can be special cases of Conditioned S1. A key advantage of ECSI is its efficiency: while SI need to estimate two terms: E[r ar. and E[ar wr). ECSI only estimates B[ra ze; vl.
C Proofs
There are infinitely many pinned processes characterized by the Gaussian transition kernel Prjo, (x, Xo. XT) (X;;;; XxXo + B,xT, B,xr,,21).. Specifically, we formalize the pinned process as a linear Itō SDE, as presented in Lemma
Lemma C.1. There exist linear SDE
dX, [fiX, stxT)dl + g.dWe, Xo- Xo,
(18)
St = distribution N (x: 04X0 8,x,, T2.
where f. ds ax
بu sr,
gt Veluet an 4n)) that has Gaussian marginal
as
Proof. Let m, denote the mean function of the given Itô SDE, then we have dm. f.m, + S,XT. Given the transition kernel, the mean function m, 0,Xo 8xx,, therefore,
6,80 + 3.XT fe(axxo + BiXT) + 8,X..
(19)
Matching the above equation:
3, 3r de
8t
(20)
04
att
Further, For the variance of of the process, the dynamics given by:
15

---

d
25r +
94
(21)
dt
de Solving for g2 we substitute ft
di
dt Yt
gi
(22)
at
Therefore,
âg 72)) 2
1
(23)
0g
Given the pinned process solving the reverse SDE or ODE from f to 0:
18 we can sample from the conditional distribution Po xxxo(xT) by
dX, [AX,, + SXXT 96 V log [x)) dt + g.dWe, Xx = xx,
(24)
dx,
fix1 + 9xXT 2 v. log (X: dt X. -X::
(25)
where the score V log X, (x) can be estimated by score matching objective 5
For dynamics described by ODE dX; u,dt, can identify the entire class of SDEs that maintain the same marginal distributions, as detailed Lemmal C.2 This enables us to control the noise during sampling by appropriately designing 62.
Lemma C.2. Consider a continuous dynamics given by ODE of the form: dX; ,tdt,, with
the density evolution PX.).. Then there exists forward SDEs and backward SDEs that match
the marginal distribution Pt- The forward SDEs are given by: dX, = (u; + 6 V.Vlog log pt ddt +
v 22,,W O. The backward SDEs given by: dX, (u, c, V log Pr) + 22 t,
Proof. For the forward SDEs, the Fokker-Plances equations given by:
(Xr(
V [uu + ez (o
(26)
Ø1.
V [urPr(X.)] V (V + PxPP.
(27)
V [X,P((X))
)] <. V
Vp X + V.
(28)
= V (X,).
(29)
This exactly the Bokker-Planche equation for the original deterministic ODE dX; u, dt. Therefore, the forward SDE maintains the same marginal distribution P. (X,) as the original ODE.
Now consider the backward SDEs, the Fokker-Planck equations become:
(X,) Ø1
V .[(u ez V log P. (X, - V
(30)
V [uppr (X,
+ V [Ea lo. - V' (X.))
(31)
[u.p. (X.))
(32)
This is again the Fokker-Planck equation corresponding the original deterministic ODE dX, u, dl. Therefore, the backward SDE also maintains the same marginal distribution pa (X.).
16

---

Lemma C.3. Let (T) ~ To To(xo.X1) X ~ Xx xT). Given the transition kernel: p(x; | Xo, XT) = N (xx; 04X0 + B,XT; TI., if (x(x,,,; xr,t) is denoiser function that minimizes the expected L2 denoising enor for samples drawn from To (x,, xr):
x XT, 1) arg min Ex., ((xx,x..t)
D(x, xo11
(33)
then the score has the following relationship with X (Xx, XT, t):
Vx, lxgp((x:|xT)= (x XT, t) + 3exT Xt
(34)
Proof.
L(D) = E(xo, xxxxxxx)))) (x))))o(()x)))))) Bex ppp (x) [xxx)) ) xol12
(35)
Pr(xe Xy To To XT. D(x,) - xul12 dx dxydx;,
(36)
.d
-:((Dxx(.xT)
((D:x1,XT) = / (x(xx(200 XT) (Xn, xT)][ -xoll dxo.
(37)
JRc
we can minimize (D) by minimizing X,, Xx iindeeeennny for each [x,,X} pair.
D*(x,, = arg min L(D; xT) D(x:)
(38)
0 V D(x,xx)) (x,
Xx, XT)]
(39)
/ pi (xxx//00 X To (x0. XT (x) xT) xol dxo
(40)
2x|((x,,x7) xT) / S20 (T))) XT) dxo S pe (xx)xx0, xo) Xx XT) dxo)
(41)
(xx, - / (x.
xo. X To (xo, x Xxo dxol:
(42)
(x(xxXXT)= - / Pe(xx xo. XT (oO XT XT) dxo.
(43)
px(x;, XT)
V Xx)
Vx. (x,
(44)
(XL, XT)
Vx,Po (x1 Xx, Xo) (Xo. xT ddx
(45)
Pr (x XT)
X;
0x - B.xT (x Xo, XT) (oo xT) dxn
(46)
222
Pt (X,, XT)
Ox D" (x,:Xr)
+ B.xx X
(47)
72
Thus conclude the proof.
Proof of Prop.
17

---

Table 8: Specify design choices for different model families. In the implementation, or t for EDM, 0; = t. 1 for DDBM-VE, Vet and a = 1/ 1/V e 4 +mmin for DDBM-VE VP, where 3a and Buin are parameters. We include details and proofs in Appendix
12SB
DDBM
DBIM
EDM
Ours
o
a a(L-o30) ( a(L-ahot/ (
1-t
Transition kernel
8.
0
aroi/
t z t(1
2
ai
oi /
ai(1-ajoi (of))) (
oi
Sampling SDEs G
d- 24t
t=0 0
n(
At
== oorn=1
0,
1
1 0
n E [0, 1]
Base distribution.T
Toond (0,
Euler
Euler
Euler Eq. 12
Heun
Euler
Discretization
Eq. 12
Eq. 133
Eqs.
13 and
12
Proof. Recall Eqs. 24
225
and Lemma C.2
[ X2
di
âg
dX; =
+
(3:
3,)xT
(T
- 7/ + Et)
Vx. log (xg dt + /2 (48)
a2
Next take the reparameterized score in Eq.
(34) into Eq. 48
du X,
áu (()xx] - (nii
ā 2
0: + B:xT Xi
dX.-
(3:
+ Ec) 71
dt 22
at
at
at
(49)
0,880 + +BxTT Xx
0:
BIXT (zrÿe + +x))
dl + V2e,dWe
(50)
BeX (ire + E
ax50 + BiXT - -X,
dl + 22r,dw;
(51)
76
7/
â:ã0 BiX (ir + E ) dt 22,,d..
(52)
7t
D Reframing previous methods in our framework
We draw link between our framework and the diffusion bridge models used in DDBM.
D.I DDBM-VE
DDBM-VE be reformulated in our framework set:
0; s1(1 of
5:0ī
of
)
7t 0
(53)
oT
Proof. the origin DDBM paper, the evolution of conditional probability ((x, (x)) has a time reversed SDE of the form:
dx, f(())) - (X,,) sis(
dt + g.dW..
(54)
18

---

and an associated probability flow ODE
dx.
(x,,) (/hhex,,
20i dl.
1
(55)
Compare Eqs.
(44 and 55
with Lemma C.I We only need to prove:
((,),) - g (X,) fiX;; + SEXT, 9t = 9r-
(56)
In the original paper,
(X,)) =0, 00. d oi h (X,) 0T o
(57)
dt
Therefore,
20101(XT - -x1)
(x,) (X,)
I = -2100
(58)
of og
In our framework, ft. st. of can be calculated:
d
d
of
oz
- 00
ft 0t dt
log az
log
(59)
dt
o
0
oi
àt 3:
20,0
20,0:
o
20,0:
8t B1
oo
+
01
2
o0
01
2
01
2
of
(60)
ái 2)
11
ài
(o
-20
20.0:
9. 2(1071
27
20,0r
(61)
7/
at
(oū oz)
o7 07
Therefore,
20,01(XT - xx))
f.X + s,XT
o - o7
-X())) -9. (X,),
9t = 9t.
(62)
which matches the formulation in DDBM.
D.2 DDBM-VP
DDBM-VP can be reformulated in our framework as we set:
of ofan afa?" ofa;"
o7 ofa;
a, a,(1=
3,
7: ===11 oo?1
).
(63)
Proof. In the original DDBM-VP setting,
d d at
(X,,)
X
(64)
dt
96 = 20,0; - 2 àt 2
20,000 - 200
- 0
(65)
a2
at
la//ai]x- Xx,
010:X7 x2
h (X.)
(66)
of /SNR 1)
.. - o
19

---

Therefore,
de
20 0(dr)
20 (ōr0: -((0))
(((,)) 96 (X,))
ailofat - 07
Xt +
ojaî 0j
XI.
(67)
at
In our framework, ft, St. of can be calculated:
log at
(68)
at
dt
d
0u 07
log
(69)
dt
ofat
20 2070000 dt
(70)
d 20;or(a;0: d,01)
(71)
4t
arfoaa - -3,0])
3 04
8t
3.
9r =
3e(
(72)
3.
0t
ofai
20
2070rd 20/0r0r
(73)
o7ar 01
oaae 0700 a
20 (djat arût).
(74)
ofaî oñoo
11
de
d6 = 7/
7t
2
2
(75)
71
d
7tt
log
(76)
dt
a0
22 d
of
log
(77)
dt
ofa
ofa
0.a
of oa
ai
of
(78)
010
2
o.
o7 a
0,010t - oîū
(79)
Therefore,
f.X, + 8,89 = f((x,) (X,, 9: = gt:
(80)
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 0z 1,8 0, n dz-
Proof. Recall 25 the ODE formulation is given by:
1
dX,- = fiX + s:XT 5
V lo Pe (x()x)) dt Xx = xT
(81)
20

---

where f== = a.: St = B:
as Sc,
9: = /2(nir - ar As a 1, = 0,7 = ot. The
sampling ODE is given by:
dx. -0 Vx, (og
(82)
Sampling SDEs with noise added. Recall Proposition 4.1 as at 1, = 0,7 = os then the SDE has the form:
dX; (-o,ō; + Vx, log (X,,d( + 2e,d
(83)
Now we recover the stochastic sampling SDE in original EDM paper.
D.4
12SB
I2SB can be reformulated in our framework as we let:
of
of
of 03(1 =10011
at 1
2 2.
B1
2.
7t
1
(84)
o
where of = 50 Brdr.
When 21 = -7-ai 82 7 /8, the coefficient of or in Eq. be simplified
12)
vanishes. Thus, Eq.
12 can
31-A1
Bi-s1
Bi-An?
7
It Ar (at-At an
)00 +
21 +
2 V
2t
(85)
3r
Be.
13
Using discretization in Eq.
85
Xt-tt= = (04-A1 - Ot
Bi-Ai
31
180 +
31-
Be
X1
+
7-
2
18 At
$3
TE
2t
(86)
Bi-A!
31-D1
BP-An Vi
= -1
31
)xn
+
Br
X1 +
7--i
2
87
2r
(87)
c-
o² o
07
)
of
07
(1
d2
=(1
+
X1 +
of
Z: (88)
2
(1
i- )xn
+
o- Xx +
o so - Zt
(89)
of
of
In the 12SB paper, define az flata G,dr, 0² = -0 ddd Therefore,
02
o2
X,
X0
Xn+1 +
Z1
(90)
a + o
a + o
ag + to
Thus, reproduce the sampler of I2SB.
21

---

k 0.5
k 0.7
0.20
20
== 1.0
0.15
= 2.0
4.0
0.10
10
0.05
0.00
0.2
0.6
1.0
Iterations (x (x105)
(a) Ye-
(b) Ablation study the shape of yt-
Figure 7: Ablation study on the shape of y.-
Ymax 0.025
Vmax 0.125
Ymax = 0.25
Vmax = 0.5
Ymax =
Figure 8: Sampling paths with dfferent choices of Yt. As Y/t extreamly low, e.g, 7max 0.025, the model will be failed to construct details of images.
E Additional design guideline
Og and 3r. Theoretically, a; and B, can be freely designed, and future work may explore alternative design choices. However, in this paper, we focus on the simple case where = 1- and B3 = L. The
rationale is as follows: consider the scenario where a; 1 3r, which represents interpolation (1) along the line segment between 2y and 21. For the path P: (x) = WV 3)) + 3. I11) where 3r is invertible, it is straightforward to construct another path p (x) A2((1 t) tæ 23 II),
which achieves the same objective function but uses a different distribution of during training. Based on this equivalence, setting a; f and 3, is reasonable choice.
The shape of 7t. We conducted an ablation study on with different shapes. Specifically, we assumed " has the form = 2/max / - t²), as shown in Fig. 7 7 will have different shape as set different k. The results indicate that the best performance is achieved when & = which is the exact setting used in this paper.
Tmaxx- Our ablation studies "max demonstrate that the optimal values of 7max are approximately 0.125 or 0.25. Furthermore, the sampling paths corresponding different choices 7/4 are shown in Fig. [8] Adding an appropriate amount of noise to the transition kernel helps in constructing finer details.
Ec. We use the setting e = 17 ( 44,²) The ablation studies on é, demonstrate that the optimal choice of for the DDBM-VP model is approximately 0.3, while the best choice for the ECSI model with Linear Path around 1.0. Additionally, we present sample paths and generated images under different settings to illustrate heuristic parameter tuning techniques. The results are shown in Figures TO, IT 12 Too small value 1) results in the loss of high-frequency information, while too large value of 7) produces wharsher and potentially noisy sampled images.
22

---

Linear (Ymax 0.5)
Linear (Yixx 1.0)
DDBM-VP
DDBM-VE
X0
N(axx0 B,x,, 1xI
XT
Linear (Ymax 0.5)
Linear (Ymax
1.0)
DDBM-VP
DDRM-VE
1.0
1.0
1.0
40
a2
Pr
Vt
0
0
0
0 50
(1)
(t)
Figure 9: An illustration of design choices of transition kernels and how they affect the 121 translation process. ar and Be define the interpolation between two images, while "/ controls the noise added to the process. ntuitively, the DDBM-VE model introduces excessive noise in the middle stages, which unnecessary for effective image translation and may explain poor performance. In contrast, our Linear path results in symmetrical noise schedule, ensuring more balanced process. On the other hand, the DDBM-V VP path adds more noise near X7, indicating that during training, more computational resources are focused around Xy-
17
17 = 0.3
17 = 0.5
17 = 0.8
1] 1.0
n == 1.5
Figure 10: Sampling path with dfferent choices of e,, As O. the generated images lack details, as é, too large, the sampled images are over-sharpening. The best choices of e, are around e; = 0.8 and e, 1.0.
23

---

n=0
n= 0.3
= 0.5
= 0.8
= 1.0
= 1.2
Figure 11: Comparison of sampled images with different e for ECSI model, where é = 7(76% de (io ), 7max = 0.25, b = 0.
F Impact Statement
Our method can improve image translation and solving inverse problem, which may benefit ap- plications 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.
24

---

n=0
T 0.3
n 0.5
= 0.8
=
11 = 1.5 1.5
Figure 12: Comparison sampled images with different e, for DDBM-VP pretrained model, where
& = W7n $s
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 192 256 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 x 64 resolution.
Training. We include additional pre- and post-processing steps: scaling functions and loss weighting, the same ingredient [17] Let Dg(x,, xr, Cakip + (t (t) Cuolse (t)).
25

---

where F6 is neural network with parameter 6, the effective training target with respect to the raw network Fg is: xx, [Al|eatip(xx + Fo Croise) X xo Scaling scheme are chosen
by requiring network inputs and training targets to have unit variance (Cin Cout), and amplifying errors in Fg as little as possible. Following reasoning in [18].
Cin (1)
Cskip (t) = (ar + (B000)) c.
(91)
V o0 + 0/ + 204 + +
Cout (1) v BRogot -Bg + + gf
Cnoise (t) 4
1
log (t),
(92)
Cout
where 0g, oj. and oor denote the variance of Xp, variance of Xr and the covariance of the two, respectively.
We note that TrigFlow [23] adopts the same reparameterizatioo and pre-conditioning techniques. can be considered a special case our framework by setting at = cos(t), Sr = 0, Yt = oo sin t [0, 2) In this case, dy 0, our 0,
1
1
Ciu(t) Vajağ + +
(93)
V (t)of ctso
Ctkip(t) = (at0)) cos(t) of og
cos(t),
(94)
(our(t) = VnHog gg Cin sin(t)oo-
(95)
Do(xi; Cskip + Cout Fo (Ciniity Cuoise) cos(t)2; sin(t)oo Fo
Cnoise)
(96)
0g
Then we recover TrigFlow.
In our implementation, we set do = oT = 0.5, our = = for all training sessions. Other setting are shown Table 9
Table 9: Training settings
Dataset
edges edges-+handiee +handbags
edges handdbas
edges hhandbags
Model
0
0.5
Tmax
0.125
0.25
0.125
GPU
A6000 48G
H100 96G
H100 96G
Batch size
32
128
200
Setting
Learning rate
1 x 10-5
5 x 10-5
x 10-4
epochs
2078
2106
1443
Training time
42 days
8 days
11 days
Dataset
DIODE (256 x 256)
DOIDE (256 x 256)
Model
r]
0
Tinax
0.125
0.25
GPU Batch size
1 H100 96G
H100 96G
16
Setting
Learning rate
2 1
2 x 10-5
epochs
2617
1745 25 days
Training time
17 days
Sampling. We use the same timesteps distributed according EDM [7]: (1
(t2/e -tiee ))
where Lmin = 0.001 and Imax 10-4 The best performance achieved by setting p = for Edges2handbags and p 0.8 for DIODE datasets.
26

---

H
Licenses
Hanndaass
[6:
BSD license.
DIODE-Outdoor 37
MIT license.
Figure 13: ECSI model and sampler ('max 0.125, "7 1, b 0. NFE=5, FID=0089).
27

---

I
Additional visualizations
Figure 14: DDBM model and Our sampler (NFF==20, FID=1.53).
28

---

Figure 15: DDBM model and ECSI sampler (n = 0.3, NFE=20, FID=4.12). Samples for DIODE dataset (conditoned on depth images).
29

---

Figure 16: ECSI model and sampler (?max 0.25, 17 1.0, O. NFE=5, FID 4.16).
30

---

Figure 17: ECSI model and sampler (Yiaa 0.25, 17 1.0, = O. NFE-20, FID 3.27).
31

---

MAR
T8ON
Figure 18: DDBM model and DBIM sampler (NFE=10, FID = 2.46, AFD=5.20).
32

---

VieM Hey
190
INSSTAAN
PANK
IROM
Figure 19: DDBM model and sampler (NFE=118, FID = 1.83, AFD=6.99).
33

---

Figure 20: ECSI model and sampler ("imax 0.125, b 1.0, NFE=10, FID = 2.07, AFD=9.35).
34

---

TPM
Figure 21: DDBM model and ECSI sampler on 446 test images. (NFE=20, FID 52.01, AFD-5.60).
35

---

Figure 22: ECSI model and sampler on 446 test images. (7max = 0.125, 0.5, NFE=20, FID 55.93, AFD=7.39).
36

---

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43


Reviews:
[{"review_id": "IEKiqTHsPW", "reviewer": "Reviewer_WLqt", "summary": "This 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.", "questions": "1. 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?", "limitations": "Clarification of the paper is limited.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 1, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "After reading all the review comments, I found the proposed bridge model is unified approach and novel enough.", "rebuttal": "**Response to Weakness 1**\n\nThank you for your feedback. We agree that noise augmentation in diffusion sampling, as explored in works like EDM and Stochastic Interpolants, shares similarities with our approach. However, a key aspect of ECSI is ensuring the marginal distribution remains consistent by satisfying the Fokker-Planck equation through our derived SDEs, which we believe is a significant contribution.\n\nAdditionally, we argue that ECSI’s novelty extends beyond noise scheduling. By modifying the base distribution, we introduce a new strategy to enhance diversity, which is not sufficiently addressed by adding noise alone, as shown in our experiments. This approach is distinct and, to our knowledge, novel, with results demonstrating its effectiveness in improving diversity for diffusion bridge models.\n\n**Response to Weakness 2**\n\nWe appreciate your insights and the opportunity to clarify the distinctions between ECSI and DDBM. While ECSI shares some conceptual similarities with DDBM, its training scheme diverges in key ways, particularly through the integration of noised stochastic sampling and a novel modification of the base distribution. This modification enhances diversity beyond what endpoint conditioning or x0-prediction achieves in the original DDBM framework.\n\nWe view ECSI as an advancement akin to an \"EDM\" version of DDBM, with additional innovations. Specifically: (1) ECSI unifies multiple diffusion bridge models (I2SB, DDBM, DBIM, and EDM), as demonstrated in Table 8; (2) it decouples parameters in the transition kernel, enabling a broader design space; and (3) it supports a more flexible noise schedule during inference. Besides,  we specifically designed a noise scheduling at inference time, i.e., $\\epsilon_t= \\eta(\\gamma_t \\dot{\\gamma}_t - \\frac{\\dot{\\alpha}_t}{\\alpha_t} \\gamma_t^2)$, which achieves better results in the experiments. \n\nFurthermore, We also proposed a novel strategy that modifies the base distribution. Intuitively, this strategy interpolates between standard bridge models and unconditional diffusion models. We found that this strategy diversifies the sampling path in a complementary manner to other stochasticity controls. This strategy significantly improved the sampler diversity.\n\n**Response to Weakness 3**\n\nIn our paper, we provide a rigorous proof for Proposition 1, which establishes the theoretical foundation of ECSI and demonstrates that I2SB [1], DDBM [2], DBIM [3], and EDM can be viewed as special cases within our unified framework, see Appendix B. Due to space constraints, detailed proofs are provided in Appendix B. Please let us know if you had a different type of convergence result in mind that we failed to address in the Appendix.\n\n**Response to Question 1**\n\nWe adopt the same $x_0$ prediction and preconditioning as DDBM [2]. The training details could be found in Appendix F.\n\n**Response to Question 2**\n\nWe acknowledge the importance of minimizing discretization errors in diffusion models. However, like other samplers for continuous diffusion and diffusion bridge models, the ECSI sampler does not fully eliminate sampling discretization error. To our knowledge, no existing sampler for these models can completely mitigate this issue. To reduce discretization error, we recommend increasing the number of sampling steps or adopting higher-order discretization schemes, as these approaches have proven effective in practice.\n\n\n**Response to Question 3**\n\nThank you for your question regarding the marginal distribution of the ECSI sampler. We confirm that the marginal distribution of the ECSI sampler coincides with the true underlying marginal distribution, as established through a rigorous Fokker-Planck equation analysis. Specifically, Lemma B.2 in Appendix B provides a detailed proof demonstrating how the inclusion of a divergence-free term ensures that the marginal distribution remains consistent with the true distribution. We are happy to provide further clarification or expand on this analysis if needed.\n\n[1] Liu G H, Vahdat A, Huang D A, et al. I $^ 2$ SB: Image-to-Image Schr\\\" odinger Bridge[J]. arXiv preprint arXiv:2302.05872, 2023.\n\n[2] Zhou L, Lou A, Khanna S, et al. Denoising diffusion bridge models[J]. arXiv preprint arXiv:2309.16948, 2023.\n\n[3] Zheng K, He G, Chen J, et al. Diffusion bridge implicit models[J]. arXiv preprint arXiv:2405.15885, 2024.\n\n[4] Karras T, Aittala M, Aila T, et al. Elucidating the design space of diffusion-based generative models[J]. Advances in neural information processing systems, 2022, 35: 26565-26577."}, {"review_id": "GbcywTwFUG", "reviewer": "Reviewer_Bkd1", "summary": "This 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.", "questions": "Please see the weakness.", "limitations": "Please see the weakness.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths (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.", "was_revised": true, "final_justification": "The paper unifies a range of previous diffusion bridge models and introduces effective techniques for improving distribution matching. I believe the work has merit, and I will therefore keep my score at \"Borderline accept.\"", "rebuttal": "**Response to Weakness 1**\n\nWe provided additional design guidelines in the Appendix D. The rationale and ablation studies are as follows:\n\n$\\alpha_t$ and $\\beta_t$. Theoretically, $\\alpha_t$ and $\\beta_t$ can be freely designed, and future work may explore alternative design choices. However, in this paper, we focus on the simple case where $\\alpha_t = 1 - t$ and $\\beta_t = t$. The rationale is as follows: consider the scenario where $\\alpha_t = 1 - \\beta_t$, which represents an interpolation along the line segment between $x_0$ and $x_1$. For the path \n\n$p_t^{(1)}(x) = \\mathcal{N}((1 - \\beta_t)x_0 + \\beta_t x_1, \\gamma_t^2 \\mathbb{I})$, where $\\beta_t$ is invertible, it is straightforward to construct another path \n\n$p_t^{(2)}(x) = \\mathcal{N}((1 - t)x_0 + t x_1, \\gamma_{\\beta_t^{-1}}^2 \\mathbb{I})$, which achieves the same objective function but uses a different distribution of $t$ during training. Based on this equivalence, setting $\\alpha_t = 1 - t$ and $\\beta_t = t$ is a reasonable choice. \n\nThe shape of $\\gamma_t$. We also conducted an ablation study on $\\gamma_t$ with different shapes. Specifically, we assumed $\\gamma_t$ has the form $\\gamma_t=2 \\gamma_{\\max} \\sqrt{t^k (1 - t^k)}$, the results are shown in Fig. (7). The results indicate that the best performance is achieved when k = 1, which is the exact setting used in this paper.\n\n$\\gamma_{\\max}$. Our ablation studies on $\\gamma_{\\max}$ demonstrate that the optimal values of $\\gamma_{\\max}$ are approximately $0.125$ or $0.25$. Furthermore, the sampling paths corresponding to different choices of $\\gamma_t$ are shown in Fig. 8. Adding an appropriate amount of noise to the transition kernel helps in constructing finer details. This setting achieves the best performance across 3 different datasets: Edges 2 handbags, DIODE and ImageNet, demonstrating its generalization.\n\n$\\epsilon_t$. We use the setting $\\epsilon_t = \\eta \\left( \\gamma_t \\dot{\\gamma}_t - \\frac{\\dot{\\alpha}_t}{\\alpha_t}\\gamma_t^2 \\right).$ The ablation studies on $\\epsilon_t$ demonstrate that the optimal choice of $\\eta$ 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 $\\eta$ settings to illustrate heuristic parameter tuning techniques. The results are shown in Figures 10, 11, and 12. Too small a value of $\\eta$ results in the loss of high-frequency information, while too large a value of $\\eta$ produces over-sharpened and potentially noisy sampled images.\n\nIn our experiments, across different tasks, the superior performance is achieved by setting:\n\n $$ p_{t|0,T}(x_t | x_0, x_T) = \\mathcal{N}\\left(x_t; (1-t)x_0 + t x_T, \\frac{1}{4}\\gamma_{\\max}^2 t(1-t)\\mathbb{I}\\right) $$\n\nThe best results are achieved by setting $\\gamma_{\\max}=0.125$ and $\\gamma_{\\max}=0.25$, and the best sampler is achieved by setting $\\epsilon_t = \\dot{\\gamma_t} \\gamma_t - \\frac{\\dot{\\alpha_t}}{\\alpha_t}\\gamma_t^2=\\gamma_{\\max}^2 / 8$. This setting can be adapted to different kinds of image-to-image translation problems, e.g., edges to handbags, depth image to RGB image, deblurring.\n\n**Response to Weakness 2**\n\nWe appreciate the reviewer’s feedback regarding the scalability of ECSI. Our experiments were conducted on three datasets—64x64 Handbags, 256x256 DIODE, and 256x256 ImageNet—which are standard benchmarks widely used in prior works, including I2SB [1], DDBM [2], and DBIM [3]. These datasets were chosen to ensure fair comparisons with existing methods. While we acknowledge the importance of evaluating scalability on higher-resolution or more complex industrial applications, the consistent performance improvements observed across these datasets provide strong evidence of ECSI’s efficiency and quality. To further address scalability concerns, we are open to extending our evaluation to larger-scale, higher-resolution datasets in future work and can include a discussion of this in the paper to highlight potential applicability in such settings.\n\n**Response to Weakness 3**\n\nWe thank the reviewer for highlighting the need for deeper analysis of the interaction between path stochasticity (controlled by $\\gamma_t$) and sampler stochasticity (controlled by $\\epsilon_t$). In our experiments, we explicitly address this interplay by selecting the sampler stochasticity based on the path stochasticity, as defined by the relation $\\epsilon_t= \\eta(\\gamma_t \\dot{\\gamma}_t - \\frac{\\dot{\\alpha}_t}{\\alpha_t} \\gamma_t^2)$, where $\\eta$ controls the stochasticity strength at inference time. This is also an innovation of ECSI and this selection has not been demonstrated in previous work. We observed that for different noise levels of path stochasticity (controlled by $\\gamma_t$), the best $\\eta$ is achived at $\\eta=1$. We will include these details in the main text and emphasize our contribution on specifically designing of $\\epsilon_t$.\n\n\n[1] Liu G H, Vahdat A, Huang D A, et al. I $^ 2$ SB: Image-to-Image Schr\\\" odinger Bridge[J]. arXiv preprint arXiv:2302.05872, 2023.\n\n[2] Zhou L, Lou A, Khanna S, et al. Denoising diffusion bridge models[J]. arXiv preprint arXiv:2309.16948, 2023.\n\n[3] Zheng K, He G, Chen J, et al. Diffusion bridge implicit models[J]. arXiv preprint arXiv:2405.15885, 2024."}, {"review_id": "Iq0mr6sXhP", "reviewer": "Reviewer_NpZo", "summary": "The 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.", "questions": "- 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\"", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### 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.", "was_revised": true, "final_justification": "The reviewer assigns a score of 4 rather than 5, as it remains unclear why the authors do not compare their Endpoint-Conditioned Stochastic Interpolants framework to the Stochastic Interpolants framework without endpoint conditioning. Nonetheless, even without this comparison, the reviewer considers the work a valuable contribution that advances the theory of diffusion bridge models and is supported by strong empirical results.", "rebuttal": "**Response to Weakness 1 and Question 2, 3**\n\nWe sincerely thank the reviewer for their insightful comments and for recognizing the novelty of ECSI compared to Albergo et al. [1]. We acknowledge the strong connection between Proposition 4.1 in our work and the derivations for linear interpolants in Albergo et al. [1, Section 4]. We will revise the text to more explicitly emphasize the similarities as well as the conceptual differences.\n\n> 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 $X_t = \\alpha(t) X_0 + \\beta(t) X_1 + \\gamma(t) Z$. a detailed discussion of the choice of $\\alpha(t)$, $\\beta(t)$, $\\gamma(t)$ in [Section 4, 1].\n\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\nIn short, 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: $\\mathbb{E}[x_0 \\mid x_t]$ and $\\mathbb{E}[x_1 \\mid x_t]$, ECSI only estimates $\\mathbb{E}[x_0 \\mid x_t, y]$ (where $y=x_1$).\n\nFirst, SI can be extended to a conditional version. Conditioned Stochastic Interpolants build a marginal probability path $p_{t \\mid y}$ using a mixture of interpolating densities: $p_{t \\mid y}(x) = \\int p_t(x_t \\mid x_0, x_1) \\pi(x_0, x_1 \\mid y) dx_0 dx_1$, where $\\pi(x_0,x_1 \\mid y)$ is a joint distribution with marginals $\\pi_{0 \\mid y}(x_0 \\mid y)$ and $\\pi_{1 \\mid y} (x_1 \\mid y)$. For linear interpolants given by: $X_t=\\alpha_t X_0 + \\beta_t X_1 + \\gamma_t z$. The conditional kernel $p_t(x_t \\mid x_0, x_1)$ is  given by a Gaussian distribution: $p_t(x_t \\mid x_0, x_1) = \\mathcal{N}(\\alpha_t x_0 + \\beta_t x_1, \\sigma_t^2 I), \\forall ~ t \\in [0, 1]$. Then we can sample from the conditional distribution $p_{0 \\mid y}(x_0 \\mid y)$ by running a stochastic process $p_{t \\mid y} (x_t \\mid y)$ from time $t = 1$ to $t = 0$, which is given by the following SDE:\n\n$$dx = b(t, x, y)dt + \\sqrt{2 \\epsilon_t} dW_t, \\quad x_1 \\sim p_{1 \\mid y},$$\n\nwhere the drift term $b(t, x, y)$ is:\n\n$$b(t, x, y) = \\dot{\\alpha}_t \\mathbb{E}[x_0 \\mid x, y] + \\dot{\\beta}_t  \\mathbb{E}[x_1 \\mid x, y] +( \\dot{\\gamma}_t + \\frac{\\epsilon_t}{\\gamma_t}) \\mathbb{E}[z \\mid x, y]$$\n\nAs $y$ represents null conditioning, we recover the original sampler of Stochastic Interpolants. In the drift term, $\\mathbb{E}[x_0 \\mid x, y]$, $\\mathbb{E}[x_1 \\mid x, y]$ and $\\mathbb{E}[z \\mid x, y]$ are unknown, but we only need to estimate two of them, since $$\\mathbb{E}[x_t \\mid x_t, y]=\\alpha_t \\mathbb{E}[x_0 \\mid x_t, y] + \\beta_t \\mathbb{E}[x_1 \\mid x_t, y] + \\gamma_t \\mathbb{E}[z\\mid x_t, y]=x_t$$\n\nWe can further reduce the number of unknown term by endpoint-conditioning. Here as we replace condition $y$ to be endpoint $x_1$, the term $\\mathbb{E}[x_1 \\mid x, x_1]=x_1$. So we have:\n\n$$b(t, x, y)=\\dot{\\alpha}_t \\mathbb{E}[x_0 \\mid x, x_1] + \\dot{\\beta}_t x_1 + (\\dot{\\gamma}_t + \\frac{\\epsilon_t}{\\gamma_t}) \\mathbb{E}[z \\mid x, x_1]$$\n\nThis is exactly the sampler for ECSI in Proposition 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: $\\mathbb{E}[x_0 \\mid x_t]$ and $\\mathbb{E}[x_1 \\mid x_t]$, ECSI only estimates $\\mathbb{E}[x_0 \\mid x_t, x_1]$.\n\n\n**Response to Weakness 2 and Question 4, 5**\n\nWe thank the reviewer for this insightful feedback on the evaluation of diversity in diffusion models, including some works we were not aware of. Below, we discuss the applicability of each method. We find that Vendi score is a good fit for our evaluation, and we will add this metric to our diversity evaluation results.\n\n\n(1) $\\beta$-Recall, Improved-Recall and Feature Likelihood Divergence are not applicable. In our work, we focus on the *conditional diversity* of images given a certain condition. For example, given a particular edge image, can we generate diverse images with different colors and textures. Therefore, we want to calculate the diversity of the generated set of images without knowing any prior distribution. Therefore, $\\beta$-Recall, Improved-Recall and Feature Likelihood Divergence may not be applicable in our case. While we totally agree that the Vendi Score is well-suited in our case. \n\n\n(2) A comparison between AFD and other 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. For example, consider two sets of feature vectors: set $a=[2, 1, 1, 1]$, set $b=[1, 1.2, 1.3, 1.6]$, For AFD, the average pairwise Euclidean distance in set $a$ is larger due to the outlier $2$, resulting in $AFD(a)>AFD(b)$. For the Vendi Score, set $b$ has greater diversity due to its more varied feature values, so $VS(a)<VS(b)$.\n\n\n(3) Additional experiments. We have included an evaluation on different denoisers and samplers using AFD and VS, as shown in the table below. The results demonstrate that ECSI, particularly with the modified base density, consistently outperforms the DDBM checkpoint with DBIM or ECSI samplers across both AFD and VS metrics at varying NFEs (5, 10, 20). \n\n  \n|                                   |   AFD | AFD    | AFD    | VS    | VS     | VS     |\n| --------------------------------- | ----: | ------ | ------ | ----- | ------ | ------ |\n|                                   | NFE=5 | NFE=10 | NFE=20 | NFE=5 | NFE=10 | NFE=20 |\n| DDDBM checkpoint + DBIM sampler   |  5.63 | 5.20   | 5.84   | 1.16  | 1.23   | 1.26   |\n| A: DDBM checkpoint + ECSI sampler |  5.11 | 5.70   | 6.04   | 1.15  | 1.20   | 1.23   |\n| B: ECSI checkpoint + sampler      |  6.00 | 6.05   | 6.25   | 1.22  | 1.25   | 1.28   |\n| B + Modified base density         |  **8.53** | **9.35**   | **9.65**   | **1.48**  | **1.63**   | **1.69**   |\n\n**Response to Question 1**\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\n\nWe will re-word this claim in a clearer way. 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.\n\n**Response to Question 6**\n\n> What is the computational cost of calculating the Average Feature Distance (AFD)?\n\n\nThe computational cost of calculating the Average Feature Distance (AFD) is $O(n^2)$, compared to VS $O(n^3)$. AFD is more time efficient, but VS is less sensitive to outliers, as described above.\n\n**Response to Minor comments**\n\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\nIn the original DDBM paper, they use the same notation for both forward process and reverse process. But we agree that it's better to use different notations for better clarification. We will revise this typo in the revised paper.\n\n> There is a reference missing for the derivation of the reverse model in eq. (3) and eq. (4)\n\n> There is a reference missing for the derivation of the denoising bridge score matching objective in eq. (5)\n\nWe will add appropriate references to the derivations of the reverse model in eqs. (3) and (4), and the denoising bridge score matching objective in eq. (5), citing the relevant source in the revised manuscript.\n\n\n> $\\log p_{t \\mid 0, T}$ is missing it's input in eq. (5)\n\n> line 105, $t x_1$ instead of $t x_0$\n\n> (line 113): \"by the SDE\" instead of \"by SDE\"\n\nThanks for the reviewer. We will revise those typos or mistakes in the revised version.\n\n> There is no reference in the main paper to the proof of Proposition 4.1 provided in the Appendix.\n\nWe will include a reference in the main paper to direct readers to the proof of Proposition 4.1 in the Appendix.\n\n\n\n\n\n\n\n[1] Albergo et al. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint arXiv:2303.08797, 2023.\n\n  \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  \n\n[3] Kynkäänniemi et al. Improved Precision and Recall Metric for Assessing Generative Models. NeurIPS 2019.\n\n  \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  \n\n[5] Friedman et al. The Vendi Score: A Diversity Evaluation Metric for Machine Learning. Transactions on Machine Learning Research, 2023.\n\n  \n\n[6] Jiralerspong et al. Feature Likelihood Divergence: Evaluating the Generalization of Generative Models Using Samples. NeurIPS 2023."}, {"review_id": "msNQ6frjrv", "reviewer": "Reviewer_CniD", "summary": "This 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.", "questions": "see above Strengths and Weaknesses.", "limitations": "yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "I still affirm that this work fills some theoretical gaps. However, I am concerned that this writing style is overly exaggerated. Therefore, I maintain my rating unchanged.", "rebuttal": "We thank the reviewer for the detailed feedback. We are pleased to hear that you believe the paper introduces new ideas and makes useful contributions.\n\nOur claims are mostly combinations or reformulations of existing ideas. Surprisingly, though, there are gaps in the bridge literature that largely stem from the difficulty of combining ideas conceived under different conceptual frameworks (e.g. Stochastic interpolants[1] versus DDBM [2]). We don't believe it is too bold to ask \"Have the bridge paths been fully explored?\", to which the answer is a simple \"no\". However, we don't want to give the impression that our paper provides the final grand unified theory of bridge paths, and we will clarify the language accordingly by focusing and narrowing the scope of our statements in several places. Nevertheless, there are significant gaps in the existing literature that our work helps to fill. \n\n> 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\nWe provided additional design guidelines in the Appendix D. The rationale and ablation studies are as follows:\n\n$\\alpha_t$ and $\\beta_t$. Theoretically, $\\alpha_t$ and $\\beta_t$ can be freely designed, and future work may explore alternative design choices. However, in this paper, we focus on the simple case where $\\alpha_t = 1 - t$ and $\\beta_t = t$. The rationale is as follows: consider the scenario where $\\alpha_t = 1 - \\beta_t$, which represents an interpolation along the line segment between $x_0$ and $x_1$. For the path \n\n$p_t^{(1)}(x) = \\mathcal{N}((1 - \\beta_t)x_0 + \\beta_t x_1, \\gamma_t^2 \\mathbb{I})$, where $\\beta_t$ is invertible, it is straightforward to construct another path \n\n$p_t^{(2)}(x) = \\mathcal{N}((1 - t)x_0 + t x_1, \\gamma_{\\beta_t^{-1}}^2 \\mathbb{I})$, which achieves the same objective function but uses a different distribution of $t$ during training. Based on this equivalence, setting $\\alpha_t = 1 - t$ and $\\beta_t = t$ is a reasonable choice. \n\nThe shape of $\\gamma_t$. We also conducted an ablation study on $\\gamma_t$ with different shapes. Specifically, we assumed $\\gamma_t$ has the form $\\gamma_t=2 \\gamma_{\\max} \\sqrt{t^k (1 - t^k)}$, the results are shown in Fig. (7). The results indicate that the best performance is achieved when k = 1, which is the exact setting used in this paper.\n\n$\\gamma_{\\max}$. Our ablation studies on $\\gamma_{\\max}$ demonstrate that the optimal values of $\\gamma_{\\max}$ are approximately $0.125$ or $0.25$. Furthermore, the sampling paths corresponding to different choices of $\\gamma_t$ are shown in Fig. 8. Adding an appropriate amount of noise to the transition kernel helps in constructing finer details. This setting achieves the best performance across 3 different datasets: Edges 2 handbags, DIODE and ImageNet, demonstrating its generalization.\n\n$\\epsilon_t$. We use the setting $\\epsilon_t = \\eta \\left( \\gamma_t \\dot{\\gamma}_t - \\frac{\\dot{\\alpha}_t}{\\alpha_t}\\gamma_t^2 \\right).$ The ablation studies on $\\epsilon_t$ demonstrate that the optimal choice of $\\eta$ 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 $\\eta$ settings to illustrate heuristic parameter tuning techniques. The results are shown in Figures 10, 11, and 12. Too small a value of $\\eta$ results in the loss of high-frequency information, while too large a value of $\\eta$ produces over-sharpened and potentially noisy sampled images.\n\nIn our experiments, across different tasks, the superior performance is achieved by setting:\n\n $$ p_{t|0,T}(x_t | x_0, x_T) = \\mathcal{N}\\left(x_t; (1-t)x_0 + t x_T, \\frac{1}{4}\\gamma_{\\max}^2 t(1-t)\\mathbb{I}\\right) $$\n\nThe best results are achieved by setting $\\gamma_{\\max}=0.125$ and $\\gamma_{\\max}=0.25$, and the best sampler is achieved by setting $\\epsilon_t = \\dot{\\gamma_t} \\gamma_t - \\frac{\\dot{\\alpha_t}}{\\alpha_t}\\gamma_t^2=\\gamma_{\\max}^2 / 8$. This setting can be adapted to different kinds of image-to-image translation problems, e.g., edges to handbags, depth image to RGB image, deblurring.\n\n[1] Albergo M S, Boffi N M, Vanden-Eijnden E. Stochastic interpolants: A unifying framework for flows and diffusions[J]. arXiv preprint arXiv:2303.08797, 2023.\n\n[2] Zhou L, Lou A, Khanna S, et al. Denoising diffusion bridge models[J]. arXiv preprint arXiv:2309.16948, 2023."}]

Author rebuttal:

Meta-review: This paper presents a unified framework, ECSI, for diffusion bridge models, aiming to improve image-to-image translation. The key idea is to expand the design space by preconditioning, endpoint conditioning, and a flexible sampling algorithm. The main strengths are the unification of prior models and strong empirical results showing better image quality, sampling speed, and output diversity. A primary weakness is that some design principles are heuristic. However, the paper's solid performance and its novelty to improving sample diversity are convincing reasons for acceptance.

During the rebuttal, reviewers raised several points. Key issues included the method's novelty compared to prior work (WLqt), the lack of comparison to the unconditional stochastic interpolants framework (NpZo), and an overclaimed writing style (CniD). The authors effectively addressed most concerns. They clarified the theoretical novelty and differences in the training scheme, convincing Reviewer WLqt. For Reviewer NpZo, they provided a conceptual comparison to stochastic interpolants. They also added new experiments using the Vendi score to validate their diversity claims. After reading all discussions, I agree with the reviewers and recommend accepting the paper, as the authors thoroughly addressed the reviewers’ concerns.
Decision: Accept (poster)