Diffusion Bridge Implicit Models
Authors: Kaiwen Zheng
Guande He
Jianfei Chen
Fan Bao
Jun Zhu
Venue: ICLR 2025 Poster (2025)
Keywords: Diffusion Bridge Models
Image Translation
Fast Sampling
Primary area: generative models

Abstract: Denoising diffusion bridge models (DDBMs) are a powerful variant of diffusion models for interpolating between two arbitrary paired distributions given as endpoints. Despite their promising performance in tasks like image translation, DDBMs require a computationally intensive sampling process that involves the simulation of a (stochastic) differential equation through hundreds of network evaluations. In this work, we take the first step in fast sampling of DDBMs without extra training, motivated by the well-established recipes in diffusion models. We generalize DDBMs via a class of non-Markovian diffusion bridges defined on the discretized timesteps concerning sampling, which share the same marginal distributions and training objectives, give rise to generative processes ranging from stochastic to deterministic, and result in diffusion bridge implicit models (DBIMs). DBIMs are not only up to 25$\times$ faster than the vanilla sampler of DDBMs but also induce a novel, simple, and insightful form of ordinary differential equation (ODE) which inspires high-order numerical solvers. Moreover, DBIMs maintain the generation diversity in a distinguished way, by using a booting noise in the initial sampling step, which enables faithful encoding, reconstruction, and semantic interpolation in image translation tasks. Code is available at \url{https://github.com/thu-ml/DiffusionBridge}.

Published as a conference paper at ICLR 2025
DIFFUSION BRIDGE IMPLICIT MODELS
Kaiwen Zheng 12- Guande He': Jianfei Chen', Fan Bao", Jun Zhu" f 123 'Dept. of Comp. Sci. & Tech., Institute for AI, BNRist Center 'Tsinghua- -oathhha---hhh-ae Joint ML Center, Tsinghua University, Beijing, China "Shengshu Technology, Beijing Pazhou Lab (Huangpu), Guangzhou, China zkwthu@gmaill com; guande hel 7@outlook com; fan. het.bao@@shngghu..aa (jianfeic, dcsz (iazz)@ttsin (etttinnnnuae edu. cn
ABSTRACT
Denoising diffusion bridge models (DDBMs) are powerful variant of diffu- sion models for interpolating between two arbitrary paired distributions given as endpoints. Despite their promising performance in tasks like image transla- tion, DDBMs require computationally intensive sampling process that involves the simulation of (stochastic) differential equation through hundreds of network evaluations. In this work, take the first step in fast sampling of DDBMs with- out extra training, motivated by the well-established recipes in diffusion mod- els. We generalize DDBMs via class of non-Markovian diffusion bridges de- fined on the discretized timesteps concerning sampling, which share the same marginal distributions and training objectives, give rise generative processes ranging from stochastic to deterministic, and result in diffusion bridge implicit models (DBIMs). DBIMs are not only up to 25 x faster than the vanilla sam- pler of DDBMs but also induce novel, simple, and insightful form of ordinary differential equation (ODE) which inspires high-order numerical solvers. More- over, DBIMs maintain the generation diversity in a distinguished way, by using booting noise in the initial sampling step, which enables faithful encoding, recon- struction, and semantic interpolation in image translation tasks. Code is available at hhtttps://githht wwt.ccommottcacecceeeeee mllummm1/Diee http://fffuiiiorrrr.ooo
INTRODUCTION
et al. 2015 Ho ct al. 2020) represent a family of powerful generative models, with high-quality generation ability, stable training, and scalability to high dimensions. They have consistently obtained state-of-the-aat performance in various do- mains, including image synthesis (Dhariwal & Nichol [021]] Karras al. [022), speech and video generation (Chen et al. [2021a] Ho et al. (022). controllable image manipulation (Nichol et al.)
Diffusion models (Song et al. 2021c
2022 Ramesh et al.] [2022 Rombach al. 2022 Meng al. 2022), density estimation (Song 2022a Zheng et al. 2023b) and inverse problem solv-
et al. 2021b] Kingma et al. 2021 Lu et al.
ing (Chung et al.) [0022] Kawar et al. 2022). They also act as fundamental components of modern
text-to-image (Rombach et al., (002)) and text-to-video (Gupta et al. 2023 Bao et al. 2024) synthe- sis systems, ushering the era Al-generated content.
However, diffusion models are not well-suited for solving tasks like image translation or restoration, where the transport between two arbitrary probability distributions is to modeled given paired endpoints. Diffusion models are rooted in stochastic process that gradually transforms between data and noise, and the prior distribution is typically restricted to the "non-informative" random Gaussian noises. Adapting diffusion models to scenarios where more informative prior natu- rally exists, such as image translatioo/restoraaiin, involves modifying the generation pipeline (Meng et al. 2022 Su al. 2022) adding extra guidance terms during sampling (Chung et al.) 2022 Kawar et al. 2022). On the one hand, these approaches are task-agnostic at training and adaptable multiple tasks inference time. On the other hand, despite recent advances in accelerated inverse problem solving (Liu et al. 2023a Pandey et al. 2024), they inevitably deliver either sub-par perfor- mance or slow and resource- inttnrieinteessve inference compared training-base ones. Tailored diffusion
"Equal contribution;
The corresponding author.

---

Published as a conference paper at ICLR 2025
Condition
DDBM
(c)
DBIM (1) O) (Ours) ND (FID 4.51)
DBIM (3rd order) (Ours) NFF=10 (FID 4.34)
Figure I: Inpainting results on the ImageNct 2 256 dataset Deng et al. 2009) by DDBM (Zhou et al. 2023
with 100 number of function evaluations (NFE), and DBIM (ours) with only TO NFB.
model variants become essential in task-specific scenarios where paired training data are available and fast inference is critical.
Recently, denoising diffusion bridge models (DDBMs) (Zhou al., 2023 have emerged as a scal- able and promising approach to solving the distribution translation tasks. By considering the reverse- time processes of diffusion bridge, which represent diffusion processes conditioned on given end- points, DDBMs offer general framework for distribution translation. While excelling in image translation tasks with exceptional quality and fidelity, sampling from DDBMs requires simulating (stochastic) differential equation corresponding to the reverse-time process. Even with the introduc- tion of their hybrid sampler, achieving high-fidelity results for high-resolution images still demands over 100 steps. Compared to the efficient samplers for diffusion models (Song et al. 202 Zhang & Chen 2022 Lu et al.] 2022b), which require around 10 steps to generate reasonable samples, DDBMs are falling behind, urging the development of efficient variants.
This work represents the first pioneering effort toward accelerated sampling of DDBMs. As sug- gested by well-established recipes in diffusion models, training-free accelerations of diffusion sam- pling primarily focus on reducing stochasticity (e.g., the prominent denoising diffusion implicit models, DDIMs) and utilizing higher-order information (e.g., high-order solvers). We present diffu- sion bridge implicit models (DBIMs) as an approach that explores both aspects within the diffusion bridge framework. Firstly, we investigate the continuous-timg cretized timesteps and generalize them to a series of non-Mart non-Markovian diffusion bridges controlled by a variance parameter, while maintaining identical marginal distributions and training objectives as DDBMs. Secondly, the induced reverse generative processes correspond to sampling procedures of varying levels of stochasticity, including deterministic ones. Consequently, DBIMs can be viewed as bridge counterpart and extension of DDIMs. Furthermore, in the continuous time limit, DBIMs can induce a novel form of ordinary differential equation (ODE), which is linked to the probability flow ODE (PF-ODE) in DDBMs while being simpler and significantly more efficient. The induced ODE also facilitates novel high-order numerical diffusion bridge solvers for faster convergence.
forward process of DDBMs on dis-
We demonstrate the superiority of DBIMs by applying them in image translation and restoration tasks, where they offer up to 25 x faster sampling compared to DDBMs and achieve state-of-the-ar performance on challenging high-resolution datasets. Unlike conventional diffusion sampling, the initial step in DBIMs is forced to be stochastic with a booting noise to avoid singularity issues arising from the fixed starting point on bridge. By viewing the booting noise as the latent variable, DBIMs maintain the generation diversity of typical generative models while enabling faithful encoding, reconstruction, and semantically meaningful interpolation in the data space.
BACKGROUND
2.1 DIFFUSION MODELS
Given d- d-dimensional data distribution go (xx), diffusion models (Song et 2021c Sihl-Dickstein (et al. 2015 Ho et al. 2020) build diffusion process by defining a forward stochastic differential equation (SDE) starting from 2o co:
da. f())x,d g())du
2

---

Published as a conference paper at ICLR 2025
where & [)] for some finite horizon T, f.g: [0, - IR is the scalar-valued drift and diffusion term, and we E IR a standard Wiener process. As a linear SDE, the forward process owns an analytic Gaussian transition kernel
(010 = -((az20,0
(2)
by Ito's formula Ito al de (²(()
1951 where Or are called noise schedules satisfying f(L)
do
2dlaar de
2 log ai of (Kingma et al. 2021). The forward SDE is accompanied
de
by a series of marginal distributions { of [= and f: are properly designed so that the terminal distribution is approximately pure Gaussian, i.e., @r (IT) s (0,
To sample from the data distribution (o())), we can solve the reverse SDE or probability flow T to
ODE (Song al. (202cc) from
dæ. [f(t) - (t) lo (2: + g(t)dw;
(3)
1 dx. f(t)zz (t) log (xx) dt.
(4)
They share the same marginal distributions fq with the forward SDE, where w the reverse-time Wiener process, and the only unknown term Vz, log q: is the score function of the marginal density gr. By denoising score matching (DSM) (Vin-) cent 2011). a score prediction network so(0,, t) can be parameterized to minimize ) (.jo:) [w - Va, log (=)o(()(<>))))
where 4:j0 is the analytic forward transition kernel and w((1 is a positive weighting function. 89 can be plugged into the re-
verse SDE and the probability flow ODE obtain the parameterized diffusion SDE and diffusion There are various dedicated solvers for diffusion SDE or ODE (Song et al. 2021a la Zhang &
ODE.
Chen 2022
Lu al. 2022b Gonzalez et 2023).
2.2
DENOISING DIFFUSION BRIDGE MODELS
Denoising diffusion bridge models (DDBMs) (Zhou et al. 2023) consider driving the diffusion process in Eqn. (L) to arrive a particular point y F2d almost surely via Doob's 's h-transsormm (Doob
& Doob 1984):
dæ;; = [(1)an + (1) logg(ær = y (22)) + g(L)dur, Fg 4o = Pdata ET y.
(5)
The endpoint y is not restricted to Gaussian noise as in diffusion models, but instead chosen as informative priors (such as the degraded image in image restoration tasks). Given a starting point To. the process in Eqn. (5) also owns analytic forward transition kernel
0i SNRT
by = a0 SNRT SNR, ) = of SNRT,
(a /20, æT) = Naa.ttr & I)
SNR,
SNR,
(6)
which forms diffusion bridge, and SNR, = of fo the signal-to-noise ratio time t. DDBMs
show that the forward process Eqn. 5 is associated with reverse SDE and a probability flow ODE starting from Ør = y:: dæ; [f(t)ar (+)) (Va, log ((2 v) Va, log (r (ar y)(r:))]] dt + g())du; (7)
dæ, f(t)xx - (t)
(ev,,
Vx. log g(xr]//T = v. log ar = y (z:z)) dt.
(8)
They share the same marginal distributions [g(2;/kTT = y) with the forward process, where w. is the reverse-time Wiener process, 2ry is analytically known similar to Eqn. ! and the only unknown term V log q( Ø = y) is the bridge score function. Denoising bridge score matching (DBSM) proposed learn the unknown score term q( aaT = y) with a parameterized network so (,, y). by minimizing
(.) = E, w) papaa ( Be (e
[wo(t]]so f, V log q(x, [mo, ør y
where ax Øo. Øт y) is the forward transition kernel in Eqn. 66) and w(t) is positive weighting function. To sample from diffusion bridges with Eqn. 77) and Eqn. (8. DDBMs propose high-order hybrid sampler that alternately simulates the ODE and SDE steps to enhance the sample quality, inspired by the Heum sampler in diffusion models (Karras et al.) 2022). However, is not dedicated to diffusion bridges and lacks theoretical insights in developing efficient diffusion samplers.
3

---

Published as a conference paper at ICLR 2025
3 GENERATIVE MODEL THROUGH NON-MARKOVINE DIFFUSION BRIDGES
We start by examining the forward process of the diffusion bridge (Eqn. ( on a set of discretized timesteps to < 81 the bridge score vz log q(<;) (T)) only depends on the marginal distribution ((x;|2T), we can construct alternative probabilistic models that induce new sampling procedures while reusing the learned bridge score sa(x;, uT), as long they agree on the N marginals {q( (T)) [T)) N-1
< tN-1- iN = 'T' that will be used for reverse sampling. Since
3.1 NON-MARKOVIAN DIFFUSION BRIDGES AS FORWARD PROCESS
We consider a family of probability distributions (z (T), controlled by a variance param- eter p E R 1-
g(0) (Foow (TT) (3ta)
II
(P)) (xt [0 Lt, 1;
(10)
where go is the data distribution at time 0 and for 1 S n < N N-1 -
((d) ( zo Bints xT) Nai,, xT bt. 2D
V /o ph
Utnt1
at,+1 2T
bi 11
20
P
Cta-1
(11)
where Pn is the n-th element of p satisfying PN-1 CtN-1 and at, by, Ct are terms related to the noise schedule, defined in the original diffusion bridge (Eqn. (6)). Intuitively, this decreases the variance (noise level) of the bridge while incorporating additional noise components from the last step. Under this construction, we can prove that g(o) maintains consistency in marginal distributions with the original forward process 9 governed by Eqn.
Proposition 3.1 (Marginal Preservation, proof in Appendix B.I For 0 n N we have ((o) (It,, [T) (t,, (T))
The definition of g(o) in Eqn. 100 represents the inference process, since it is factorized as the distribution of Ht,, given Ttn+1 at the previous timestep. Conversely, the forward process gio) PD, , TT) can be induced by Bayes' rule (Appendix C.I). As æt,, in g(p) can si- multaneously depend øt, and øo. refer to it non-Markovian diffusion bridges, in contrast to Markovian ones (such as Brownian bridges, and the diffusion bridge defined by the forward SDE in Eqn. 5 which should satisfy (3 (m, Ft 2T) q(2t (t,, TT)-
3.2 REVERSE GENERATIVE PROCESS AND EQUIVALENT TRAINING OBJECTIVE
Eqn. 10 can be naturally transformed into a parameterized and learnable generative model, by replacing the unknown Zy in Eqn. (TO) with a data predictor xo t, 2T). Intuitively, æ; the diffusion bridge is a weighted mixture of ær; 2o and some random Gaussian noise according to Eqn. (6) where the weightings at, b or are determined by the timestep The network an is trained to recover the clean data zo given I,, TT and t.
Specifically, we define the generative process starting from æT as
[N(x(( Er, P)I)), (()) (zt,, (2 (Z tn+1, TT). I Dintn: RT), 1 isns
Pe (T Fla+1 FT
(12)
and the joint distribution as po (oto ( = I po (at., 1 RT). To optimize the network parameter 0, we can adopt the common variational inference objective as in DDPMs (Ho et al.) 2020), except that the distributions are conditioned on æT:
J(o) = Eq(mr) B (atto.x (rr) log q (Itt.N, 120, xT)
log pe (Itm [r)
(13)
It seems that the DDBM objective C., in Eqn. (9) is distinct from J (p)); respectively, they are defined on continuous and discrete timesteps; they originate from score matching and variational inference; they have different parameterizations of score and data predictior However, we show they are equivalent by focusing on the discretized timesteps and transforming the parameterization.
'The diffusion bridge models are usually parameterized differently from score prediction, but can be con- verted score prediction. See Appendix [. for details.
4

---

Published as a conference paper at ICLR 2025
Table Comparison between different diffusion models and diffusion bridge models.
Difrasion Models
Diffusion Rridge Models
DDPM Ho et all 2020
ScureSDE Song 2021e
DDIM Song et al, poz2aa]
JSB (Liu et al. 20235
DDBB Zhou et al. 2023
DBIM (Ours)
Noise Schedule
VP
Any Diserete Ve (an- ao) a (90a))
Vid
Any
Timestops
Furward Distribution
glatz ita)
di lia sex) No (P (st)) x
v lns, aby
olar Mo iia (Py) 19
Inference Procoss
(in (
SD5O0D:
SDEDDE
Non-Markovian
For O, there exists certain weights 7 so that (O) C. (0)+ on the discretized timesteps Enlan. where C is a constant irrelevant to Besides, the bridge score predictor so (0) has the following relationship with the data predictor ag in (())
Proposition 3.2 (Training Equivalence, proof in Appendix B.2
so so( (tr = Dt CgT bezr xz, t, xT)
(14)
Though the weighting 7 may not precisely match the actual weighting for training Sa. this dis- crepancy doesn't t affect our utilization of 8g (Appendix C.2). Hence, it is reasonable to reuse the network trained by L while leveraging various p for improved sampling efficiency.
SAMPLING WITH GENERALIZED DIFFUSION BRIDGES
Now that we have confirmed the rationality and built the theoretical foundations for applying the generalized diffusion bridge pa to pretrained DDBMs, range of inference processes now at our disposal, controlled by the variance parameter This positions us to explore the resultant sampling procedures and the effects of p in pursuit of better and more efficient generation.
4.1 DIFFUSION BRIDGE IMPLICIT MODELS
Suppose we sample in reverse time on the discretized timesteps = to < f < ... fx-1 <
iN T. The number N and the schedule of sampling steps be made independently of the original timesteps on which the bridge model trained, whether discrete (Liu et al.) 2023b) or continuous (Zhou al. 2023). According to the generative process pa Eqn. 12 the updating rule from f1+1 to in is described by
Ft. at, FT be, to + Gr -Ph Dtat1 at.41 2T bt To +pnc, N(0, (15)
Ct predicted noise
where to = xe (Wtt,, (. 11+1, 2T) denotes the predicted clean data at time O.
Intuition of the Sampling Procedure Intuitively, the form of Eqn. (15) resembles the forward transition kernel the diffusion bridge in Eqn. (6) (which can be rewritten as æ; a,rr by + qeC, ~ (O, I)). In comparison, 2g is substituted with the predicted &.. and portion of the standard Gaussian noise now stems from the predicted noise é. The predicted noise é is derived from 1 at the previous timestep and can be expressed by the predicted clean data abo.
Effects of the Variance Parameter We investigate the effects of the variance parameter from the theoretical perspective by considering two extreme cases. Firstly, we note that when
On ot., v SNR,,+1 for each S n < N - 1, the Ør term Eqn. 15 is canceled out. In this scenario, the forward process in Eqn. (4.1 becomes a Markovian bridge (see details in Appendix C.I.. Besides, the inference process will get rid of xT and simplify to Po (x). akin to the sampling mechanism in DDPMs (Ho et al.) 2020). Secondly, when Pn. 0 for each S n < N -, the inference process will be free from random noise and composed of deterministic iterative updates, characteristic of an implicit probabilistic model (Mohamed & Lakshminarayanan,
SNR.,
5

---

Published as a conference paper at ICLR 2025
== 1=T
condition
deterministic
booting
Figure 2: Illustration of the DBIM's deterministic sampling procedure when 0.
2016). Consequently, name the resulting model diffusion bridge implicit models (DBIMs), draw- ing parallels with denoising diffusion implicit models (DDIMs) (Song et al. 2021a). DBIMs serve as the bridge counterpart and extension of DDIMs, as illustrated in Table
When we choose that lies between these two boundary cases, we can obtain non-Markovian dif- fusion bridges with intermediate and non-zero stochastic levels. Such bridges may potentially yield superior sample quality. We present detailed ablations in Section 6.]
The Singularity at the Initial Step for Deterministic Sampling One important aspect note regarding DBIMs is that its initial step exhibits singularity when = 0. a property essentially distinct from DDIMs in diffusion models. Specifically, in the initial step we have fx+1 = T, and Ct in the denominator in Eqn. (T5) equals O. This phenomenon can be understood intuitively: given a fixed starting point ær, the variable x; for t < T typically still stochastically distributed (the marginal po (z)( is not a Dirac distribution). For instance, in inpainting tasks, there should be various plausible complete images corresponding to a fixed masked image. However, fully deterministic sampling procedure disrupts such stochasticity.
To be theoretically robust, we employ the other boundary choice Pr. ot. V SN in the SNR. initial step which is aligned with our previous restriction that PN-1 = Ctw-1 This will introduce an additional standard Gaussian noise 6 which we term as the booting noise. It accounts for the stochasticity of the final sample 2p under given fixed or and can be viewed as the latent variable. We illustrate the complete DBIM pipeline in Figure 2
4.2 CONNECTION TO PROBABILITY FLOW ODE
It is intuitive to perceive that the deterministic sampling can be related to solving an ODE. By setting P = 0, fn+1 = and fn+1 - tn At in Eqn. (13). the DBIM updating rule can be reorganized as c--ár functions of time t defined in Eqn. (6). the ratios de and la also remain continuous functions of Therefore, DBIM (p = 0) can be treated as an Euler discretization the following ordinary differential equation (ODE):
AL EL ai
e1 xT +
At d x,( t, As ae b,, Ct are continuous b. Ce-ar ee)
ce-at <
d xt xrd
at cc
be (ct)
(be)
+ xo (2,,
(16)
Ct
Though it does not resemble conventional ODE involving dt, the two infinitesimal terms d (
and d can be expressed with dt by the chain rule of derivatives. The ODE form also suggests ct that with a sufficient number of discretization steps. we can reverse the sampling process and obtain encodings of the observed data, which can be useful for interpolation or other downstream tasks.
In DDBMs, the PF-ODE (Eqn. (D) involving dæ, and dt is proposed and used for deterministic sampling. We reveal in the following proposition that our ODE in Eqn. (16) can exactly yield the PF-ODE without relying on the advanced Kolmogorov forward (or Fokker-Planck) equation.
Proposition 4.1 (Equivalence to Probability Flow ODE, proof in Appendix [.3]. Suppose
s0( t, IT) is learned as the ground-truth bridge score Va. log q( (xT) ), and wa is related to 88 through Eqn. (14, then Eqn. (6) can be converted the PF-ODE (Eqn. (8)) proposed in DDBMs.
"Wtt this choice, the initial step n N-1, we have pr. Ge. v
SNKEY SNR
Can V p2, 00
Ctat in the denominator in Eqn. (15 will be canceled out.

---

Published as a conference paper at ICLR 2025
Condition
Ground-truth
DDBM (NFE=20)
DDBM (NHE=100)
DBIM (NFE=20)
(NFE=100)
Figure Image translation results the DIODE-Outdoor dataset with DDBM and DBIM.
Though the conversion from our ODE to the PF-ODE is straightforward, the reverse conversion can be non-trivial and require complex tools such as exponential integrators (Calvo & Palencia 2006 Hochbruck et al.) 2009) (Appendix C.4). We highlight our differences from the PF-ODE in DDBMs: (I) Our ODE has novel form with exceptional neatness. (2) Despite their theoretical equivalence, our ODE describes the evolution of rather than 2t, and its discretization is performed with respect
d (()) and
instead of dt. (3) Empirically. DBIMs (p 0) prove significantly more
efficient than the Euler discretization of the PF-ODE, thereby accelerating DDBMs by substantial margin. (4) contrast the fully deterministic ODE, DBIMs are capable of various stochastic levels to achieve the best generation quality under the same sampling steps.
4.3 EXTENSION TO HIGH-ORDER METHOОS
The simplicity and efficiency of our ODE (Eqn. (16) also inspire novel high-order numerical solvers tailored for DDBMs, potentially bringing faster convergence than the first-order Euler discretization.
Specifically, using the time change-of-variabe A log h2 = (SNR; SNRr), the solution
of Eqn. [6) from time time s < can be represented as
Co xc + 0g Cs as
ØT + (/me e ex 1x Br)dd
Fg
(17)
where tx is the inverse function of A₁. The intractable integral can be approximated by Taylor ex- pansion of xy and finite difference estimations high-order derivatives, following well-established numerical methods (Hochbruck & Ostermann, 2005) and their extensive application in diffusion
models (Zhang & [hen][022] Lu 2022b) Gonzalez al., 2023). We present the derivations of our high-order solvers in Appendix D and the detailed algorithm in Appendix
5 RELATED WORK
We present detailed related work in Appendix A including diffusion models, diffusion bridge mod- els, and fast sampling techniques. We additionally discuss some special cases of DBIM and their connection to flow matching, DDIM and posterior sampling in Appendix C.3
EXPERIMENTS
In this section, we show that DBIMs surpass the original sampling procedure of DDBMs by large margin, in terms of both sample quality and sample efficiency. We also showcase DBIM's capabil- ities in latent-space encoding, reconstruction, and interpolation using deterministic sampling. All comparisons between DBIMs and DDBMs are conducted using identically trained models. For DDBMs, we employ their proposed hybrid sampler for sampling. For DBIMs, we control the vari- ance parameter by interpolating between its boundary selections:
SNR 1,41
Pn 1,
"7 E [0
(18)
SNR,
where 7 = 0 and 77 = correspond to deterministic sampling and Markovian stochastic sampling.
We conduct experiments including (1) image-to-imagee translation tasks on Galges>Handbbgee (Isola cet al. 2017) (64 x 64) and DIODE-Outdoor (Vasiljevic et al. 2019) (256 x 256) (2) image restora-
tion task of inpainting ImageNet (Deng al. 2009) (256 256) with 128 x 128 center mask.

---

Published as a conference paper at ICLR 2025
Table 2: Quantitative results in the image translation task. Baseline results are taken directly from DDBMs, where they did not report the exact NFE. Gray-colored rows denote methods that do not require paired training but only a prior diffusion model trained on the target domain.
Edges
HHnddbgg (64 x 64)
THODE-Outdooe (256 x 256)
NFE
FID 4
15
LPIPS
MSE
FID
IS
LPIPS
MSE
DDIB St 2022
2 40"
186.84
2.04
0.869
1.05
242.3
4.22
0.798
0.794
SDEair Mens
2022
40
26.5
3,58
0.271
0.510
31.14
5.70
0.714
0.534
Pix2Pix ef al.
2017
74.8
3.24
0.356
0.209
82.4
4.22
0.556
0.133
FSB lib el ml 20
2. 40
7.43
3,40
0.244
0.191
9.34
5.77
0.373
0.145
Zhou al.
0.142
0.040
6.21
0.084
DDBM
Zbou et al. 2023
200
0.88
3.69
0.110
0.006
3.34
5.95
0.215
0.020
DBIM (OWIS)
20
1.74
3.63
0.095
0.005
4.99
6.10
0.201
0.017
DBIM (Ours)
100
0.89
3.62
0.100
0.006
2.57
6.06
0.198
0.018
Table 3: Quantative results in the image
Table 4: Ablation of the variance parameter controlled by for image restoration, measured by FID.
restoration task.
Impuinting
ImageNNt
256)
Center (128%) 128)
NFF
FID
Sampler
NFE
DDRM
Kawar et al
2022
20
24.4
62.1
10
20
50
100
200
500
RGDM
S0n2 a1.
20238
100
7.3
72.6
DENM
Want 01
2013
100
15.1
55.9
Inpainting, ImageNet (256 x 256).
Center (128 x 128)
Palete
(Smarii
5,24
63.0
66.1
0.0
6.08
4.51
3.95
3.91
3.91
3.91
PSB
33233
20
4.98
63.9
01.
6.12
4.48
4.09
3.95
3.92
3.90
3.88
PSB
30036
1000
4.9
66.1
17
0.5
6.25
4.52
4.07
3.92
3,90
3.84
3.86
DDBM
Zhou al. 2025
500
4.27
71.8
0.8
6.81
4.79
4.16
3,91
3.88
3.84
3.81
DRIM
4.48
71.3
1.0
8.62
5.61
4.51
4.05
3.91
3.80
3.80
DBIM (Ours)
4.07
72.3
DDBM
275.25
57.18
10.63
4.27
DBIM (Ours)
100
3,88
72.
We report the Fréchet inception distance (FID) (Heusel al. 2017) for all experiments, and addi-
tionally measure Inception Scores (IS) (Barratt & Sharmal 2018), Learned Perceptual Image Patch Similarity (LPIPS) (Zhang al., 2018), Mean Square Error (MSE) (for image-to-image translation)
and Classifier Accuracy (CA) (for image inpainting). following previous works (Liu et al. 2023b] Zhou et al. 2023). The metries are computed using the complete training set for Bigges-Haandbags and Outdoo, and 10k images from validation set for ImageNet. We provide the inference time comparison in Appendix G. Additional experiment details are provided in Appendix
6.1 SAMPLE QUALITY AND EFFICIENCY
We present the quantitative results of DBIMs in Table 2 and Table 3 compared with baselines including GAN-based, diffusion-based and bridge-based methods We set the number of function evaluations (NFEs) of DBIM to 20 and 100 to demonstrate both efficiency at small NFEs and quality large NFEs We select r) from the set [0.0, 0.3, 0.5, 0.8. 1.0] for DBIM and report the best results.
In image translation tasks, DDBM achieves the best sample quality (measured by FID) among the baselines, but requires NFE > 100. In contrast, DBIM with only NFE 20 already surpasses all baselines, performing better than or on par with DDBM at NFE 118. When increasing the NFE to 100, DBIM further improves the sample quality and outperforms DDBM with NFE = 200 on DIODE-Outdoor. In the more challenging image inpainting task on ImageNet 256 256, the superiority of DBIM is highlighted even further. In particular, DBIM with NFE = 20 outperforms all baselines, including DDBM with NFE 500, achieving 25 speed-up. With NFE 100, DBIM continues to improve sample quality, reaching FII lower than 4 for the first time.
The comparison of visual quality is illustrated in Figure I and Figure 3 where DBIM produces smoother outputs with significantly fewer noisy artifacts compared to DDBM's hybrid sampler. Additional samples are provided in Appendix H
Ablation of the Variance Parameter We investigate the impact of the variance parameter p (con- trolled by 7)) identify how the level of stochasticity affects sample quality across various NFEs, as shown in Table and Table 5 For image translation tasks, we consistently observe that employing
'It is worth noting that, the released checkpoints of IS are actually flow matching/interpolant models instead of bridge models, as they (1) start with noisy conditions instead of clean conditions and (2) perform straight interpolation between the condition and the sample without adding extra intermediate noise.
8

---

Published as a conference paper at ICLR 2025
Table Ablation of the variance parameter controlled by v) for image translation, measured by FID.
Sampler
NFE
20
50
100
200
500
10
20
50
100
200
500
Image Translation,
(64 x 64)
Image Translation, DIODE-Outdorr (256 x 256)
0.0
3.62
2.49
1.76
1.17
0.91
0.75
0.65
14.25
7.96
4.97
3.18
2.56
2.26
2.10
3.64
1.81
1.21
0.94
0.76
0.65
8.25
5.22
3.37
0.5
3.69
2.61
1.91
1.30
1.00
0.81
0.67
14.93
8.75
5.68
3.71
2.92
2.47
2.17
2.25
1.58
1.23
16.41
10.30
6.98
4.63
2.41
1.0
4.21
3.38
2.72
1.96
1.50
1.15
0.85
19.17
12.59
8.85
5.98
4.55
3.59
2.82
DDBM
317.22
137.15
46.74
7.79
2.40
0.88
0.53
328.33
151.93
41.03
15.19
6.54
3.34
2.26
a deterministic sampler with 77 = yields superior performance compared to stochastic samplers with 7) O. We attribute it to the characteristics of the datasets, where the target image is highly correlated with and dependent on the condition, resulting in a generative model that lacks diver- sity. In this case, straightforward mapping without the involvement of stochasticity is preferred. Conversely, for image inpainting on the more diverse dataset ImageNet x the parameter exhibits significance across different NFEs. When NFE 20, 77 0 is the optimal choice, with FID steadily increasing as ascends. However, when NFE 2 50, a relatively large level of stochas- ticity at 17 0.8 even 7) yields optimal FID. Notably, the FID of n converges 3.91 at NFE = 100, with no further improvement at larger NFEs, indicating convergence to the ground- truth sample by the corresponding PF-ODE. This observation aligns with diffusion models, where deterministic sampling facilitates rapid convergence, while introducing stochasticity in sampling enhances diversity, ultimately culminating in the highest sample quality when NFE is substantial.
Table 6: The effects of high-order methods, measured by FID.
Sampler
NFE
20
50
100
Image Translation
Inpainting
Edges Handbags (61
64)
DIODE-Outdoor (256 x 256)
ImageNet (256 256)
DBIM (1) = 0)
3.62
2.49
1.76
1.17
0.91
14.25
7.96
4.97
3.18
2.56
6.08
4.51
4.11
3.95
3,91
(2nd-order)
3.44
2.16
0.99
0.79
13.54
2.87
2.41
DBIM (3rd-order)
3,40
2.12
1.45
0.97
0.79
13,41
7.01
4.20
2.84
2.40
5.50
4.34
4.07
3,93
3.91
High-Order Methods We further demonstrate the effects of high-order methods by comparing them to deterministic DBIM, the first-order case. As shown in Table high-order methods consis- tently improve FID scores in image translation tasks, well in inpainting tasks when NFES 50, resulting in enhanced generation quality in the low NFE regime. Besides, the 3rd-order variant performs slightly better than the 2nd-order variant. However, in contrast to the numerical solvers in diffusion models, the benefits of high-order extensions are relatively minor in diffusion bridges and less pronounced than the improvement when adjusting 7) from I to O. Nevertheless, high-order DBIMs are significantly more efficient than DDBM's PF-ODE-based high-order solvers.
As illustrated in Figure ! our high-order sampler produces images of similar semantic content to the first-order case, using the same booting noise. In contrast, the visual quality is improved with finer textures, resulting better FID. This indicates that the high-order gradient information from past network outputs benefits the generation quality by adding high frequeryy visual details.
Generation Diversity We quantitatively measure the generation diversity by the diversity score, calculated as the pixel-level variance of multiple generations, following CMDE (Batzolis et al., 2021) and BBDM (Li et al. 2023). As detailed in Appendix G.2, increasing NFE or decreasing can both increase the diversity score, confirming the effect of the booting noise.
6.2 RECONSTRUCTION AND INTERPOLATION
the deterministic nature of DBIMs at 17 and its connection to neural ODEs enable faithful encoding and reconstruction by treating the booting noise as the latent variable. Furthermore, employing spherical linear interpolation in the latent space and subsequently decoding
As discussed in Section 4.2

---

Published as a conference paper at ICLR 2025
22
(a) cecodiin/Reconsstuuctiin
(b) Semantic Interpolation
Figure 4: Illustration generation diversity with deterministic DBIMs.
back to the image space allows for semantic image interpolation in image translation and image restoration tasks. These capabilities cannot achieved by DBIMs with 1/ > O, or by DDBM's by- brid sampler which incorporates stochastic steps. We showcase the encoding and decoding results in Figure 4a| indicating that accurate reconstruction is achievable with sufficient number sampling steps. We also illustrate the interpolation process in Figure 4b
CONCLUSION
In this work, we introduce diffusion bridge implicit models (DBIMs) for accelerated sampling of DDBMs without extra training. In contrast to DDBM's continuous-time generation processes, we concentrate on discretized sampling steps and propose series of generalized diffusion bridge mod- els including non-Markovian variants. The induced sampling procedures serve as bridge counter- parts and extensions of DDIMs and are further extended develop high-order numerical solvers, filling the missing perspectives in the context of diffusion bridges. Experiments on high-resolution datasets and challenging inpainting tasks demonstrate DBIM's superiority in both the sample quality and sample efficiency, achieving state-of-the-ee FID scores with 100 steps and providing up to 25 acceleration of DDBM's sampling procedure.
Despite the notable speed-up for diffusion bridge models, DBIMs still lag behind GAN- methods in one-step gen- eration. The generation quality is unsatisfactory when NFE is small, and blurry regions still exist even using high-order methods (Figure [T. This is not fast enough for real-time applications. Besides, training-free infer- ence algorithm, DBIM cannot surpass the capability and quality upper bound of the pretrained diffusion bridge model. difficult and delicate inpainting scenarios, such as human faces and hands, DBIM fails to fix the artifacts, even under large NFEs.
Limitations and Failure Cases
Figure 5: DBIM case (n = D, NFE=500).
ACKNOWLEDGMENTS
This work was supported by the NSFC Projects (Nos. 62350080, 62106120, 92270001), the Na- tional Key Research and Development Program of China (No. 2021ZD01 10502), Tsinghua Institute for Guo Qiang, and the High Performance Computing Center, Tsinghua University. J.Z was also supported by the XPlorer Prize.
REFERENCES
Michael Albergo, Nicholas Boffi, and Eric Vanden-Fijnden. Stochastic interpolants: unifying framework for flows and diffusions. arXiv preprint arXiv:2303.0879) 2023.
Fan Bao, Chendong Xiang, Gang Yue, Guande He, Hongzhou Zhu, Kaiwen Zheng, Min Zhao, Shilong Liu, Yaole Wang, and Jun Zhu. Vidu: highly consistent, dynamic and skilled text-to- video generator with diffusion models. arXiv preprint arXiv:2405.0425 04233, 2024.
Shane Barratt and Rishi Sharma. A note on the inception score. arXiv preprint arXiv:1801.01977 2018.
Georgios Batzolis, Jan Stanczuk, Carola- Carola-Bibiane Bibiane Schönlieb, and Christian Etmann. Conditional image generation with score-based diffusion models. arXiv preprint arXiv:211 131606, 2021.
10

---

Published as a conference paper at ICLR 2025
Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning, vol- ume 4. Springer, 2006.
Mari Paz Calvo and César Palencia. A class explicit multistep exponential integrators for semi- linear problems. Numerische Mathematik, 102:367-381, 2006.
Nanxin Chen, Yu Zhang, Heiga Zen, Ron Weiss, Mohammad Norouzi, and William Chan. Wave- grad: Estimating gradients for waveform generation. In International Conference on Learning Representations, 2021a.
Tianrong Chen, Guan-Horng Liu, and Evangelos A Theodorou. Likelihood training of schr\," odinger bridge using forward-backwand sdes theory. arXiv preprint arXiv:2110.11291 2021b.
Zehua Chen, Guande He, Kaiwen Zheng, Xu Tan, and Jun Zhu. Schrodinger bridges beat diffusion models on text-to-speech synthesis. arXiv preprint arXiv:2312.0349, 2023.
Zixiang Chen, Huizhuo Yuan, Yongqian Li, Yiwen Kou, Junkai Zhang, and Quanquan Gu. Fast sampling via discrete non-markov diffusion models with predetermined transition time. In The Thirty-eight Annual Conference on Neural Information Processing Systems, 2024.
Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Marc Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. The Eleventh Interna- lional Conference on Learning Represenlations, 2022.
Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score- based generative modeling. Advances in Neural Information Processing Systems, 34:17695 2021.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A large-scale hier- archical image database. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248-255. IEEE, 2009.
Wei Deng, Weijian Luo, Yixin Tan, Marin Biloš, Yu Chen, Yuriy Nevmyvaka, and Ricky TQ Chen. Variational schr\(" odinger diffusion models. arXiv preprint arXiv:2405.04795. 2024.
Prafulla Dhariwal and Alexander Quinn Nichol. Diffusion models beat GANs image synthesis. In Advances in Neural Information Processing Systems, volume 34, pp. 7780-8794, 2021.
Joseph L Doob and JI Doob. Classical potential theory and its probabilistic counterpart, volume 262. Springer, 1984.
Martin Gonzalez, Nelson Fernandez, Thuy Tran, Elies Gherbi, Hatem Hajri, and Nader Masmoudi. Seeds: Exponential sde solvers for fast high-quality sampling from diffusion models. arXiv preprint arXiv:2305. 144.:2335..4267, 2023.
Agrim Gupta, Lijun Yu, Kihyuk Sohn, Xiuye Gu, Meera Hahn, Li Fei-Fei, Irfan Essa, Lu Jiang, and José Lezama. Photorealistic video generation with diffusion models. arXiv preprint arXiv:2312.06662. 2023.
Guande Hc, Kaiwen Zheng, Jianfei Chen, Fan Bao, and Jun Zhu. Consistency diffusion bridge models. arXiv preprint arXiv:2410.226:: 2024.
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs trained by two time-scale update rule converge to a local Nash equilibrium. In Is- abelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. Wallach, Rob Fergus, N. Vishwanathan, and Roman Garnett (eds.), Advances in Neural Information Processing Systems, volume 30, pp. 6626 6227-6637,22 2017.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, volume 33, pp. 6840-6851, 2020.
Jonathan Ho. William Chan, Chitwan Saharia, Jay Whang. Ruigi Gao, Alexey Gritsenko, Diederik P Kingma, Ben Poole, Mohammad Norouzi, David J Fleet, et al. Imagen video: High definition video generation with diffusion models. arXiv preprint 20. 2022.
11

---

Published as a conference paper at ICLR 2025
Marlis Hochbruck and Alexander Ostermann. Explicit exponential Runge- Kutta methods for semi- linear parabolic problems. SIAM Journal on Numerical Analysis, 43(3):1069-1090, 2005.
Marlis Hochbruck, Alexander Ostermann, and Julia Schweitzer. Exponential rosenbrock-typ type meth- ods. SIAM Journal on Numerical Analysis, 47(1):786-80) 2009.
Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and patter recognition, pp. 1125-1134 2017.
Kiyosi Itô. On formula concerning stochastic differentials. Nagoya Mathematical Journal, 3: 55-65, 1951.
Tero Karras, Miik Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in Neural Information Processing Systems, 2022.
Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song. Denoising diffusion restoration models. In Advances in Neural Information Processing Systems, 2022.
Dongjun Kim, Chich-Hsin Lai, Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: Learning prob- ability flow ode trajectory of diffusion. In The Twelfth International Conference on Learning Representations, 2023.
Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. In Advances in Neural Information Processing Systems, 2021.
Bo Li, Kaitao Xue, Bin Liu, and Yu-Kun Lai. Bbdm: Image-to-image translation with brownian bridge diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern Recognition, pp. 1952-1961, 2023.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022.
Gongye Liu, Haoze Sun, Jiayi Li, Fei Yin, and Yujiu Yang. Accelerating diffusion models for inverse problems through shortcut sampling. arXiv preprint arXiv:2305.16965 2023a.
Guan Guan-Homg Liu, Arash Vahdat, De-An Huang, Evangelos Theodorou, Weili Nie, and Anima Anandkumar. Ibs Image-to-image schrödinger bridge. In International Conference on Ma- chine Learning, pp. 22042-22062. PMLR, 2023b.
Cheng Lu, Kaiwen Zheng, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Maximum likelihood training for score-based diffusion odes by high order denoising score matching. In International Conference on Machine Learning, pp. 14429-14460, PMLR, 2022a.
Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. In Advances in Neural Information Processing Systems, 2022b.
Chenlin Meng, Yang Song, Jiaming Song, Jiajun Wu, Jun- Yan Zhu, and Stefano Ermon. SDEdit: Image synthesis and editing with stochastic differential equations. In International Conference on Learning Representations, 2022.
Shakir Mohamed and Balaji Lakshminarayanan. Learning in implicit generative models. arXiv preprint 16Xiv:1610.00 003433 2016.
Alexander Quinn Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob Mcgrew, Ilya Sutskever, and Mark Chen. Glide: Towards photorealistic image generation and editing with text- text-guided diffusion models, In International Conference on Machine Learning, pp. 16784-16804. PMLR, 2022.
Kushagra Pandey, Maja Rudolph, and Stephan Mandt. Efficient integrators for diffusion generative models. arXiv preprint 10..4:2310.07894 2023.
12

---

Published as a conference paper at ICLR 2025
Kushagra Pandey, Ruihan Yang, and Stephan Mandt. Fast samplers for inverse problems in iterative refinement models. arXiv preprint arXiv:240S.17673, 2024.
Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical text- conditional image generation with CLIP latents. arXiv preprint arXiv:2204.06/12 2022.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition, pp. 10684-10695, 2022.
Chitwan Saharia, William Chan, Huiwen Chang, Chris Lee, Jonathan Ho, Tim Salimans, David Fleet, and Mohammad Norouzi. Palette: Image-to-ine -omegg diffusion models. In ACM SIGGRAPH 2022 Conference Proceedings, pp. 1-10, 2022.
Axel Sauer, Dominik Lorenz, Andreas Blattmann, and Robin Rombach. Adversarial diffusion dis- tillation. arXiv preprint arXi::2311.17042 2023.
Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion schrödinger bridge matching. Advances in Neural Information Processing Systems, 36, 2024.
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrimm thermodynamics. In International Conference on Machine Learn- ing, pp. 2256-2265, PMLR, 2015.
Vignesh Ram Somnath, Mattco Pariset, Ya-Ping Hsich, Maria Rodriguez Martinez, Andreas Krause, and Charlotte Bunne. Aligned diffusion schrödinger bridges. In Uncertainty in Artificial Intelli- gence, pp. 1985-1995. PMLR, 2023.
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In Interna- tional Conference on Learning Representations, 2021a.
Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pseudoinverse-guided diffusion models for inverse problems. International Conference on Learning Representations, 2023a.
URL http:::://ppeeeeeiieeeeeee httpeeewwwwwweeeeeeeeeeeeeeeeeee momeet/oorumiddi 9 gsMA8MRKQ
Yang Song. Conor Durkan, lain Murray, and Stefano Ermon. Maximum likelihood training of score- based diffusion models. In Advances in Neural Information Processing Systems, volume 34, pp- 1415-1428 2021b.
Yang Song. Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations, lc.
Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consisteney models. In International Conference on Machine Learning, pp. 3221 -32252. PMLR, 2023b.
Xuan Su, Jiaming Song, Chenlin Meng, and Stefano Ermon. Dual diffusion implicit bridges for image-to-image translation. arXiv preprint arXiv:2203.08332. 2022.
Igor Vasiljevic, Nick Kolkin, Shanyi Zhang, Ruotian Luo, Haochen Wang, Falcon Z. Dai, Andrea Danicle, Mohammadreza Mostajabi, Steven Basart, Matthew R Walter, et al. Diode: A dense indoor and outdoor depth dataset. arXiv preprint arXiv:1908.0000 2019.
Pascal Vincent. A connection between score matching and denoising autoencoders. Neural compu- tation, 23(7):1661-165 2011.
Yinhuai Wang, Jiwen Yu, and Jian Zhang. Zero-shot image restoration using denoising diffusion null-space model. In The Eleventh International Conference on Learning Representations, 2023.
URL htt.hhtts///ppeeeeeeee wwwppeenreeiww..mm.. net wttpee.nee//fourreeeeeeeeee forum? wwt.mmiddmmRe@@eeee
Yuang Wang. Siyeop Yoon, Pengfei Jin, Matthew Tivnan, Zhennong Chen, Rui Hu, Li Zhang, Zhiqiang Chen, Quanzheng Li, and Dufan Wu. Implicit image-to-image schrodinger bridge for image restoration. arXiv preprint arXiv:2403.06065 2024a.
13

---

Published as a conference paper at ICLR 2025
Yufei Wang, Wenhan Yang, Xinyuan Chen, Yaohui Wang, Lanqing Guo, Lap-Pui Chau, Ziwei Liu, Qiao, Alex C Kot, and Bihan Wen. Sinsr: diffusion-based image super-resolution in a single step. In Proceedings the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp. 25796-25805, 2024b.
Jintao Zhang, Haofeng Huang, Pengle Zhang, Jia Wei, Jun Zhu, and Jianfei Chen. Sageatten- tion2: Efficient attention with thorough outlier smoothing and per-thread int4 quantization. arXiv preprint arXiv:2411. 10958, 2024.
Jintao Zhang. Jia Wei, Pengle Zhang, Jun Zhu, and Jianfei Chen. Sageattention: Accurate 8-bit attention for plug-and-play inference acceleration. International Conference on Learning Rep- resentations (ICLR), 2025a.
Jintao Zhang, Chendong Xiang, Haofeng Huang, Jia Wei, Haocheng Xi, Jun Zhu, and Jianfei Chen. Spargeattn: Accurate sparse attention accelerating any model inference. arXiv preprint 251.:22502.18137, 2025b.
Qinsheng Zhang and Yongxin Chen. Fast sampling of diffusion models with exponential integrator. In The Eleventh International Conference on Learning Representations, 2022.
Qinsheng Zhang, Molei Tao, and Yongxin Chen. gddim: Generalized denoising diffusion implicit models. arXiv preprint arXiv:2206.05555 05564, 2022.
Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features perceptual metric. In Proceedings the IEEE conference on computer vision and pattern recognition, pp. 586-595, 2018.
Kaiwen Zheng, Cheng Lu, Jianfei Chen, and Jun Zhu. Dpm-ssover-v3: Improved diffusion ode solver with empirical model statisties. Thirty-seventh Conference on Neural Information Pro- cessing Systems, 2023a.
Kaiwen Zheng, Cheng Lu, Jianfei Chen, and Jun Zhu. Improved techniques for maximum likelihood estimation for diffusion odes. In International Conference on Machine Learning, pp. 42363- 42389. PMLR, 2023b.
Kaiwen Zheng, Yongxin Chen, Hanzi Mao, Ming-Yu Yu Liu, Jun Zhu, and Qinsheng Zhang. Masked diffusion models are secretly aime-sin masked models and exploit inaccurate categorical sampling. arXiv preprint arXiv:2409.02900 02908, 2024.
Linqi Zhou, Aaron Lou, Samar Khanna, and Stefano Ermon. Denoising diffusion bridge models. arXiv preprint arXiv:2309.1694 2023.
14

---

Published as a conference paper at ICLR 2025
A
RELATED WORK
Fast Sampling of Diffusion Models Fast sampling of diffusion models can be classified into training-free and training-based methods. A prevalent training-free fast sampler the denoising diffusion implicit models (DDIMs) (Song et al. 222 that employ alternative non-Markovian gen- eration processes in place of DDPMs. discrete time diffusion model. ScoreSDE (Song et al. 2021ch further links discrete-time DDPMs to continuous- time score-based models, unrevealing the generation process to be ordinary and stochastic differential equations (ODEs and SDEs). DDIM can be generalized to develop integrators for broader diffusion models (Zhang et al., 2022 Pandey jet al. 2023). The concept of implicit sampling, in a broad sense, can also be extended to discrete diffusion models (Chen et al. 2024 Zheng al. 2024). although there are fundamental differences in their underlying mechanisms. Subsequent training- samplers concentrate on developing ded- icated numerical solvers to the diffusion ODE or SDE, particularly Heun's 's methods (Karras al.,
2022) and exponential integrators (Zhang & Chen 2022 Lu et al.] 2022b Zheng et al. [2023a] Gon- zalez et al. 2023). These methods typically require around 10 steps for high-quality generation. In contrast, training-based methods, particularly adversarial distillation (Sauer et al., (023) and consis-
tency distillation (Song al.] 20236] Kim al. 2023), become notable for their ability achieve high-quality generation with just one or two steps. Our work serves as thorough exploration of
training- fast sampling of DDBMs. Exploring bridge distillation methods, such as consistency bridge distillation (He et al., 2024), would be promising future research avenues to decrease the inference cost further. Infrastructure improvements, such as quantized or sparse attention (Zhang 2024), can also be used to accelerate the inference of diffusion bridge models.
al. 2025a b
Diffusion Bridges
Diffusion bridges (De Bortoli et al. 2022 Chen et al. 2021b Liu et al. 2023b
Somnath et al. 2023 Zhou et al. 2023] Chen al.) [023] Shi et al.) 2024 Deng et al., 2024 are
promising generative variant of diffusion models for modeling the transport between two arbitrary
distributions. One line of work is the diffusion Schrodinger bridge models (De Bortoli et al. |021] Chen et al. 202 Shi et al. 2024 Deng et al. (2024). which solves an hetropy-rrgularizee opti- mal transport problem between two probability distributions. However, their reliance on expensive iterative procedures has limited their application scope, particularly for high- sigh-ddmeesionall data. Sub- enhance the tractability of the Schrodinger bridge problem by making assumptions such as paired data (Liu et al.) [023b)] Sommath et al. 2023 Chen et al. 2023). On the other hand, DDBMs (Zhou et af.) 2023) construct diffusion bridges via Doob's h-transform, offering reverse-time perspective of diffusion process conditioned on given endpoints. This ap- proach aligns the design spaces and training algorithms of DDBMs closely with those of score-based generative models, leading to tfae-of-the-aa performance in image translation tasks. However, the sampling procedure of DDBMs still relies on inefficient simulations of differential equations. lack- ing theoretical insights to develop efficient samplers. BBDM (L et al. 2023) and SS (Wang et al. 2024a) extend the concept of DDIM to the contexts of Brownian bridge and 1SSB (Liu et al. 2023b), respectively. SinSR (Wang et al. 2024b) is also motivated by DDIM, while the application is concentrated on the mean-reverting diffusion process, which ends in Gaussian instead of delta distribution. In contrast to them, our work provides the first systematic exploration of implicit sam- pling within the broader DDBM framework, offering theoretical insights and connections while also proposing novel high-order diffusion bridge solvers.
sequent works have endeavored
B PROOFS
B. PROOF OF PROPOSITION 3.1
Proof. Since glo) in Eqn. 10 is factorized as ((p))((a. (IT) = (0(w))g(1) (x (Fo, ØT)
where q(o) (2) (oo, ØT) = II g(o) (zx, (20, Dt Di 2T), we have gie) (xo(xx) = qo((x)) = q( (r),, which proves the case for O. For 1 S n 5 N 1, we have
(Kr, (TT) / glo) (x (20, xT)g (x ( )dzo
(x
(19)
and
g(x;,, (xT) = ʃ g( (20, 2r )g( ddæo
(20)
15

---

Published as a conference paper at ICLR 2025
Since ( ( g(Zo) we only need to prove ql)) Zu. xT) q(2t,, (1,,
Firstly, when n = N - 1, we have En+1 T. Note that p is restricted by PN-1 = Ctx-1 and Equ. II becomes
( No. TT) (att-- oT er + b R. o2 cx-,
(21)
which is exactly the same as the forward transition kernel of q in Eqn. 66. (xr, Fo, 2T) = q( . (0, TT) holds for n
Therefore,
Secondly, suppose q t (0 TT) q [p TT) holds for k, we aim to prove that holds for = k 1. Specifically, ql0) (I Do. 2r) can be expressed
(2tz-1 Fo. Ør)
gle) ( Fg Tt 2t (P) (1., xg. Tr dæ
/ 0 (I 2o, Lt )g( Tzh 20. 2y da
(22)
/ NI Mi 1 11 Pk N an øT b xu. ei d d
where
012_1 DT + bie_, Do + / /e 1.r1-11 - -Pp_1 St.
ae, xT b 2o
Pk 1|
(23)
Ct2
From
Bishop & Nasrabadi
2006) (2.115), (10 13)
is a Gaussian, denoted as
( 1, Ek-1), 1), where
ar DT + br, 20 - ar, DT by xo
1k-1 = aie_, ET + bia-1 + V / -Ph-1
Cte
(24)
= aty DT + btg 30
and
2
P
Σx- Pk-
2
c22
I
(25)
Cle
Ct
2
Therefore, (a 0o, TT) = g( aag, 2T) M(tt aT + btc Fo 2 fx-1 I). By math-
ematical deduction, (()) (xx,, (00, .T)) = ( (2u, 2r) holds for every S S N 1, which
completes the proof.
B.2 PROOF OF PROPOSITION 3.2
Proof. Substituting Eqn. 10) and the joint distribution into Eqn.
13 have
f(m)(0)
-By(srr (Bron
log q (P) (xt (00, xT) - log pe (Ito [TT)
N
N-1
-By(or) Byto) (2010. ( (KT)
log (00) (20t.. (o0,, st,, xT)
log po (20,, De,, 1: OCT)
R=
N-1
E Egiar) Eegte mi (
DRI (x)), (oo, ( || po( 11 xTT))
n=1
E Eg(rrr ) (err) (ao) [log Po (z (0,, xT))
(26)
where
DKL(g' (I (2o, Diati FT) po(x,, Fenms xr))
=DKI (P) (xx, aoo, Lt,, xT) || g(0) (ar,, (1 lati, 2TT) 11, aTT)
(27)
de, (2 bn+1, xT) zoll
2p
16

---

Published as a conference paper at ICLR 2025
where we have denoted d, == bi,
/22 - Pnn Mnt1, Besides, we have cta+1
log po (2ol TT)
log (ogg(o) (zo ( t1, (1,)).
It,
FT)
log (mng (3ie 11 (T,,
(28)
| (t. L1, aT) zo +
2p
where C' irrelevant 0, According to Eqn. (6), the conditional score
3t U,TT
6,30
V Ve, log (x,
xT)
(29)
cf
Therefore,
m Pta I, 22r) xroll
4
Dt. 2t, DT
be, xo (It,, In, xTr)
Lt, - Ot,,, ET
bt, Zn
by
ot
G
(30)
c.
In xTr) V on
log ((z,, (00 xr)11
where 89 is related to we by
8o s0 (T))
3t 0,TT
bi (1,, xT)
(31)
c7
Define do = 1, the loss (0) is further simplified to
((P)(() C
N-1
del zo (I tn+1, ØT)
xo
2p²
n=0
de
E Byg (egeeraacaolerrolee keo)))) [1a20| (a0e, las xTT) 20112]
2ph-1
Bg (e wT)) [[l80 (Be,, las 2T)) - mr log ((r, (00, 2T )
2p b2
(32)
Compared to the training objective of DDBMs in Eqn. 9 T (0) is totally equivalent up to
constant, by concentrating the discretized timesteps (tn, In= choosing g( (r)) as the
d2 ot
paired data distribution and using the weighting function y that satisfies 7(ka) 2p--, %
B.3 PROOF OF PROPOSITION
Proof. We first represent the PF-ODE (Eqn. (8)
dæ; = ((0) - ((()) (EV log ((rg(a)))))) - Væ. log Vrrj(arr (z2))) )] de
(33)
with the data predictor By (x, t, æT). We replace the bridge score V., log (xx (x)) with the net- work so (2 1, RT), which is related to an (zr,, 1, ar) by Eqn. (14.. Besides, Va, log gra (xrla))))
17

---

Published as a conference paper at ICLR 2025
can be analytically computed as V aa log 9r ( (xT((x)
g xT) ( (00)
V n, log
(( x00)
Va, log q
(00 2T)
Vv, log g 20))
I, a
6,000
2t 3,
(34)
SNRT SNR:
(æz;
ot arr)
OT
of(1 - SNRr
SNR.
Ot TT)
Substituting Eqn. (4) and Equ. (34) into Equ. (33.
the PF-ODE is transformed to
dæc
5(1)x - (²(1)
2t
U,TT
b,o0 (2r, xT)
ag aT It xT))
dl
2c
&
((()) + +()
1- 22; Oi
3t + (1)
at
ET
b (2)
wa (2te, L, xT
dl
(35)
20q
20g
2r
[1r0) + (Q(
g (L)
+ g (t)
4t Fr
b (²)
ao at, xTT) dt
of
2c
2c
2c,
On the other hand, the ODE corresponding to DBIMs (Eqn.
16
can be expanded as
dæ
c
a,dt
(
ae
+
b
Dg (cr, (0), (T))
dt
(36)
C
where we have denoted
(.)² = d(-)
Further simplification gives
dt
dx;
at 4 c x; + (dg-ai Ct
xT
(% by b,
_
L Ce
xe (0, t, (.) dt
(37)
The coefficients de by. Gr are determined by the noise schedule ot or in diffusion models. Comput- ing their derivatives will produce terms involving f(t), g(t,, which are used to define the forward SDE. As revealed diffusion models, f(t), g(t) are related to a, or by f(t) d a de
g =
de? 2 2dlog at o.. We can derive the reverse relation of ae, and f(t).g(t): dt dt
/ (T) dr
02 = e f(r)dr
of =a
Jo
(38)
a?
which can facilitate subsequent calculation. We first compute the derivative of a common term in a, by, Ce:
1 SNR,
(1) a
(39)
a
For Ct since of = =f
SNR ), we have SNR,
d 4t
1 (loge4) (log = (log of log((1
1
SNRT ))
(40)
SNR,
where
07 (logo?) (log y y (log (f) (²) at 0f
(2) + 25(1)
-2((1)
(41)
of of
of
and
SNRT
SNRT
SNRT
(t)
(t) SNRT
(log(1
SNR,
))² = 1
SNR SNR,
SNR,
oi
0 2
a
of SNR,
(42)
18

---

Published as a conference paper at ICLR 2025
Substituting Eqn. (41 and Eqn.
(4
into Eqn. (40). and using the relation
SNRI SNR,
1
r we have
4 = s(1)
((())
(1)
SNRT
((1) + (
(1)
(43)
Ct
20
20
SNR;
oi 20
For ag, since 02
al SNRT, SNR,
we have
d
SNRI
= f(t) + SNR, (
= f(1) + (
(log ax)² = (loga:) + (log SNR,
(44)
at
dj
of
For bt, since by are(1
SNRZ) SNR,
). we have
b/2 b
SNRY
)) [(1) (t) SNRT
1(1+ 9 (()
(t(()
(logbi) (log ar) /o''++(ll +logg(1
(45)
SNR,
d SNR,
of
Therefore,
d/
d = ar
d
-)
c
gt(
26
at
(46)
Ct
and
d
b1
9 (t) (t)
8/
be(
(47)
Ct
bt
Ct
26
Substituting Eqn. 44
Eqn.
(46)
and Equ.
(47) into the ODE of DBIMs in Eqn. (37). we obtain
exactly the PF-ODE in Eqn. 35
C MORE THEORETICAL DISCUSSIONS
C.I MARKOV PROPERTY OF THE GENERALIZED DIFFUSION BRIDGES
We aim to analyze the Markov property of the forward process corresponding to our generalized diffusion bridge in Section 3.1 The forward process of ((P) can be induced by Bayes' rule as
glo) (I Zo. Lenti DT ( (Ito4 Dg. TT)
gi | Lt, (T)
(48)
(x,, (2o 2T)
where glo) (ox xo, ΦT) = (00 ØT) is the marginal distribution of the diffusion bridge in Eqn. and (z,,, [oo ((,,11 wT) is defined in Eqn. (II as
(0))) (at, (mog Dtatr BT) N(as, ør + be, xo + V pa alati at.+1 DT
b 20
P I).
Chat1
(49)
Due to the marginal preservation property (Proposition 3.1 p. we have (( |00; aT)
a( [Fo 2T) and ( (Zo, (0, = ((=,,, (20 TT), where g(xx/200, xT) = Na + byzo, T) is the forward transition kernel in Eqn. (6. To identify whether q(e) (att.. (mo, Ft,,, FT) is Markovian, we only need to examine the dependence of Zt.., on TD. To this end, we proceed to derive conditions under which V log (xtr.. |2o Fr,, ØT) involves terms concerning Wo.
Specifically, V
a1a1 log (10)) (att,, [o, Lt,,, 2T) can be calculated as Ftati log ((0) (It..4 Co. Ft.. TT)
- Vanti log (ogg((( (I0, (o, D ØT) + V Btrt log g
(I) (20, xT) p. ar, BT Dr,, 2o x
V C2 - p (a:, 2T + br, xo +
CA
Ctn11 Pa
(50)
Ltnt1 ala+1 ET bent: xo
be O2
br, Ct, v 12, 12.
38n + C(vt., Bt,,, 12 xT))
co Pd
19

---

Published as a conference paper at ICLR 2025
where Cz Lt [.) are terms irrelevant to To. Therefore,
bt Gn
be, Cta+1
22 pz
glo) (2t.-1 Rt,,, 2T) ) Markovian
catz P
(51)
SNR
Pn Ote
SNR,
which is exactly boundary choice of the variance parameter p. Under the other boundary choice
Pn ota / S, SNR
Pn 0 and intermediate ones satisfying glo) (Iints (oo It, TT) is non-Markovian.
the forward process
C.2 THE INSIGNIFICANCE OF Loss WEIGHTING IN TRAINING
The insignificance of the weighting mismatch in Proposition 3.2 can be interpreted from two aspects. On the one hand, L consists of independent terms concerning individual timesteps (as long as the network's ' parameters are not shared across different t), ensuring that the global minimum remains the same as minimizing the loss each timestep. regardless of the weighting. On the other hand, L min: we (0) S (o S max. wi C., (0). Be-
under different weightings are mutually bounded by max,
min,
sides, it is widely acknowledged that in diffusion models, the weighting corresponding to variational inference may yield superior likelihood but suboptimal sample quality (Ho et al. 2020 Song et al.) 2021c), which is not preferred in practice.
C.3 SPECIAL CASES AND RELATIONSHIP WITH PRIOR WORKS
Connection to Flow Matching When the noise schedule at = 1,T = and a; = Bt, the forward process becomes g( (2o æi) .V (twr + (1 f) 3t(11 - t)) which is a Brownian bridge, When 3 + 0, there will be no intermediate noise and the forward process is similar to flow matching (Lipman et al.) 2022 Albergo al., 2023). In this limit, the DBIM (n 1) updating rule from time to time < will become I, SØT +
() t, xT) = ç + (1 (2);..) the velocity
sa Here we define vy (z,, function of the probability flow (i.e., the drift of the ODE) in flow matching methods. Therefore, in the flow matching case, DBIM is simple Fuler step of the flow.
) x,
Connection to DDIM In the regions where t is small and SNRT is close to 0, we have de 0, SNR, Qr, C dr. Therefore, the forward process of DDBM in this case approximately ( ( (a of I). which is the forward process of the corresponding diffusion model. Moreover, in this case, the DBIM (17 = 0) step approximately
CC R
Os xt + 0s
Oee
)
04
20 (10t,,,,,, (,
(52)
which is exactly DDIM (Song et al. 2021a), except that the data prediction network æa is dependent on ær. This indicates that when t is small so that the component of ør in œ, is negligible, DBIM recovers DDIM.
Connection to Posterior Sampling The previous work ISBB (Liu et al. 2023b) also employs diffusion bridges with discrete timesteps, though their noise schedule is restricted to the variance exploding (VE) type with [())
the forward process. For generation, they adopt a sim- ilar approach DDPM (Ho al. 2020) by iterative sampling from the posterior distribution Po (=( which is a parameterized and shortened diffusion bridge between the endpoints 20 (Zn, ln, .NN) and on. Since the posterior distribution is not conditioned on ør (ex- æto xt
cept through the parameterized network), the corresponding forward diffusion bridge is Markovian. Thus, the posterior sampling in I2SB is special case of DBIM by setting /) 0 and f(t) 0.
C.4 PERSPECTIVE OF EXPONENTIAL INTEGRATORS
Exponential integrators (Calvo & Palencia 2006; Hochbruck et al. 2009) are widely adopted in recent works concerning fast sampling of diffusion models (Zhang & Chen 2022 Zheng et al.,
20

---

Published as a conference paper at ICLR 2025
2023a
Gonzalez et al. 2023). Suppose have ODE
dæ, [a(t) + F(t) t)]df
(53)
where Fg is the parameterized prediction function that want to approximate with Taylor pansion. The usual way of representing its analytie solution ey at time & with respect to an initial condition æ, at time is
21 2. J.100 (a(T)a=- + (x),
(54)
By approximating the involved integrals in Eqn. Eqn. (53) such as Euler's method. The key insight of exponential integrators is that, it is often bet-
we can obtain direct discretizations of
ter to utilize the "semi-linear" structure of Eqn. (53) and analytically cancel the linear term a()) obtain solutions that only involve integrals of Fg and result in lower discretization errors. Specifically, by the crariation-of-consaaee formula, the exact solution Eqn. (53) can be alternatively given by
This way,
( elt alr)dr (())))
Φt = el. a(r)dr vg +
(55)
We can apply this transformation to the PF-ODE in DDBMs. By collecting the linear terms w.r.t.
Eqn. (8) can be rewritten (already derived in Appendix B.3
dæ, [110) [11))+
(t)
gt)
(t) be x (2r, 20) dt
w. 2²)
(56)
of
20
2cg
20
By corresponding it to Eqn.
(33
we have
a(t) = ft) + g²(()
(L)
(1(t) = g² at
b²(t) = -g2(t)) 20,
(57)
of
2ct
From Eqn.
43
Eqn.
(46) and Eqn.
47
we know
log C
d dlog(b.//c) by (1) = dlog(ae/ca) b2(1) = by
a(I)
(58)
dt
dt
dt
Note that these relations are known in advance when converting from our ODE to the PF-ODE.
Otherwise, finding them in this inverse conversion will be challenging. The integrals in Eqn. 555
can then be calculated as ((T))dr= = log Ct log C.. Thus
el a(T)dr
elt a(r)dr Ca
(59)
Co
CT
Therefore, the exact solution in (55) becomes
Eqn.
Ct x, xt
a
ar
+ ci I by by o0, aT
b
+
Ct
log
Ca
Cr
C
or
(60)
C x, +
Ct ag
OT + ae
{ by
x0( T, T, dd
br
which is the same as Eqn. after exchanging and t and changing the time variable in the integral
X, = log (x)
6x)
Lastly, we emphasize the advantage of DBIM over employing exponential integrators. First, deriving our ODE via exponential integrators requires the PF-ODE as preliminary. However, the PF-ODE alone cannot handle the singularity at the start point and presents theoretical challenges. Moreover, the conversion process from the PF-ODE to our ODE is intricate, while DBIM retains the overall simplicity. Additionally, DBIM supports varying levels of stochasticity during sampling, unlike the deterministic nature of ODE- based methods. This stochasticity can mitigate sampling errors via the Langevin mechanism (Song al. 2021c), potentially enhancing the generation quality.
21

---

Published as a conference paper at ICLR 2025
D DERIVATION OF OUR HIGH-ORDER NUMERICAL SOLVERS
High-order solvers of Eqn. 17 be developed by approximating ao in the integral with Tay- lor expansion. Specifically, as function of X, we have Ta (Ti lx- TT) s (g t, xr) +
Ex=1 (A-da)* (1)) (x2, t, ær), where zg (z,, t, xr) d dm tx.
is the k-th order derivative w.r.t. A, which can be estimated with finite difference of past network outputs.
k! 2g
dA*
A=d,
2nd-Order Case With the Taylor expansion zo 20 tx. xT)
(x(,,, + (A
) (at, t, ar), we have
(N/² edax) ao aro (,07) (f/² (A Aolerda (ae, (TT
exmolaee lx, wT)d 2s
2 e (1-- e "=)=))) + (h 1 e - |a0111]
(61)
where we use é, denote the network output at time and h X A 0. Suppose have used a previous timestep u (s < t < u), the first-order derivative can be estimated by
Dt
Ba
hn = de- Au
(62)
h1
3rd-Order Case With the Taylor expansion wo (2 (TT ) (x,, (T)) + (A-2)) To (wr, 1, æT), have
2olw,, 1, (, + (A
(1) (0000)))0,))))))))))
(2)
Cama(aas (x, (T)
JTan e'd1) (,
2 20 (xn,,
(63)
(1=2 eddd 2 (2) (2
h22
t (1 ch (h- 1 )2 +
h c-h ((2)
Similarly, suppose we have two previous timesteps 11 U2 (s t #1 < U2), and denote h = X Xuy he = Xua Xag: the first-order and second-order derivatives can be estimated by
Ruj (2h1 + h22) Su, Buz h1
Buy
(1)
b1
(2)
(64)
h1 + l12
h1 + h22
The high-order samplers for DDBMs also theoretically guarantee the order of convergence, similar to those for diffusion models (Zhang Chen 2022 Lu et al. 20226 Zheng et al. 2023a). We omit the proofs here they deviate from main contributions.
22

---

Published as a conference paper at ICLR 2025
E ALGORITHM
Algorithm DBIM (high-order)
Require: condition æ7, timesteps 0 I to < l <.. < 1N- iN data prediction model
Le. booting noise ~ (0, I), noise schedule ar, be, Ct. d log(br/cz), order (2 or 3).
1: Øt f Tg T, aT)
2: Ftx
0,2r + bgār CtE
3: for i - to do
4: s,t 1, h f- X₂- X2
5:
i; - no [1. 1, TT)
6:
if 2or N
then
7:
+ t+11 h1 + xe- Xa
Estimate ai (1) with Eqn. (62)
8:
i ed [11- (1-e-h)a (h 1 ) (1)
9:
10:
else
11:
U1. U2 f tit1, t442 h1 d Aun ; h2 x Aus
Auz
Estimate 2.11) (2(2) with Eqn. (64
12:
i- ex (1 (1- - h + (h + e )ot (1) + ( 122 - -hh+1 1- s,,00
14: end
if
(a. gay xr c
15:
Go xt a
16: end for
return Tto
F EXPERIMENT DETAILS
F.1 MODEL DETAILS
DDBMs and DBIMs are assessed using the same trained diffusion bridge models. For image trans- lation tasks, we directly adopt the pretrained checkpoints provided by DDBMs. The data pre- diction model æ,( ØT) mentioned in the main text is parameterized by the network Fg as Zo (a, t, xr) = Caip (f)x, : + Cour Fo
Cnoise (t), IT), where
(m(() Va707 + b,o0 + 2a.bioor +
Com (t) v ot (07.08 obr) + afcrem(1)
(65)
Cskp (t) (b.og a.our ) (t),
Croise (t)
log
4
and
af Var(zo), 07 Var[ær], oor = Cov Cov[xn,30 Ør]
(66)
For the image inpainting task on ImageNet 256x256 with 128x 128 center mask, DDBMs do not provide available checkpoints. Therefore, we train new model from scratch using the noise sched- ule of I'SB (Liu et al. 2023b). The network initialized from the pretrained class-conditional diffusion model on IetgeNet 256x256 provided by (Dhariwal & Nichol, 2021), while addition- ally conditioned on Sr. The data prediction model in this case is parameterized by the net- work Fe æa (x
Dt Fe t, #T) and trained by minimizing the loss L(()
+ (10 ł, xT) = We train the model on 8 NVIDIA A800 GPU cards with
batch size of 256 for 400k iterations, which takes around 19 days.
F.2 SAMPLING DETAILS
We elaborate the sampling configurations of different approaches, including the choice of timesteps [ and details of the samplers. In this work, we adopt tiin = 0.0001 and tmax following (Zhou al. 2023). For the DDBM baseline, the hybrid, high-order Heun sampler
23

---

Published as a conference paper at ICLR 2025
proposed in their work with Euler step ratio of 0.33, which the best performing configuration for the image-to to to-image translation task. We use the same timesteps distributed according to EDM (Kar-
ras et al. 2022 's scheduling 1/e + Nv (1 ) consistent with the official implementation of DDBM. For DBIM, since the initial sampling step is distinctly forced to stochastic, we specif- ically set it to transition from max to fmax 0.0001, and employ simple uniformly distributed timestep scheme in mim max 0.0001) for the remaining timesteps, across all settings. For inter- polation experiments, enhance diversity, increase the step size of the first step from 0.0001 to 0.01.
F.3 LICENSE
Table 7: The used datasets, codes and their licenses.
Name
URL
Ciaaion
License
Edges - Handhags
actps: / www.tbtttbbb...........t..bbb.bt con/ wwnyana (pttrchh CycleCAN- MalloooN--addd and--add----e pix2pix
Lola et al
2015
BSD
DIODE Outdoor
Vathevie
2019
MIT
ImageNett
Dene
Guille--DifTssioo
TE Fat nub, con/ Geens Lagi ded-d.
Dhanwal Nichul
2021
MIT
F'BB DDBM
ScEps:/ ritaub, con/ XV 1256
Lia al 20236
CC-BY-NC-SA-LD
Setp5:/
alexchou90 77DBM
We list the used datasets, codes and their licenses in Table
G ADDITIONAL RESULTS
G.1 RUNTIME COMPARISON
Table 8 shows the inference time of DBIM and previous methods single NVIDIA A100 under different settings. We use torch. cuda Event and torch. cuda. wwnchronicziii iee to accu- rately compute the runtime. We evaluate the runtime on batches (dropping the first batch since it contains extra initializations) and report the mean and std. We can see that the runtime is pro- portional to NFE. This because the main computation costs are the serial evaluations of the large neural network, and the calculation of other coefficients requires neglectable costs. Therefore, the speedup for the NFE is approximately the actual speedup of the inference time.
Table Runtime of different methods to generate single batch (second batch, std) on a single NVIDIA A100, varying the number of function evaluations (NFE).
NFE
Method
Center 128 128 Inpainting, ImageNet 256
256 (batch size /6)
PSB Liu et al. 2023b
2.8128 + 0.0111
5.6049 + 0.0152
8.3919 + 0.0166
11.1494 + 0.0259
DDBM (Zhou et al. 2023
2,8711 + 0.0318
5.7283 1 0.0572
8.3787 1:20 0.
11.0678 +: 0.3061
DBIM
2.8755 + 0.0706
5.7810 + 0.1494
8.5890 = 0.2730
11.1613 0.3372
DBIM (2nd-order)
2.8859 0.0675
5.7884 0.1731
8.6284 + 0.1907
11.5898 0.2260
DBIM (3rd-order)
2.9234 0.0361
5.8109 0.2982
8.6449 + 0.2118
11.3710 0.3237
G.2 DIVERSITY SCORE
We measure the diversity score (Batzolis et al., 2021 Li et al. 2023) the ImageNet center inpaint- ing task. We calculate the standard deviation of generated samples (numerical range ~ 255) given each observation (condition) ær, averaged over all pixels and 1000 conditions.
As shown in Table 9 the diversity score keeps increasing with larger NFE. DBIM (n = 0) consis- tently surpasses the flow matching baseline I'SB, and DDBM's hybrid sampler which introduces diversity through SDE steps. Surprisingly, we find that the DBIM r7 case exhibits a larger di- versity than the 7) case. This demonstrates that the booting noise introduce enough
24

---

Published as a conference paper at ICLR 2025
Table 9:
Diversity scores on the ImageNet center inpainting task, varying 7) and the NFE. We exclude statistics for DDBM (NFEş10) as they correspond to severely degraded nonsense samples.
NFE
Method
5
10
20
50
100
200
500
FSB Liu al. 2023b
3.27
4.45
5.21
5.75
5.95
6.04
6.15
DDBM Zhou et al.
2023
2.96
4.03
4.69
5.29
5.83
DBIM
71 77 =
3.74
4.56
5.20
5,80
6.10
6.29
6.42
2.62
3.40
4.18
5.01
5.45
5.81
6.16
stochasticity to ensure diverse generation. Moreover, the = 0 case tends to generate sharper images, which may favor the diversity measured by pixel-level variance.
H ADDITIONAL SAMPLES
25

---

Published as a conference paper at ICLR 2025
(a) Condition
(b) Ground-truth
(c) DDBM (NFE=20), FID 46.74
(d) DBIM (NFE=200.. 1 0.0), FID 1.76
(c) DDBM (NFF=100), FID 2.40
(i) DBIM (NFE=100, = 0.0). FID 0.91
g DDBM (NFE=200), FID 0.88
DBIM (NFF=200) 7) 0.0), FID 0.75
Figure 6: Handbags samples on the translation task.
26

---

Published as a conference paper at ICLR 2025
(a) Condition
(b) Ground-truth
(c) DDBM (NFE=20), FID 41.03
(d) DDBM (NFE=500), FID 2.26
(e) DBIM (NFE=20, // 0.0), FID 4.97
(f) DBIM (NFE=200, 17 = 0.0), FID 2.26
Figure 7: DIODE- samples on the translation task.
27

---

Published as a conference paper at ICLR 2025
(a) Condition
(b) Ground-truth
(c) DDBM (NFF=10), FID 57.18
(d) DDBM (NFE=500), FID 4.27
(e) DBIM (NFE=10,, 0.0), FID 4.51
(f) DBIM (NFE=100,, 17 = 0.0), FID 3.91
(g) DBIM 1NFE=100, 0.8), FID 3.88
(h) DBIM (NFE=500, 1.0), FID 3.80
Figure 8: ImageNet 256 x 256 samples on the inpainting task with center 128 x 128 mask.
28


Reviews:
[{"review_id": "P1Pfw5f5ub", "reviewer": "Reviewer_Wtri", "summary": "The paper presents a theoretically justified method for fast sampling from DDBMs. The authors introduce non-Markovian diffusion bridges and derive the sampling procedure with the same marginals for discrete time steps. They show that the training objective of the proposed methods is equivalent to the DDBM objective on the discretized timesteps. The method is validated on image-to-image translation problems using edges2handbags and DIODE-Outdoor datasets and inpainting tasks using the ImageNet dataset. The proposed method achieves comparable results with DDBM on NFE > 100 while using only NFE = 20 for these problems regarding FID, LPIPS, and MSE.", "questions": "Questions:\n1. Can you compare your method with I2SB for inpainting with freeform masks and evaluate I2SB on small NFEs?\n2. Can you discuss the limitations of your method or provide failure cases? \n3. Can you comment on how much slower high-order methods are than first-order methods regarding inference time? Does the quality improve further for bigger orders like 4?\n4. Can you additionally provide inference time for DBIM and other methods to show the inference time acceleration?\n5. I suggest the authors consider the other method of DDBM acceleration using consistency distillation and include its discussion in the literature review; see Consistency Diffusion Bridge Models (NeurIPS-2024). \n \nTypos:\n1. It looks like in Eq. 7, it should be \\bar{w}_{t} instead of w_{t}.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes fast sampling for the DDBM method, which achieves impressive results for image-to-image translation problems. Enhancing and accelerating DDBM is essential since these models provide a solid alternative to Flow Matching and Schrodinger Bridge methods for image-to-image problems. \n2. The proposed method is training-free and doesn't require any model parameter fine-tuning. Proposition 3.2 establishes equivalence between DDBM and DBIM objectives up to the constant.\n3. Experimental results, which prove that deterministic sampler with $\\eta = 0$ leads to better results than stochastic samplers with $\\eta > 0$ is novel for DDBM, which suggests operating with stochastic samplers and may provide \"averaged\" or blurry outputs given initial conditions. This result allows further improvement of DDBM with distillation methods for deterministic samplers like consistency distillation (Consistency Models, ICML-2023).", "weaknesses": "In my opinion, the paper doesn't have significant weaknesses. However, some comments should be discussed additionally:\n1. I think comparing the best competitor on the inpainting problem according to Table 3 - the I2SB method - is not enough. According to the I2SB paper, this method can repaint the masked region with semantic structures with only 2-10 NFEs for the inpainting problem with freeform masks. I suggest that the authors compare their method with I2SB for the inpainting problem with freeform masks and show the dependence of its quality (LPIPS, FID) on small NFEs (1-20) following setup from I2SB.\n2. The idea of incorporating non-Markovian diffusion bridges seems not novel. For example, the I3SB method (Implicit Image-to-Image Schrodinger Bridge for Image Restoration, https://arxiv.org/abs/2403.06069) shares similar ideas and parameterization for I2SB. I suggest the author add the discussion of this paper and the main differences between their method and I3SB.\n3. The diversity of the proposed method should be evaluated quantitatively. It is worth showing how small $\\eta$ and NFE values degrade diversity compared to stochastic models. I suggest that the authors compute the diversity score from CMDE (Conditional Image Generation with Score-Based Diffusion Models, https://arxiv.org/abs/2111.13606) as in BBDM (BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models, CVPR-2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SHPMl5Lkgh", "reviewer": "Reviewer_hmv5", "summary": "The paper proposes an extension of DDBM by adopting the ideas from DDIM for the bridge problem (translating one distribution to another with diffusion).", "questions": "1. Appendix C.1, lines 892-897. Why do you take gradient $\\nabla_{x_{t_{n+1}}} \\log q^{\\rho} (x_{t_{n + 1}} \\vert x_0, x_T)$, not $\\nabla_{x_{t_{n+1}}} \\log q^{\\rho} (x_{t_{n + 1}} \\vert x_0, x_{t_n}, x_T)$. Also, the eq. 50 is incorrect, while the result (lines 904 - 906) is correct. Please check the derivation.\n2. Line 256 - The statement that $x_T$ is cancelled out under the choice of $\\rho_n$ seems to be correct, but a link to the proof should be given.\n3. Do I understand correctly, that different variants of variance parameter $\\rho$ appear only at inference, i.e., training with eq. 13 is agnostic to $\\rho$? \n4. You mention that eq. (8) (from DDBM) is equivalent to (16). But on par with (16) you have additional booting noise required by theory. How previous works (e.g., DDBM) manage to avoid the necessity of this noise? \n5. Lines 427 - 428: ‘straightforward mapping without the involvement of stocahsticity’ - but you have booting noise still, yes?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper novel in the sense that it proposes to adopt the ideas from DDIM (fast sampling, non-markovian diffusion) to the bridge problem, thereby enhancing DDBMs and expanding the practitioner’s toolset for such kind of problems. So, “creative combinations of existing ideas” - is the best description of the presented research. Also, I would like to notice good experimental section.", "weaknesses": "The majority of my concerns are in the \"Questions\" section. To be honest, I didn’t check Proposition 3.2 - but it seems to be correct. Proposition 4.1 - also not carefully checked by me, but seems to be correct.\nWhat I do not like much about the paper is that some particular formulas and results (see point 2 in my questions) are stated as is, without proof/reference to the proof. It complicates reading and checking the results.\nSome other weaknesses:\n1. You have \"Related works\" section (A) in the Appendix. Ok, but I didn't find a reference to this section from the main text. Furthermore, I think it is important to somehow cover related works in the main text (maybe, with some references to the appendix)\n2. No source code in the submission. It is important for reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "67UQ9mrHmt", "reviewer": "Reviewer_Jfwr", "summary": "The authors construct an intriguing process that aligns with the marginal distributions and the well-known diffusion bridge. Building on the success of DDBM in modeling paired distributions at given endpoints, the authors introduce Diffusion Bridge Implicit Models (DBIMs), emphasizing discrete steps over continuous diffusion. Robust empirical techniques are employed in real-world experiments, significantly enhancing prediction performance.", "questions": "1. DDBM proposed the pred-x parameterization motivated by the EDM framework to enhance the predictions and derive a set of scaling functions for distribution translation.  Is your work applicable to this extension? If not, what are the challenges?\n\n2. Any intuitions on why a non-Markovian property is preferred?\n\nNIT: some discussions on OT-accelerated diffusions, such as [1,2], may be helpful.  \n\n[1] https://arxiv.org/pdf/2110.11291\n[2] https://arxiv.org/pdf/2405.04795", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Clear derivation to extend DDBM with DDIM properties. \n\n2. Nice connections to existing literature such as I2SB, flow matching, and exponential integrators.", "weaknesses": "the notation of $q_{tT}$ in Eq.(7) is confusing.\n\n\nNIT - the literature section should be revised significantly, for example, Schrodinger in line 502: Schr\\\"odinger, schrodinger in line 509, 600, 606, sde line 521, rosenbrock in line 543, Dpm-solver in line 576, ode in line 644 should have properly capitalized letters.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rXHhzhlxYT", "reviewer": "Reviewer_W5Ra", "summary": "The authors consider the denoising diffusion bridge models (DDBMs [1]) recently proposed as a variant of diffusion models useful for interpolating between two paired distributions. The authors propose a way to accelerate this type of diffusion model without retraining them. The proposed method and the paper are mostly analogical to the Denoising Diffusion Implicit Model (DDIM) [2], which is used to accelerate classical diffusion models for generation such as DDPM [3]. In the experimental section, the authors use the same setups and models as in DDBMs and show that the proposed sampling scheme outperforms the sampling scheme of DDBMs.", "questions": "I do not have any particular questions. In my opinion, this is a borderline paper which can be accepted if the space permits.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The approach is presented and sufficiently supported by the theoretical derivations.\n- The authors support their method by showing that it indeed achieves significant speed-up of the inference on the same setups as were considered in the DDBM paper while preserving the quality.\n- The method does not require retraining and is implemented just by a different sampling of the DDBMs models.", "weaknesses": "The general approach of obtaining a new sampling scheme in analogy to DDIM is known and has already been used for a different type of bridge process in SinSR [4], CVPR 2024, used for super-resolution problems. Specifically, the difference is that in SinSR, the bridge process $q(x_t|x_0, x_T)$ for a given low-resolution image $x_{T} = y$ and a given high-resolution image $x_0$ is given by:\n$$q(x_t|x_0, x_T) = \\mathcal{N}(x_t|x_0 + \\eta_t(y-x_0), k^2\\eta_t I), $$\nand $\\eta_t \\rightarrow 1$, while in the current paper, the bridge process $q(x_t|x_0, x_T)$ is given by:\n$$\nq(x_t \\mid x_0, x_T) = \\mathcal{N}(a_t x_T + b_t x_0, c_t^2 I),\n\\quad a_t = \\frac{\\alpha_t}{\\alpha_T} \\frac{\\text{SNR}_T}{\\text{SNR}_t},\n\\quad b_t = \\alpha_t \\left( 1 - \\frac{\\text{SNR}_T}{\\text{SNR}_t} \\right),\n\\quad c_t^2 = \\sigma_t^2 \\left( 1 - \\frac{\\text{SNR}_T}{\\text{SNR}_t} \\right),\n$$\nwhere $SNR_t = \\frac{\\alpha_t^2}{\\sigma_t^2}$ and $c_t^2 \\rightarrow 0$ for $t \\rightarrow T$.\n\nThus, there are two differences. The first one is that in SinSR, the process ends in the Gaussian with center $x_T$, while in the current work, the process ends in the delta function. The second is the different parametrization of these processes. To conclude, the methods SynSR & DBIM (proposed here) are not directly equivalent but one can be derived analogously to the other one with some effort.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AVXnW2dlkr", "reviewer": "Reviewer_YfPj", "summary": "The authors present a fast sampling approach to Denoising Diffusion Bridge Models (DDBMs, Zhou et al.) by constructing a DDIM (Song et al.) analogue of DDBMs and hence denote it as Denoising Bridge Implicit Models (DBIMs). The authors empirically demonstrate the advantages of the proposed method on several image translation and restoration benchmarks. The authors also illustrate the application of DBIMs for reconstruction and interpolation using latent space traversal.", "questions": "See above.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I found the paper relatively straightforward to follow. Unsurprisingly, the empirical results support the claims in the paper as DDIM leads to significant improvements in sampling efficiency over vanilla DDPM. Therefore, it is to be expected that similar variants of other stochastic processes could achieve similar gains (See [1]).", "weaknesses": "Please see below.\n\n**Reinventing the wheel**: My main concern with the paper is that the authors try to reinvent the DDIM framework in the context of a very specific class of stochastic processes (DDBMs in this case) when previous works already define a framework for generalizing over DDIM (as also noted by the authors in the paper in Sec 5.1, higher order methods and Appendix D). More specifically, Pandey et al.  [2] present a generic framework, Conjugate Integrators, for efficient diffusion model sampling of which DDIM and exponential integrators are special cases (See Section 3.1). As a consequence, their framework can be used to enable fast sampling from any pretrained diffusion/flow/interpolant models. Similarly, [1] presents generalized DDIM, which is a special case of Conjugate Integrators and can be applied to diffusion models like CLD or Blurring diffusion models. The main idea of all these methods is to modulate the drift of the reverse diffusion/flow process to enable fast sampling during inference (See also [3] for a similar approach). Since DDBMs also possess a semi-linear drift, extending these generic frameworks to obtain the corresponding DDIM sampler for DDBMs is trivial without delving into the non-markovian formulation of the original DDIM paper.\nConsequently, I feel the paper does not offer any new perspective and the flow is very similar to the DDIM paper (defining the non-markovian process followed by setting rho=0 to enable deterministic sampling). Can the authors clarify how their work presents any novel insights compared to the original DDIM paper (or other works mentioned here [1,2]) since this is not obvious to me from the current explanation in the main text?\n\n**Empirical Comparisons**: Although not my main concern, the empirical comparisons seem a bit unfair. For instance, in Table 3, the authors use $\\Pi$-GDM and DDRM as comparison baselines, while these methods are task-agnostic and can be adapted for solving different inverse problems using a single pretrained diffusion model prior. However, following Eq. 9, DDBMs/DBIMs require paired data (x_0, y) for training and thus task-specific. Comparing task-specific models with task-agnostic models seems unfair. Moreover, other important baselines based on flow matching [4] or stochastic interpolants [5] are missing, which have advanced quite a lot in terms of sample quality recently [6]. I would recommend updating Table 3 to remove some baselines like DDRM and Pi-GDM and instead compare DBIM with the latest flow matching or interpolant models.\n\n**Dubious claims**: The authors state the following in the Introduction: \n```\nAdapting diffusion models to scenarios where a more informative prior naturally exists, such as image translation/restoration, requires cumbersome techniques like adding conditions to the model (Ho & Salimans, 2021; Saharia et al., 2022), modifying the generation pipeline (Meng et al., 2022; Su et al., 2022) or adding extra guidance terms during sampling (Chung et al., 2022; Kawar et al., 2022). These approaches inevitably result in either sub-par performance or slow and resource-intensive inference processes.\n```\nI agree that task-agnostic models like DPS or $\\Pi$-GDM require extra guidance terms in their sampling process, making inference slow due to expensive Jacobian-vector products. However, in my opinion, a slower sampling process (though there is some recent work to mitigate slower sampling in solving inverse problems using diffusion models [7,8]) will likely be acceptable over training new models for each task. Moreover, these methods are more scalable in practice due to their adaptiveness to different restoration tasks while requiring no paired data. On the other hand, task-specific methods will likely require large paired datasets, which can be expensive to acquire. Therefore, certain models are better suited for specific applications (as also implied by the No-Free lunch). Therefore, this claim in the introduction seems misleading to me. It would be great if the authors could update this claim to more accurately reflect the nuances of different approaches, including recent advancements in mitigating slower sampling for task-agnostic models. Moreover, it would be nice to discuss how their method fits into the broader context of these trade-offs and under what scenarios it might be preferable to use DBIM versus a more general, task-agnostic method.\n\nReferences:\n\n[1] gDDIM: Generalized denoising diffusion implicit models, Zhang et al.\n\n[2] Efficient Integrators for Diffusion Generative Models, Pandey et al.\n\n[3] Bespoke Solvers for Generative Flow Models, Shaul et al.\n\n[4] Flow Matching for Generative Modeling, Lipman et al.\n\n[5] Stochastic Interpolants: A Unifying Framework for Flows and Diffusions, Albergo et al.\n\n[6] SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers, Ma et al.\n\n[7] Accelerating Diffusion Models for Inverse Problems through Shortcut Sampling, Liu et al.\n\n[8] Fast Samplers for Inverse Problems in Iterative Refinement Models, Pandey et al.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)