SFBD-OMNI: Bridge models for lossy measurement restoration with limited clean samples
Authors: Haoye Lu
Yaoliang Yu
Darren Lo
Venue: ICLR 2026 Poster (2026)
Keywords: ambient diffusion
diffusion models
generative modeling
density deconvolution
Primary area: generative models

Abstract: In many real-world scenarios, obtaining fully observed samples is prohibitively expensive or even infeasible, while partial and noisy observations are comparatively easy to collect. In this work, we study distribution restoration with abundant noisy samples, assuming the corruption process is available as a black-box generator. We show that this task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm. We further provide a test criterion to determine whether the true underlying distribution is recoverable under per-sample information loss, and show that in otherwise unrecoverable cases, a small number of clean samples can render the distribution largely recoverable. Building on these insights, we introduce SFBD-OMNI, a bridge model-based framework that maps corrupted sample distributions to the ground-truth distribution. Our method generalizes Stochastic Forward-Backward Deconvolution (SFBD; Lu et al., 2025) to handle arbitrary measurement models beyond Gaussian corruption. Experiments across benchmark datasets and diverse measurement settings demonstrate significant improvements in both qualitative and quantitative performance.

Published as a conference paper at ICLR 2026
SFBD-OMNI: BRIDGE MODELS FOR LOSSY MEASURE- MENT RESTORATION WITH LIMITED CLEAN SAMPLES
Haoye Lu Yaoliang Yu & Darren Lo!
School of Computer Science, University of Waterloo
Vector Institute fhaoye lu, yaoliang. yu,
yu, htg.://ggrooslloo/attttroooooooo dlslo)@uwaterloo.ca
ABSTRACT
In many real-world scenarios, obtaining fully observed samples is prohibitively expensive or even infeasible, while partial and noisy observations are comparatively easy collect. In this work, we study distribution restoration with abundant noisy samples, assuming the corruption process is available as black-bo boxk-box generator. We show that this task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm. We further provide test criterion to determine whether the true underlying distribution recoverable under per-sample information loss, and show that in otherwise unrecoverable cases, small num- ber clean samples can render the distribution largely recoverable. Building on these insights, we introduce SFBD-OMNI, a bridge model-based framework that maps corrupted sample distributions to the ground-truth distribution. Our method generalizes Stochastic Forward-Backward Deconvolution (SFBD; Lu et al., 2025) to handle arbitrary measurement models beyond Gaussian corruption. Experi- ments across benchmark datasets and diverse measurement settings demonstrate significant improvements in both qualitative and quantitative performance.
INTRODUCTION
Diffusion-based generative models (Sohl-Dickstein al. 2015 Ho et al. 2020 Song et al. 202 la b.
2023) have attracted growing interest and are now regarded as one of the most powerful frameworks for modelling high -igheninnnsi distributions. They have enabled remarkable progress across various domains (Croitoru et al.) (2023). including image (Ho et al. 2020 (Song et al. 202 b Rombach al., (022b). audio (Kong et al., 2021] Yang et al., 2023), and video generation (Ho al.[ Today, most state-of-the-art image and video generative models are diffusion-
or their variants,
such as flow matching (Lipman et al., (203)) and consistency models (Song et al. (023).
While much of their success is attributed stable training dynamies, diffusson models (DMs), like nearly all other generative frameworks, also depend on large collections of high-quality training data. In many practical domains, however, such data are costly or even infeasible to obtain, whereas large volumes of corrupted samples are readily available. For example, in medical imaging, acquiring cleaner X-ray scans requires higher radiation doses, which can endanger patient health (Seibert 20088. making most available scans inherently noisy. Likewise, in ground-based astronomical imaging, clean deep-space observations demand long exposures under ideal atmospheric conditions, yet most telescope images are degraded by atmospheric turbulence, sensor noise, and light pollution (Chimitt & Chan 2023
Given this reality, a natural question arises: With only a limited number of clean samples but an abundance of corrupted ones, can we train model to recover the clean sample distribution? Under suitable identifiability conditions on the corruption process, [oraa et al. (2018) demonstrated that generative model can indeed be trained using only corrupted samples, by leveraging the GAN framework (Goodfellow 2016). Building on this idea and the remarkable success of diffusion models, subsequent works have sought recover data distributions under specific corruption processes- for example, Ambient Diffusson for pixel masking (Daras & Dimakis (023), Tweedie Diffusion (Daras et al. 2023), and Stochastic Forward-Backward Deconvolution (SFBD, Lu et al. 2025) for

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additive Gaussian noise. However, to the best of our knowledge, there is existing framework that both accommodates general corruption processes and theoretical guarantees, while exploiting the advantages of diffusion models. A more detailed review of related literature is provided in Sec B
In this work, we address this gap by proposing a principled framework for the distribution recovery problem through diffusion- -based models. Instead of formulating distribution learning as a min-max game via the variational representation of the Kullback-Leibler (KL) divergence, as in GANs, we show that an alternative variational form, provided by the Donsker- Varadhan principle (Donsker & Varadhan ) reveals the problem to be essentially equivalent one-sided entropic optimal transport objective. This reformulation naturally yields an alternative minimization pipeline that fully leverages the design advantages of diffusion-based models. Importantly, our approach avoids adversarial training, making it both simpler to implement and more stable in practice. Since the method can be viewed generalization of the SFBD algorithm, we refer it SFBD-OMNI.
Under suitable identifiability conditions on the corruption process, the proposed method is theo- retically guaranteed to recover the ground-trutt truth clean data distribution. For practical corruption processes that do not satisfy these conditions, be largely recovered when a limited number of clean samples are available, and we provide conver- gence guarantees for this setting. Since the proposed alternating minimization algorithm requires training a sequence of neural networks, we also introduce an online variant that enables end-to-end training. simplifying implementation and potentially accelerating convergence, while still preserving optimality guarantees. Empirical results corroborate our theoretical analysis, and experiments across benchmark datasets demonstrate significant and consistent improvements over strong baselines under diverse measurement settings. A key strength of SFBD-OMNI is its robustness in scenarios where the identifiability condition fails: by incorporating a small number of clean samples, the method still able to effectively guide recovery toward the true data distribution. The implementation is available at http://gihhub.oommaam/////55
further show that the clean distribution can still
2 PRELIMINARY
Diffusion models and SFBD. Diffusion models learn distributions by progressively corrupting data with Gaussian noise and then training model to approximate the reverse process through successive denoising steps. Formally, given a distribution j over RRd the forward process governed by stochastic differential equation (SDE):
dx, = dw,, Xo
f
(1)
where [we]eefo,ry is the standard Brownian motion. Eq I induces transition kernel Paj. (xx|xx) (x)x(xx x = (xx,, (t s) I) for 2 s 2 O. Let (x,)
p: (x, (x)) ((xo) dx, denote the marginal
distribution of x, (in particular, p = p)) Anderson (192)) showed that the backward SDE can describe the time-reversed process corresponding to the forward SDE:
dx, s((,, l)dt dw., X, ~ Pr.
where T >> O, W; is standard Brownian motion in reverse time and s(-,t) V log (.()) is the score function. In practice, the score can be efficiently approximated via a neural network So trained by minimizing the conditional score matching loss (SsM(Se, ()) (Song et al. (0211). Crucially, this reverse SDE induces transition kernels that coincide with the posterior of the forward process:
1" (x, (x.) = Peja(xe xx) px. (x.)) (Xe)
for s < in [0.
(3)
Consequently, sampling from p" (xx xx) can be carried out by integrating Eq (2 backward from X, with / In standard diffusion models, 7 is chosen sufficiently large so that Df x (O Thus, sampling from the model amounts to drawing x, ~ NN(0 rI) followed by xo ~ Ph (Xo x+).
In contrast, SFBD (Lu et al. 2025) operates in the regime of finite specifically considering a Gaussian Corruption process realized through the forward transition kernel (r Xy)- In particular, they assume access limited set of clean samples Eekea and a large set of Gaussian corrupted ones Eneicy obtained through this forward transition kernel. For a set of samples E, let pe denote the corresponding empirical distribution. Starting from pretrained model so, by minimizing
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LesM( Ectean); the algorithm proceeds follows: for ........
Ex - { {x Enoisy; solve Eq 2 from 1 to 0, with and S0, 1.}
(4)
0% < Continue training s0.-1 to obtain so, by minimizing LCsM Ek)
(5)
Lu et al. (2025) proved that K + oc, PÊx converges to the true distribution Pdata by analyzing the evolution of the underlying stochastic processes, leveraging the relation in Eq (3) for all (s, s)t) [0, r]. However, this relation inherently tied to the Gaussian forward corruption process in Eq which
makes extending the approach to arbitrary corruption processes challenging.
Interestingly, the sampling step (4) essentially corresponds to drawing from P with / P
This observation suggests that, rather than enforcing the posterior relation in 3 for all (s. 1) [0, and learning it via score function approximation in (2), it may be sufficient to train a model that learns only the posterior pojr- In this case, may extend the forward kernel Prjo to arbitrary corruption processes. Indeed, when the corruption process satisfies suitable identifiability conditions, generalized SFBD method can be employed to recover the data distribution, as we will show in We conclude this section by presenting a unified framework to learn Poj, with bridge models.
Sec
Learning posterior distributions with bridge models. Unlike standard diffusion models, which learn to transform Gaussian noise into data samples via the backward SDE (2). bridge models generalize this idea to transformations between arbitrary distributions (Lipman et al. 2023 Peluchetti 2023 Zhou et al. 2024). Given paired samples (x.y) ~ π (x. from a joint distribution æ, bridge model constructs distributional path connecting the z-marginal π. and the y-marginal Ty by interpolating between each pair (x, y) through transition processes (Peluchetti 2023). Typical
choices include line segments in flow matching and rectified flow (Liu et al. 2022 Lipman al.
2023), or Brownian bridges in DDBM and 12SB (Liu et al.) 2023 Zhou et al. 2024). The resulting
process defines a transition path distribution Ptjo1 (Xt Xo X, Xi y whose evolution from =
to can often be expressed in closed form via a backward SDE (Peluchetti 2023):
Xo, dx. fxx; Xo, X1, t) dt + g(t) dw,
(6)
Let fo(x;; (), be the minimizer of the conditional drift matching (CDM) loss
LCDM O n) ET (xxx))) Pxp f(x;; Xo, X1, () (x,; x1, t) I|?
(7)
where U is sampling distribution over E (0, 1). It then follows that samples from Taja (Xo y) can be obtained by integrating from = 1 to 0 with
(Peluchetti 2023 De Bortoli et al. 2023
dx, = fo (x,; x1, t) df g(f) dw,
(8)
In this way, given a Markov kernel rly x) for a general corruption process and a sample distribution 1r, the joint distribution of y) is 77 (x,,) μ(() r(y x). A bridge model can then be trained to learn the posterior distribution in manner analogous to diffusion models, using CDM loss LCDM
corresponding to the chosen transition process.
KULLBACK-LEIBLER AMBIENT PROJECTION PROBLEM
Let r(. x) denote the Markov kernel for the corruption process. Define the corresponding corruption operator T,, which maps clean distribution j its corrupted counterpart:
Taj (y) = / r(y x) (x() dx.
(9)
Given the corrupted data distribution q := TT Pdaaa our objective, following the classical GAN formulation in AmbientGAN (Bora et al. 2018) is to recover Paaaa by solving
arg min, DKL. (q T.p).
(10)
The intuition is that minimizing the discrepancy between corrupted distributions drives toward the true clean distribution Paata- We refer to this optimization task as the Kullback-Leee Leibler Ambient Projection (KLAP) problem.
If x and are connected by deterministic path (i.e., - in Eq (6)) the sampling process may become ill-conditioned, as it degenerates to deterministic mapping. To mitigate this, can he perturbed with small Gaussian noise during both training and sampling. See See for details.
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3.1 IDENTIFIABILITY
Whether the recovery is possible depends on the choice of the corruption kernel r(. y). For instance, x is an image and r(: y) always outputs a white patch, then the corrupted distribution = T Pdata collapses a single point
on the white patch, regardless of Plnta In this degenerate case, every distribution p achieves the same objective value in (TO). so the minimizer p need not equal the true distribution Pdata- The next proposition characterizes when minimizing (10) recovers p".
Proposition (Identifiability Condition). Let P(X) denote the set of clean sample distributions. When the corruption kernel r(- x) depends continuously on x, the convex objective in Eq (10) admits a unique minimizer p" Pdota whenever Te is injective on P(X). Tf not injective, the objective is still convex, but all distributions satisfying T.p = TrPdoto are minimizers.
All proofs are deferred to the appendix. We highlight sev- eral common corruption operators T, together with their injectivity properties:
Additive noise. y = x with noise V, then r(y x) pyy- x). When has characteristic function without zeros (e.g., Gaussian), the induced convolution operator p > p * is injective. This setting corresponds to the classical density deconvolution problem (Meister) 2009), with SFBD (Lu et al. 2025) addressing the Gaus- sian case in particular.
Pdata
ht
P1
P(X)
Random dropout. Each pixel is masked with probability > and otherwise unchanged. It can be shown that when each pixel is masked independently. T, is injective (Bora al. 2018). (Non-injective when a = 1.)
Figure I: Effect of X on p As A 0. the first term in Eq (1 ensures that remains within S(q). while the second term selects the element ht E Sq)) clos- est to h. Consequently, PA converges to ht which represents the projection of onto the feasible set S(g).
Linear transforms. If y Ax for a linear map A, r(y x) = ō(y Ax). If A has full column rank (hence injective), then T, also injective. (Non-injective if A has a nontrivial nullspace, such as projections or grayscale conversions of images.)
3.2 AUGMENTED KLAP
As noted in Prop [ if T, is not injective, the objective is convex but not strictly convex. In this case, any distribution p E P(X) with T.p = T.PPaa is a minimizer, and we denote this solution set by S(q). Thus, Pdata cannot be uniquely identified from the noisy distribution. One way to overcome this ambiguity is to incorporate additional information. In practice, this often comes from a small number of clean samples or, more generally, from a prior distribution h over Pdata This motivates the following augmented formulation.
Given the corruption operator T, defined in Eq D. a prior distribution h over plata and regulariza- tion parameter À 0, we consider the following optimization problem:
p² arg minpe (x)), where Tx(p) = DKL (g | T.p) A (h p).
(1))
For À >>0 0, the strict convexity of the second term ensures the entire objective is strictly convex with a unique minimizer PI. whereas for À = 0 reduces to the classical ambient problem.
For intuition, consider a corruption process r that maps colour images to grayscale, with Paaa consisting of human face images. Here T, is not injective, since many different colourings yield the same grayscale distribution. In other words, S(q) contains multiple elements. Thus, when À = 0, can recover the distribution of face structures but not the colour patterns. To capture the full colour distribution, we may assume access to a few clean colour images from Pdata and encourage align with their empirical distribution by choosing À 0.
Fig Iillstrates how the additional regularization term shapes the optimal solution. As À 0, the first term in Eq [1) keeps p within S(q), while the second selects the element ht E S(q) closest to h. We formalize this observation in the following proposition.
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Proposition 2.
Let ht arg mis)) DKL (h p) denote the information-projectte of h onto the original KLAP solution set. Then the minimizer of Eq 111. p PX- converges ht as ->
Clean samples also matter under injective T, Identifiability o Recoverability. While Prop ! shows that T, is injective, then Pdat is in principle recoverable by minimizing Eq (10), this guarantee relies on having access to the true corrupted density q = P.aaa In practice, however, q must be estimated from finitely many noisy samples, and the resulting estimation error is amplified through the inverse of T... Consequently, the minimizer of Eq (TO) based on an empirical estimate of q can deviate substantially from Pdain For additive-noise corruption operators Tr, the unfavourable sample complexity of this inverse problem well documented in the density deconvolution literature (see, e.g. Meister (2009)), ) and the pessimistic rates suggest that acquiring enough noisy samples to train high-quality model is often practically infeasible (Lu et al. 2025). To overcome this issue, in Sec [ we show that even a very small number of clean samples (as few as 50) can substantially mitigate this difficulty, consistent with the findings of Lu al. 2025)..
4 TWO VARIATIONAL PERSPECTIVES OF KLAP
In this section, we present two variational perspectives for characterizing KLAP, each derived from a different variational formulation of the KL divergence. The first perspective corresponds to the classical formulation, which previously employed in training Ambient GANs (Bora et al. 2018). and is included here for completeness. The second perspective reveals that the classical KL.AP can be viewed as one-sided entropic optimal transport (OT) problem and also leads alternative minimization algorithm for solving both the classical and augmented KLAP formulation.
4.1 AMBIENT GAN'S FORMULATION
For any convex function f. a corresponding f-divergence can be defined: (m(y) gly) dy (Nowozin et al. 2016), which also admits an variational form mly)
Dy(allm)
Dy(gl|m) = max, (Byls(Y)) - Em (f"()))))))
(12)
where f is the convex conjugate of f. When f() In D, reduces the KL divergence. As result, with this choice of f, the original KLAP problem (00) can be rewritten as
min, Dy TTp) = min, max, (9))) ET., [f (o())))
(13)
This mix-max formulation can be naturally implemented in the standard GAN framework (Good- fellow] 2016), with g as the discriminator and p parameterized by the generator, [Braa et al. (2018) showed that this setup can recover Púata when T, is injective and the corruption process is differentiable with respect to the clean inputs.
To the best of our knowledge, existing KLAPP-basd frameworks cannot directly incorporate the additional identifiability term or support a more scalable, liffusiion/bidge-sttlll generator. Sec we introduce an alternative variational formulation that yields an internating-miniminee algorithm (Sec addressing both issues, Notably, the method requires only black-box access to the corruption process, without any differentiability assumptions.
4.2 ONE-SIDED ENTROPIC OPTIMAL TRANSPORT FORMULATION
Let (x) log r(y x) Rather than invoking the variational representation of KL-divergence, we apply the Donsker- Varadhan variational principle (Donsker & Varadhan, 1983):
log Ex-xp [efx(x)] maxuy Exzuy (x)] DRL. (uy p)
(14)
where By denotes distribution of x given Taking expectation over y ~ y and rearranging yields
DKL (g T,p) min., Eyng [KL (uy || p) Ex (f (x)] + CC,
(15)
where C collects the terms independent of p (see See Sec for the derivation). As a result, the augmented
KLAP problem 11 is equivalent to
arg min, min., Fx (p. Uy)
(16)
We assume r(: x) has full support; this can he enforced by injecting infinitesimal Gaussian noise to y.
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with
Fx my) == Byng [ (My p) - Exwuy (x)]] + ADRL (h p)
This nested minimization suggests an alternative strategy for solving the (augmented) KLAP problem in Sec 5 We conclude this section by noting that this observation allows KLAP to be viewed as variant of classical entropic OT, offering a new perspective to understand the KLAP problem.
Proposition Define the cost function c(x,y) - -logr(y x). Problem (6 is equivalent to arg min, p(p) + A DKL (h p)
with
/f c(x,)) yd + DKL. ( p)
(p) = min
TElly(a)
where Iyy (q) denotes the of joint distributions of (x,y) with y-marinnal fixed to q. Moreover, when À O, the optimal solution p" coincides with the x-marginal of the corresponding minimizer x" in the inner problem.
Notably, (p) in Prop coincides with the entropic OT objective (Cuturi 2013), but with constraints imposed only on the y-marginal rather than on both marginals. In particulat, lar when the cost function quadratic, as the case of Gaussian corruption kernel, the optimal coupling 'T" corresponds to the Schrödinger Bridge (Léonard 2014). Moreover, Prop 3 shows that in the absence of the regularization toward the prior distribution (i.e., when 0). the optimal solution p" induces an optimal coupling x² in the inner entropic OT problem whose marginals are precisely p" and This interpretation suggests that solving KI.AP amounts finding distribution p that minimizes the transportation cost induced by the corruption kernel, subject entropy regularization. The methods introduced in Sec 5 provide an effective approach for solving this one-sided entropic OT problem.
5 STOCHASTIC FORWARD-BACKWAN DECONVOL.UTION-OME
The variational formulation of the augmented KLAP Eq (6) suggests an alternative minimization approach for finding the minimizer y defined in Eq (TT) This leads to an algorithm that generalizes SFBD
(Lu et al. 2025) to arbitrary corruption models, which we call SFBD-OMNI.
Starting from arbitrary initialization (x), we minimize F(p, uy) in 16 by alternating updates over and uy, holding the other fixed. Specifically, at each iteration, we compute
uy arg min.,, Fx(p", Uy):
*
h+1 arg min, Fx ug),
(17)
where both subproblems admit closed-form solutions:
rx)))(( x) (y)
1
A I+ h(x),
uy (x)
(x I+
p
(18)
with +x)+ q(y) (x) dy.
Note that uy is the posterior distribution of n* (x:) under the joint distribution n T(x, = (x()x x). As described in Sec 2, by introducing a transition process connecting x and y, we can leverage bridge model to learn this posterior
manner analogous to diffusion models, by minimizing the corresponding CDM loss LCDM. Let ug denote the learnt posterior distribution. The quantity Pk is then approximated using samples from (- y) with y q(y).
We deseribe the implementation of SFBD-OMNI in Alg assuming access small set of clean samples that define the prior h, denoted heleam which also serves as the initialization pº. During training, f is approximated by pe and updated iteratively, while the mixture of ps and he is realized through a weighted sampler.
Online SFBD-OMNL. The implementation of Alg I alternates between training and sampling, which in practice demands considerable manual intervention. Moreover, because E changes drastically at each iteration, optimizers such as Adam (Kingma & Bal (015)) must reset after every fine-tuning step; otherwise, stale momentum can trigger sharp and irreversible increase in training loss. To

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Algorithm 1 SFBD-OMNI
Algorithm 2 Online SFBD-OMNI
Input:
clean data Eclean = 1 noisy data
Input: clean data Eclean = {x J==1 noisy data
Epoisy = {y (y(1) Ji=1. CDM loss LCDM
Enoisy {y Ji=1 gradient steps m,
CDM loss LCDM
// Pretrain using clean samples
// Pretrain using clean samples
1 +- Minimizing LCDM heam (x) r(ylx)
1 0 Minimizing (CDM(0, heam (x)((y)x)
2 3 <- [x(() take one sample from up (xly) for each corrupted sample y Enoisy }.
& (x(") take one sample from us (xly) for each corrupted sample y E Eaoiy! }.
// Iteratively optimize with corrupted samples
// Iteratively optimize with corrupted samples (online updates)
3 for k 1,2, do
3 for = 1,2,,..... K do
4
0 t Minimizing (0 (en))r((/i) r(ylx)
4
0 + Minimizing LCDM p(x) r(ylx)
with p = PE + 1#xheaa
with T + T
5
3 < [x(*) take one sample from uo (xly) 5
E+ (Replace ratio y of samples in & with the new ones by sampling x from un (xly) for y drawn from Egoisy)
for each corrupted sample y Enoiyy
Output: Final up
Output: Final ui
guarantee convergence in each iteration, the network must also be optimized for a sufficiently large number of steps. However, this can lead overfitting the current iterate p", making subsequent adaptation to new targets more difficult.
To address these challenges, we introduce an online variant in Alg 2 where fraction 7 of the reconstructed set E is refreshed at each iteration. This corresponds updating + (x) in Eq [18 as
+ (x) -x > ((y) (x) dy + -1-1)) (1-7) with px(x) - / q(y) uy (x) (x) dy.
(19)
When = 1, the algorithm reduces to the standard SFBD-OMNI. Because & changes only slightly after each update, we can continue optimizing Ue for additional gradient steps without resetting the optimizer state, allowing it to adapt smoothly to the new minimum. This strategy reduces manual intervention and accelerates convergence. Prop we show that this "lazy" update scheme still guarantees convergence to the optimum. Since the result covers the case 1, it also establishes the convergence of SFBD-OMNI.
Proposition 4 (Convergence to the optimum). Let the distribution sequences (u4) and {p*} evolve according to Eq [18, with i" updated by Eq 19 Starting from an arbitrary initialization pd and for (0, 1), under mild assumptions, we have pd
- P as K cc. Moreover, when X + 0, we have
lim 1imp==1 = DRL. (ht 111 DKL. (h p²)
(20)
k-+00
In addition, the following bounds hold:
DKL. (q TTpxt < DKL (ht po) ISKSK
min
(21)
where denotes the total number of iterations and TyPdata-
While Eq (21) may suggest that a smaller 7 leads slower convergence, note that with smaller 7, the set E is only partially updated, so the network requires fewer steps to converge. Thus, although larger K may needed to guarantee convergence, each step is cheaper, and the total training time does not necessarily increase. In practice, since the optimizer does not need to reset, training time can even decrease.
Comparison to existing methods. When À = and the corruption process is Gaussian noise injection. with the posterior modeled via the backward SDE in See 2 our framework reduces to SFBD (Lu et al. 2025). In EMDiffusion, Bai et al. (224)) heuristically derive an iterative rule that coincides with SHBD-OMNI's update in Eq (18) when À 0; work establishes convergence of
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CIFAR-10
Method
Pisel Morking (f)
Addiiiv Guaes. (
Oragazale (8)
Caess Rie (A)
Anngssll (9)
Noise Ratson & Rover 3019
92.06
SURB-Sexc ol
132.61
191.96
219.1
AnbleenDii
Daras
Dandli
2023
25.85
EMDiflusion
8647
115.11
91.89
$9.04
SPBD Lu et a. 2023
13.53
SPRD4ENINI (curs)
21.31
10.11
32.61
11.00
11.85
Ouline SFBD-OMN] (ours)
11.06
31.32
11.21
Table 1: FID scores across different corruption processes on CIFAR-10 and CelebA. Processes marked with v/ satisfy the identifiability condition, while those marked with X do not. Pixel masking is applied with probability p 0.6 per pixel. Additive Gaussian comuptiion adds noise with o 0.2 to each clean sample. The grayscale process converts a color image into a single-channel grayscale image, while Gaussian blur is applied with kernel size of nine and 2. All methods, except Noise2Self, are pretrained on 50 clean images randomly sampled from the training dataset.
this rule to the optimal solution, which EMDiffusion does not, and further extends it with an online formulation and the ability to handle non-identifiable corruption processes. Unlike AmbientGAN (Bora al. 2018), which requires differentiating noisy samples with respect to clean ones and cannot address non -on-inentef corruption processes. SFBD-OMNI and the online version assume black-bor box access to the corruption process. avoid adversarial training, and thus sidestep common issues such as gradient vanishing (Goodfellow et al. 2014, Miyato et al., 2018 Fedus et al.) 2018) and mode collapse (Goodfellow) 2016 Arjovsky & Bottou 2017 Mescheder et al. 2018).
6 EMPIRICAL STUDY
In this section, evaluate the proposed benchmark settings, both SFBD-OMNI and its online variant demonstrate superior performance over existing approaches for recovering the original data distribution from corrupted observations. Furthermore, our ablation studies show that the method can effectively address non-identifiable corruption processes.
framework introduced in Sec Across diverse
Datasets and evaluation metrics. Our experiments are performed on CIFAR-10 (Krizhevsky & Hinton 2009) and CclebA (Liu et al., 22022). with image sizes of 32 x 32 and 64 x 64, respectively.
CIFAR-I0 10 contains 50,000 training samples and 10,000 test samples spanning 10 object categories. CelebA is a large-scale dataset of human faces with a standard split of 162,770 training, 19,867 validation, and 19,962 test images. For CelebA, preprocessing follows the official tool released with DDIM (Song et al. 202
Models and other configurations. In our implementation, we parameterize u (x y) with a fow-matching model (Lipman et al., 2023) and apply small endpoint perturbations to y to avoid degeneracy, as described in Sec|C] We adopt flow matching because it converges faster and has lower-variance training objective than diffusion-
models, while achieving comparable or even superior sample quality. This computational efficiency is particularly important in our framework, where the bridge models are trained repeatedly against a moving target distribution. To further mitigate overfitting, we adopt the non-leaky augmentation technique (Karras et al. 2022). For the classical SFBD-OMNI, after pretraining on a small set of clean samples, we set the slmple weight A_ zero when the corruption process satisfies the identifiability condition; otherwise, we A use I+X 0.2, unless specified otherwise. For the flow variant, we fix I+x 0.2, as this setting yields more stable training. In addition, unless noted, we set the noisy-set update ratio to 7 = 0.002 and perform the update at the end of each training epoch. For sampling, we generate samples by first picking y from the noisy dataset and then sampling from the final ug (x y). Additional training configurations are provided in Sec G We evaluate image quality using the Frechet Inception Distance (FID), computed between the reference dataset and 50,000 images generated by the models.
Performance comparison. In T we compare SFBD-OMNI with representative models trained on noisy images corrupted by various processes. As discussed in See 3.1 pixel masking and additive Gaussian noise satisfy the identifiability condition, making it theoretically possible to recover the data distribution using only noisy samples. In contrast, grayscale conversion and Gaussian blur do not satisfy this condition, meaning that additional prior information is required for effective distribution recovery. (Notably, Gaussian blur discards high fighffreyeecc components of an image and can be viewed projection in the Fourier domain.)
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CelebA Grayscale (X)
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Figure 2: FID scores of Online SFBD-OMNI under different clean sample weights p p= 1++ across various corruption processes. Processes marked with // satisfy the identifiability condition, while those marked with X do not.
For the baseline models, Noise2Self (Batson & Royer 2019) is general-purpose denoising method trained with self-supervised techniques. SURE-Score (Aali et al.) (023) and EMDiffusion (Bal al. 2024) address general inverse problems, leveraging Stein's 's unbiased risk estimate and expecta- tion non-mmximiiztion, respectively. Notably, the update rules of EMDiffusion coincide with those of standard SFBD-OMNI when no additional prior information incorporated. rendering ineffective for non-identifiable corruption processes. AmbientDiff (Daras & Dimakis, (023)) in contrast, is specifically designed to train diffusion models on images corrupted by masking. We also report results from the original SFBD, which is tailored to additive Gaussian noise (Lu et al. (025). A discussion and empirical comparison with a very recent work, Ambient Diffusion OMNI (Daras et al.) 2025b), is provided in Sec[] Following Bai al. (2024), unless otherwise stated, all methods except 2ois are pretrained on 50 clean images randomly sampled from the training dataset. In SFBD-OMNI and the flow variant, these images are further used as prior information during sequential training whenever the clean-sampl weight 14X > 0. For all reported results, we consistently use the same set of 50 clean images.
As shown in Table D apart from the pixel-masking corruption process, SFBD-OMN OMDI and its flow variant consistently outperform the baselines, achieving substantially better performance on non- identifiable processes. In the pixel masking case, EMDiffusion reports a marginally lower FID than SFBD-OMN however, the difference is negligible, indicating that SFBD-OMNI performs on par with EMDiffusion in this setting. For the non-identifiable corruptions, we observe that incorporating prior information, by jointly training the model with reconstructed samples in E and clean samples, effectively guides the model toward the true data distribution, as reflected in the much lower FID scores. In addition, because the flow-variant implementation always assigns non-zero weight A to clean samples for added stability, its optimal solution P deviates from the true data distribution in identifiable cases, leading to a slightly higher FID than classical SFBD-OMNI. In contrast, for the con-identifiable processes, this additional regularization is essential and applied in both variants. Consequently, the smooth updates and end-to-end training pipeline of the flow model provide it with an additional advantage, enabling it to achieve lower FID scores.
Effect of the clean sample weights. To examine how SFBD-OMNI leverages clean samples to mitigate identifiability issues, Fig 2 reports FID curves under varying clean-sample weights and corruption types (settings follow Table []. When identifiability does not hold, using clean samples as soft prior constraint guides the model toward the correct distribution; however, overly large weights pull the solution away from the target, increasing FID. Conversely, when identifiability is satisfied, this regularization is unnecessary and may degrade performance. This phenomenon corroborates our discussion in See 3 and See In particular, in identifiable setups, clean samples mainly help initialize Po. after which training proceeds best without them (e.g., CIFAR-10 10 with Gaussian noise). When identifiability fails, clean samples must remain active (A > 0) to avoid convergence to an arbitrary element of S(q), as seen in CelebA with Grayscale and Gaussian Blur, where removing clean samples increases FID dramatically. These trends align directly with the theoretical role of identifiability. Since the clean samples are only used for initializing po when the identifiability condition is satisfied, we show in See See that it is acceptable to use samples from a similar distribution instead when clean samples are not available.
Effect of the number of clean samples. Fig reports the FID scores of Online SFBD-OMNI on CelebA under Grayscale corruption for different amounts of clean data. Increasing the number of clean samples improves performance at both the pretraining and iterative optimization stages, though with diminishing returns. This is expected and aligned with our discussions in Sec 5 more clean
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CalebA Grayscale
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905 1.0: 3.5c 2.0 2.5 5/w 1.3k 4.7x 4.3 5.0k Number of Clean Samples
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Figure FID scores of SFBD-OMNI under different settings, (a) Online SFBD-OMNI FIDs under grayscale corruption for varying numbers of clean samples. (b) FID trajectories of the online version under additive Gaussian corruption (a = 0.5) with 2k clean samples, for both the running reconstructed set & and a newly generated sample set. (c) FIDs of the classical SFBD-OME under additive Gaussian (a = 0.2) without clean samples; iteration O represents the untrained model.
samples make the empirical clean distribution helean closer to Pdata- yielding a better initialization pd = helean and a limiting distribution Pz (defined in Eq (II)) that more closely matches Pdata- Once helean is already a good approximation, however, additional samples provide only marginal benefit.
Effect of the update ratio 7. Fig 3p shows the FID trajectories of both the running reconstructed sample set & and a newly generated sample set during the iterative optimization stage of Online SFBD-OMNI, evaluated under different reconstructed-sanet
update ratios y. The experiment is conducted on CIFAR- 10 with additive Gaussian corruption (a = 0.5) and 2,000 clean samples. A larger y causes the reconstructed set E to be refreshed more frequently, which yields sharper early decrease in FID (as seen for 0.5). Yet, because & changes so rapidly, the model cannot fully adjust to the current reconstruction set before it updated again. This instability appears as growing discrepancy between the FIDs of reconstructed and newly generated samples after epoch 6, eventually degrading reconstruction quality and causing both FID curves to rise. In contrast, smaller y values make & evolve more gradually, giving the model enough time to optimize with respect the current set. This leads to more stable training, delays degradation, and achieves lower overall FIDs. Hence, in practice, relatively small 7 is generally preferable.
Identifiability vs. practical recoverability. As discussed in Sec B] although injective corruption operators in principle allow recovery of Paata from corrupted samples alone, the unfavourable sample- complexity rates make this practically infeasible. Fig 3 shows the iteration-wise FID of classical SFBD-OMNI under additive Gaussian noise with no clean samples (A = 0). The steadily decreasing FID consistent with Prop 4 which states DKL (Ptam r²) decreases monotonically (as ht = Paaa the corrpption is injective and X = 0), starting from the untrained model pº. However, even after saturating around iteration 4, the FID remains
substaaatayy worse than the 10.81 achieved when just 50 clean samples are provided. This gap supports our claim: relying solely on corrupted samples is impractical, whereas even very small clean set dramatically alleviates the issue.
Further empirical evaluation and insights on practical limitations. To further demonstrate the effectiveness of SFBD-OMNI, we evaluate it on high- nigh-resolution satellite and MRI images with Poisson and compressive sensing corruption; see Sec|K] The results support our theoretical findings, yet the remaining artifacts suggest that achieving reployment-qualin reconstructions may require dowain-aware priors or problem-specific design choices.
7 DISCCUSION
In this work, we proposed SFBD-OMNI, a principled framework for distribution recovery based on models. Unlike GAN-based approaches that rely on adversarial training. our method builds on the Donsker-Varadhan representation of the KL divergence, which reveals an equivalence to one-sided entropic optimal transport objective. This reformulation naturally yields alternating minimization scheme that is theoretically grounded and practically stable.
Our analysis shows that SFBD-OMNI can recover the clean data distribution under identifiability conditions, and with the aid of small set of clean samples, it remains effective even when these conditions fail. To address the computational challenges of sequential training, we introduced an online variant that enables end-to-end optimization without sacrificing optimality guarantees. Experiments on CIFAR- 10 and CelebA confirm that the proposed method achieves consistent improvements over representative baselines across a range of corruption processes.
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ACKNOWLEDGMENTS
We gratefully acknowledge funding support from NSERC and the Canada CIFAR AI Chairs program. Resources used in preparing this research provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute.
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A
LLM USAGE
LLMs were used assist with text refinement and data formatting, but not for generating core research content.
B RELATED WORK
Recovering the underlying clean distribution from noisy or incomplete observations has been an active line of research in recent years. [Bora et al. (2018) introduced AmbientGAN, demonstrating both theoretically and empirically that GANs can recover the true distribution even when only corrupted samples, such as randomly masked images, are available. Extending this idea, Waan et al. (2023) showed that, under mild assumptions, if corrupted real and generated samples are indistinguishable, the learned model necessarily recovers the ground-trum distribution. In this work, we present a comprehensive study of the identifiability conditions of corruption processes. When these conditions are satisfied, the clean distribution can be recovered directly from corrupted data. When they are not, we propose an effective strategy that leverages a small number of clean samples enable substantial recovery of the underlying distribution.
Building on the success of training GANs with corrupted data, several recent studies have investigated whether diffusion models can also be trained under additive Gaussian corruption (Daras & Dimakis 2023 Daras et al. 2024). Daras et al. 2024) demonstrated that when corruption is induced by a forward diffusion process, the marginal distribution at any time step constrains those all other steps through a set of consistency relations. Exploiting this property, they showed that training on distributions above the corruption noise level allows the model to infer distributions lower noise levels by enforcing consistency an approach that has proven effective for fine-tuning latent diffusion models. Nevertheless, subsequent work found that training such models from scratch impractical. as it would require an unrealistically large number of corrupted samples (Lu et al. 2025 Daras et al. 2025a). To address this, both Lu al. (2025) 2026) and Daras et al. (2025a) proposed augmenting training with a small set of copyright-free clean samples, demonstrating that diffusion models can indeed be trained from scratch to achieve strong performance, albeit through distinct methodological
routes.
Beyond recovering data distributions from corrupted observations, there has been growing interest in leveraging pretrained diffusion models to solve inverse problems, where the goal to reconstruct underlying images from corrupted inputs (Chung et al. 2023 Feng et al. 2023 Zhang et al.) 2023 Chung et 2002 [Song et al. 2022 Murata al. 2023). While these methods also perform recovery, they operate in a fundamentally different regime: they assume access to pretrained diffusion model that already encodes the ground-truth distribution. By contrast, our work addresses the from-scratch setting, where the objective is to learn the ground-trus distribution itself directly from corrupted samples, without relying on a pretrained model.
Very recently, Daras et al. (025b) proposed a complementary strategy, Ambient-o, which also seeks to align corrupted samples with the clean distribution. The method adds extra Gaussian noise to corrupted inputs so that, once sufficiently noised, they become nearly indistigguishable from clean-noisy samples and can be used directly for diffusion-model training an idea similar in spirit to SDEdit (Meng et al. 2022). Because the aligmment is achieved through noising rather than modelling the corruption, A is agnostic to the underlying corruption process and requires knowledge of it. However, this design introduces a trade-off: stronger noising improves distributional alignment but may also remove informative structure from the observations. In contrast, by assuming access the corruption process as a black-box generator, SFBD-OMNI requires no extra noise and therefore preserves the full signal.
C GAUSSIAN NOISE REGULARIZATION FOR DETERMINISTIC SAMPLING PATHS
In Sec 2 we discuss how bridge models can employed learn the posterior distribution
μ(x) r(y xx)
My (x)
5 [ r(y x')dx'
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given access samples from ((xx) and the ability to query the corruption kernel r(y x) black-box generator.
challenge arises when the interpolation path is chosen to the straight- Mine segment between and y. In this case, the backward sampling scheme in Eq (8) reduces to the special case g = 0, so sampling is performed by solving an ODE, Because the dynamics deterministic, the model can no longer represent distribution, leading to a degeneracy.
To avoid this issuc, we perturb y with small Gaussian noise before using it as the endpoint x; in both training and sampling. This restores the stochasticity of the interpolation: the model learns deterministic flow transporting dy (x) to ((, II), which aligns well with standard flow-matching formulations (Lipman al. 2023 Liu et al. 2022).
Importantly, the perturbation is applied only the endpoint used as the ODE's initial condition; the model itself is still conditioned on the original, unperturbed y. Thus, the perturbation alters only the sampling path, not the conditioning variable.
From conditional VAE perspective, this is similar adding small noise the latent code while keeping the conditioning variable fixed (e.g., a class label or observed image). Such perturbations regularize the decoder but do not change the underlying conditional distribution p(x y) being modelled. Likewise, in our setting, the flow model continues to learn the correct posterior uo [ y because y. the variable that defines the conditional law, remains unchanged. The perturbation merely prevents degeneracy in the ODE initialization and does not distort the learned conditional mapping.
D THEORETICAL RESULTS RELATED TO THE IDENTIFIABILITY
Proposition (Identifiability Condition). Let P(X) denote the set of clean sample distributions. When the corruption kernel r(- x) depends continuously
the convex objective in Eq (TO) admits a unique minimizer p" Pdoto whenever Te is injective on P(X). Tf is not injective, the objective is still convex, but all distributions p satisfying T.p = TyPdota are minimizers.
Proof. Let q = Pypaa and define
(p) = DKL (q |I Txp).
Convexity: For P1,P2 PXX) and t (0, 1),
Ti(tpi + - t)p2) = =T + (1 -)) T.p2
Since the map v + DKL (q || m)) is strictly convex,
Mlpi + (1 - (1-1)p2) = DRL. (q 1. Txp + (1-1)Trp) TTT2)) < (J(1)) + J(22).
Injective case. Assume T, is injective on P(X). pf + pa then ToP + Tep2, and by strict convexity of m DKL (m 9).
Htp + t))p)
+1 (1-)))
E (0, 1)).
Thus J is strictly convex in p. Since J(pamu = DKL (q q) = 0, Pdaaa the unique minimizer, ie., p" Pama- (Continuity of x + r(y x)dy for bounded continuous f gives the usual l.s.c/coopactness to ensure well-posedness, uniqueness comes from strict convexity.)
Non-injective case. If T, is not injective, then for any with T.p = q.
J(p) DxL (q 4) = O.
which is the global minimum. Hence every p E S(g) == (PEP(X) T,p - q} is minimizer.
of h onto the original KLAP solution set. Then the minimizer of Eq (1), py, converges to ht as X - 0.
Proposition 2. Let hl arg min peslg) DRL (h |I p) denote the Information-
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Proof. Let T,p mp and define
F(p) = DEL(a
G(p) = (kl(H))p).
For À > 0 define
{ (F(D) \.
PA E arg min p
Then the optimality against h! gives, for each X 0,
F( + +G((3) & F(h)) + +G(H)))(())) AG(h))
since F(ht)
Hence
0 < F(D)) & ((G(h))G((A)) S AG(h)) -
By compactness of P) pick subsequence À J O with PX, + p. By lower semicontinuity of F, F(p) < lim inf; F(P),) 0, hence S(q). From the same inequality, \xG(())) S AeG(()), so dividing by A > 0 and taking limsup gives limsup, G(p3,) s G(h!), By lower semicontinuity of G and convergence px. p.
G(p) < lim inf G(P..) S sum G(p).) ) - G(h!).
k
Thus G(p) minpes(() G(p). As the minimizer ht is unique, px ht.
E THEORETICAL RESULTS RELATED TO THE ONE-SIDED OT The derivation of Eq 15
DEL (q Tp) = / aly) log ply) dy = f a(x) log ((x')r(y x')dx' dy
((y)
Typly)
/ a(y) log p(x')r(y rryy) xx)dx' +
f q(y) log Ep (xx fx(xx) +C
/ aly) max En, (fs DKL (uy l1 (1 +C min Bgg [DKL (uy I) Euy (fy (x)]] +C,
where we have applied Eq (4). the Donsker- Varadhan variational principle (Donsker Varadhan 1983) in the second last equation.
Lemma 1. Given the cost function be c(x,y) -log r(y x) for some corruption kernel consider the problem
/f ((x dx dy + DKL(T p # #lll(4)
min
where (g) is the set of joint distributions fixed y- y-marginal q. q realizable under p via i.e. p E S(q), then the optimizer is
T Tx(x,") y) p(x 1y) qy))
which has marginals πミ = and क. q.
Proof. Introducing a Lagrange multiplier for the constraint f T (xx, dx = q(y), the optimal solution takes the form
(x, p(x) q(x)) e c(x.))
Z(y) = (((x) cc(x.)) dx.
Z(y)
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With c(x,y) -logr(y x), this becomes
n' "'(x y) p(x)r(y x).
If qly) f p((()) rly x) dx, then Z(y) q(y). and thus
x y) ((x)r(y x)
g(y) = pp(x y).
Therefore,
T³ (x y) = gly) p(x ly);
which indeed has marginals x p and γं q.
Proposition 3. Define the cost function c(x,y) = - llg x). Problem 16 arg min, $(p) + x DKL (h p)
is equivalent to
with
/f c(x, T(x, dxdy + DKL (T p)
(p)
min TElly(a)
where (g) denotes the set of joint distributions of (x, y) with y-marginal fixed to q. Moreover, when À = O, the optimal solution p" coincides with the x-marginal of the corresponding minimizer x² in the inner problem.
Proof. We note that, by definition, c= -y- Then starting from Eq (15) we have
min E. [DKL (My 11 - Euy (x)]] ny
/f ((x,y) log x(x,y) dxdy + If p(x)g(y)
min
yddxdy
1f m y((x DEL (rr p))..
=min
where xxx,y) y) (x) (()) consisting of all joint distributions of x and y with the y-marginal equal to
For the second part of the proposition, when A our discussion See 3.2 shows we have p" S(q). Therefore, by Lem we complete the proof.
F THEORETICAL RESULTS RELATED TO SFBD-OMNI
Proposition 4 (Convergence to the optimum). Let the distribution sequences {ug) and {p*} evolve according to Eq (18). with p" updated by Eq (19). Starting from an arbitrary initialization pd and for E 10, under mild assumptions, we have pd - P. as k + cc. Moreover, when A + 0, we have
lim x =
DKL. (ht x1) s DRL. (N² || 1*))
(20)
k+000
In addition, the following bounds hold:
DKL (ti || po) TK
min DKL (q T.pht) <
(21)
ISRCK
where K denotes the total number of iterations and q TyPdotaa
Convergence to p By collecting the terms involving uys Fx(p", My defined Eq (66 can be
written
Fa uy) = E.g (DEL (uy mt)) + A
(22)
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where (z) is defined in Eq 18) and A contains all the terms independent of dly. As result, taking the minimizer of the objective in Eq (22) gives the update rule of My in Eq 18) Note that the result also shows that:
FA(p", 1 Fxpt, ) B. DKL (uy [
(23)
In addition, according to the Donsker-Varadhar variational principle (Donsker & Varadhan 1983). when My is picked to the minimizer of the current p", we have
Fy(e" uy 1 = (Pk),,
(24)
with Ix (PR. defined in Eq
When 5t is updated in an incrementable way as shown in Eq 19 we claim p*+1 is updated by
minimizing
Fx ) + DDT.. (p* p)
(25)
with (1-7)(1+1) Notably, when updating ratio & 1, the entire sampling set & will be replaced.
and we recover the original SFBD-OMNI, In this case, O, the update of p is then obtained by taking the minimizer of Fx(p, ut) with ut fixed.
Note that
1 I+
DKL (p* I) (1+ Fx(p ut)
(mg Ah + vD") (1p) Bb:
(26)
where Bg collects all the terms not involving p and
mx(xx) = f ((y)u
(27)
we take ++1 as the minimizer of Eq 26 we have
(m + Ah + up²)..
(28)
By choosing v (1+))(++A) the update rule of ph+11 coincides with the one in Eq 18) with 5 updated according to Eq
7 x To see this, we note that, when (1-*)(++)) the weights of m f gly) ut (x) dy in Eq 18
and Eq 28) are matched and equal to I+x In addition, the weight ratios between m and m
(absorbed respectively in y² in Eq (8) and 5 in Eq (1)) are both - 7. This suggests that for both update rules, m are mixed in exactly the same way. As result, the two update rules must be equivalent.
The optimiality of + also suggest,
(Fx(p", , VDL (p* 11p²)) (Fx(, ut) + vDKL (pt (1tt1))
( (mp Nh + vpt) || 1t 1
=(1+A+ DKL
which implies
FA(p u Fx(ph+1 wx 0.
(29)
As result, according to Eq 23 and Eq 29 we have Fx(p ut) + fx(ptuut))
Fx ux+1) decreases monotonically. Combined with Eq (4), have Tx(pk) decreases monotonically well. In addition, (Ps) is bounded below, (p&) much converge to some limit JTC As result, for every subsequence of pk. it must converge to some cluster point p. where p is then fixed point under the update nules in Eq (18). That is,
1
p
mmp Ah vp),
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where mx(x)) f ( (x)dy and the posterier distribution (x)) p(x 1y) rearrangement, have
p(x)r(xl)) T.Ply)
After
1 A p.
(30)
T+
We complete the proof of the optimal convergence by showing that the only satisfying Eg (30 is PX- Lemma 2. The following equivalent for such p:
p(x) | g(y) p(xl)) dy + 1 A
h(x),
(31)
3peRS. S.L. - -f g(y)
r(yl)) dy h(x)
+ ==
(32)
Ty(y)
p(x)
Moreover Tx stricily convex on the probability simplex, any solution of 37 is the unique global minimizer of Tx.
Proof. (+)) Multiply
32 by p(z) and apply
gly)
dy p(x) / p(xl))
1
(y())
to obtain
/ p(xl)) + + = mp(x).
Integrate both sides over x:
1 1+\= / p(x) # =-=1+,
and substitute back to get 31
(E). Starting from 31 rearrange:
(1 X)p(x) - Xh(x)= - - p(x))) dy p(x)
-p((x) /
q(y)
r(ylx) dy.
T,(y)
Divide by p(y) > 0 and rearrange:
f
g(y) Ty(
r(ylx) dy h(xx)
+ +(1+ 11 = 0,
p(x)
which is 432 with = 1 X.
In addition, the Lagrangian for minp20.f p=1 Tx) L(p) = Tx(p) + + (ʃ p-pp-- -111 1). For interior
p > 0, the Gateaux derivative of Jx at p equals the left side of (32) minus μa. Thus (32 is the first-order condition vL(p) O. Since Jx is convex, any interior stationary point is global minimizer.
By Lem 2 we know that P PI is the unique minimizer of JA for > 0. Moreover, as À + O, Prop implies that px + ht, which establishes the first part of the statement.
Convergence rate when - Define H(p) = DKI. (ht p). When À 0, as (1-)((14)
1-7
Eq
28
reduces to
p+l mm + (1 7) >)pp
(33)
where mx(x) = q(v)u) (x)dy and ut updated according to Eq 18 Then by the convexity of the KL divergence, we have
(x)ddy
H(p s 7) H(()) H(m,))
(34)
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Rearrangement yields
H(m?) 2 H(p") (H(()t))-HHiC
(35)
Let Ht denote the joint distribution induced by h (x)r(y x) and likewise pk the one by p (x)r(y x).
In addition, let uy denote the posterior distribution of H Note that, as ht E S(q), we have
hl (x)r(y x) ((y) (())))(())...))))))()))))))))( Then, by the disintegration theorem (Vargas al. 202 we have
H(p²) - DKL. (h' 11 - DKL (H1 t
p*) = DKL (a T.p²) + Enr [DKL. (u)) u
2 DRL. (g |l Trp²) + DEL. (ht
m DEL (4 |l Tru²) H(p") + (H(p"))) H((1))
Cancel out H(p") and rearrange to obtain the monotonic decrease of DEL (ht || p+)
DKLL (a Tp²) H(p -H(p").
(36)
Telescoping yields
K [p(p)) H(p 27 Σ
K
H(pp)) 2
DRL. (g Tp²).
(37)
k=0
k=1
As result,
H(p¹)
DKL. (ht 11
DEL (a Tp²) (E(1.2...,.)
min
-
(38)
7K
7K
G EXPERIMENT CONFIGURATIONS
All SFBD-OMNI models were trained one to four L40 GPUs using SLURM scheduling system. With the standard SFBD-OMNI, training on CIFAR-10 takes about days and on CelebA about days. The online variant is more efficient, requiring roughly days for CIFAR-10 10 and 6 days for CelebA.
G.1 MODEL ARCHITECTURES
We implement the proposed methods using the EDM backbone (Karras et al. 2022) without precon- ditioning, and adopt this configuration throughout our empirical studies. The training pipeline is built on flow matching (Lipman et al. 2023).
Table 2: Experimental Configuration for CIFAR-10 and CelebA
Parameter
CIFAR-10
CelebA
General
Batch Size
256
256
Loss Function
Flow match ing loss man et al. 2023
Lip
Flow mat.ching loss man et al., 2023
L
Denoising Method
torchdiffeq Chen
2018
torchdiffeg Chen
2018
Sampling Method
torchdiffeg Chen 2018
torchdiffeg Chen 2018
Network Configuration
Dropout
0.3
0.3
Channel Multipliers
{2,2,2)
{2,2,
Model Channels
128
128
Channel Mult Noise
2
Optimizer Configuration
Optimizer Class
RAdam
Kingma & Ba, 2015 Liu
RAdam
Kingma Ba 2015 Liu
et al. 2020
et al. 2020
Learning Rate
0.0001
0.0001
Betas
(0.9, 0.95)
(0.9. 0.95)
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G.2 DATASETS
All experiments CIFAR-10 (Krizhessky & Hinton 2009) and CelebA (Liu et al. 2015) are performed using only the training splits. For FID evaluation, each model generates 50,000 samples, and the score is computed against the entire training set.
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H
SAMPLING RESULTS
H.1 CIFAR-10
(a) SFBD-OMNI (FID: 21.31)
(b) Online SFBD-OMNI (FID: 22.43)
Figure 4: Pixel Masking
(a) SFBD-OMNI (FID: 10.81)
(b) Online SFBD-OMNI (FID:: 11.06)
Figure Addictive Gauss.
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(a) SFBD-OMNI (FID: 32.61)
(b) Online SFBD-OMNI (FID: 31.32)
Figure 6: Grayscale
H.2 CELEBA
(a) SFBD-OMNI (FID: 11.60)
(b) Online SFBD-OMNI (FID: 10.28)
Figure 7: Gauss, Blur
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(a) SFBD-OMNI (FID: 11.85)
(b) Online SFBD-OMNI (FID: 11.21)
Figure 8: Grayscale
I DISCUSSION ON AMBIENT DIFFUSION OMNI
Ambient diffusion- Ommi (Ambient-o) incorporates corrupted samples by injecting additional Gaus- sian noise. The key idea is that once sufficient Gaussian noise is added, the corrupted-noisy distribu- tion and the coean-noisy distribution become harder to distinguish. This observation suggests that corrupted sample. after being further noised, can effectively be treated as a noised clean sample, allowing it to be used in standard diffusion-monn model training. This effect does not depend on the specific form of the corruption process, allowing AD-OMNI to operate without requiring knowledge of the corruption mechanism. However, this strategy comes with an inherent trade-off. While heavy noising helps align corrupted samples with clean ones, it also risks erasing useful structure and details within the observations. In other words, sufficient noise is needed for Ambient-o to function as intended, but excessive noise may suppress the informative signal that could otherwise benefit model learning.
In contrast, SFBD-OMNI does not inject additional noise into the samples and therefore preserves the full information of the observations. Rather than relying on excessive noising to align distributions, method leverages knowledge of the corruption process itself, avoiding information destruction while still enabling effective training.
In Table 3 following the Ambient-o setting, we apply a Gaussian blur with varying strengths a and assume access to 10% clean samples. The table shows that, by fully leveraging the information
Blur Strength (a)
Ambient-o
Online SFBD-OMNI
0.6
5.34
0.97
1.0
6.16
3.07
Table Ambient-o vs. online SFBD-OMNI: FID under Gaussian blur of varying strengths.
contained in the corrupted samples, SFBD-OMNI achieves substantially lower FID across blur levels, outperforming Ambient-o by large margin.
PRETRAINING MODELS USING SAMPLES FROM A SIMILAR DISTRIBUTION
As mentioned in Sec 6, according to our theory, when the corruption function is identifiable, a small
number of clean samples needed only obtain good initial distribution Po- This also implies
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that if clean samples from the target distribution unavailable, it is acceptable to samples from a similar distribution instead. To demonstrate this, we pretrain the model on CIFAR- using clean samples from the truck class, and then apply iterative optimization to recover the distributions of automobile, ship, and horse, where all samples are corrupted by additive Gaussian noise with noise level 0.2. The FID scores before and after iterative optimization shown in Table
Class
After Pretrain
Final
Automobile
8.36
6.19
Ship
13.96
8.78
Horse
25.87
13.55
Horse (no pretrain)
80.17
Table 4: FID comparison across CIFAR-I0 classes before and after finetuning, with pretraining conducted on the truck class.
As the table shows, for classes similar to truck - such as automobile the model successfully recovers the target distribution, as indicated by the low final FID. For classes that are less similar, pretraining still provides substantial benefits. In particular, for horse, pretraining on the truck class reduces the final FID dramatically from 80.17 (without pretraining) to 13.55, illustrating the importance of good initial distribution even when the clean samples drawn from different but related class. (Notably, the horse and truck classes still share several low-level characteristics such as edges and common background elements like grass road surfaces.)
K SUPPLEMENTARY EMPIRICAL RESULTS IN LATENT SPACE
In this section, provide additional empirical results high-resolution satellite (256 256) and MRI datasets (320 x 320) corrupted by Poisson noise and compressive sensing (CS). For MRI, the experiments are conducted in the latent space for computational efficiency. The results remain consistent with those in the main text, further validating the effectiveness of SFBD-OMNI across diverse corruption settings. At the same time, qualitative inspection reveals visible reconstruction artifacts, indicating remaining limitations and motivating future work that incorporates stronger priors or modality-specifff inductive biases.
Satellite images and Poisson noise. We satellite images from the training split of the NWPU- RESISC45 dataset (Cheng et al., 2017), which contains 45 scene classes with 600 images per class.
For this dataset, consider Poisson noise corruption. Poisson noise arises naturally in photon-limited imaging systems, including satellite and remote- sensing cameras, where the number of detected photons per pixel is inherently stochastic and follows Poisson statisties (Hasinoff 2014) Schott 2007]. Following common practice in Poisson-noise simulation studies, we vary the photon budget a E 10, 50, 100, corresponding severe, moderate, and mild shot-noise conditions, respectively. We simulate Poisson noise by interpreting each pixel intensity Tig.e E arrival rate and sampling
[0, 1] as normalized photon
Vigicc Poisson(a Ti.j.);
followed by clipping the resulting values to the valid range [0, 1] (Makitalo & Foi 2011
MRI image set and compre compressive sensing corruption. We conduct our experiments the fastMRI brain dataset (Zbontar et al. 2018), using its multicoil training subset, which provides fully sampled raw k-space data from clinical brain MRI scans. For each volume, discard the final four slices, as these typically contain little or no brain anatomy. After filtering, the dataset contains 52,778 MRI slices, from which we randomly sample 2,000 as the clean
For this dataset, consider the compressive sensing degradation, which is natural corruption model for MRL. In particular, MRI scanners do not acquire images directly; instead, they measure the spatial frequencies of the underlying anatomy in k-space (Lustig et al. 2007). The acquisition process
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Satellite - Poisson Noise
MRI
Stage
10
50
100
Compressive Sensing
After Pretrain
9.32
5.71
4.43
36.98
Final Result
7.11
4.13
3.40
28.71
Table 5: FID Scores of Online SFBD-OMNI for satellite images (Poisson noise with 10, 50, 100) and MRI scans (compressive sensing).
10)
Carr. -
Figure 9: Reconstructed Satellite Images Poisson Noise (photo budget a 10, 50, 100).
therefore corresponds to sampling the Fourier transform of the image. Because clinical MRI protocols routinely undersample k-space to shorten scan time, compressed-sensing MRI accelerates acquisition by collecting only subset of frequency coefficients and relying reconstruction algorithms to recover the missing data. Consequently, partial Fourier undersampling is not an artificial degradation, but a realistic and practically motivated corruption process for accelerated MRI.
To simulate a realistic compressive sensing degradation, we follow the standard undersampled MRI acquisition model of Lustig et al. (2007). Given an image x corrupted observation obtained by undersampling its Fourier transform:
Pa(F(z)),
where denotes the 2-D discrete Fourier transform and Po is a binary mask selecting subset Ω of frequency coefficients. We use fixed variable-density sampling mask, generated once at the beginning the experiment and reused for all samples. Following common practice in compressed- sensing MRI, the central low-frequency region of k-space (10% of the spatial extent) is fully sampled to preserve global structure, while coefficients outside this region are sampled independently with probability 0.20. This produces a realistic and reproducible compressive sensing corruption operator that retains essential low-frequency content while heavily undersampling high-frequeng, components.
Implementation. For satellite images, we continue using the model architectures described in Sec For the experiments MRI, (Rombach et al. 2022a) to encode images into the latent space and to decode the model outputs. We keep the model architectures described See unchanged, except for adjusting the input and output channels to to match the dimensionality of the latent representations.
the pretrained autoencoder (VAH) from Stable Diffusion vI.5
Results. Table 5 summarizes the FID performance of online SFBD-OMNI for satellite images with Poisson corruption and MRI scans under compressive sensing, measured after pretraining
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Corrupted
Reconstructed
Ground Truth
Figure 10: Reconstructed MRI Compressive Sensing
and after online iterative refinement. Across all evaluated settings, the online phase yields clear improvement over the pretrained model, demonstrating the effectiveness SFBD-OMNI as general reconstruction framework for real-world corruption processes.
We provide qualitative reconstructions in Figures[land 10 On satellite images, SFBD-OMNI visibly recovers large-scale structures such as building layouts, runway geometry, and aircraft outlines that are heavily disrupted by Poisson shot noise. In the MRI setting, despite severe undersampling, the method reconstructs coherent tissue boundaries and globally consistent anatomical structure.
Limitations and noise sensitivity. While our framework produces promising reconstructions, the Poisson-noise results also reveal a clear limitation. As the photon budget decreases (i.e., noise increases), output quality degrades, with more residual artifacts and reduced line-grained fidelity. This trend is reflected quantitatively in Table where performance drops moving from a=1000 to a=10. Qualitative examples in Fig further highlight these failure modes-under extreme shot noise, texture-level restoration remains challenging and fine structure is only partially recovered. Likewise, for MRI samples corrupted by compressive sensing, we still notice some visible artifacts as shown in
Fig 10
Across both Poisson-corrupte satellite imagery and compressive-sensed MRI scans, SFBD-OMNI reliably recovers the global structure of the underlying data distribution, but struggles to reconstruct fine details, particularly under severe corruption. Poisson noise reduces the amount of usable information in low www-photon settings, and similarly, heavy MRI undersampling restricts the available signal for reconstruction. In such cases, recovering the clean distribution becomes inherently more challenging and may require stronger priors or model-specific inductive biases.
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Reviews:
[{"review_id": "zWnhZkT1qG", "reviewer": "Reviewer_b1RA", "summary": "The paper proposes SFBD-OMNI, a bridge-model framework to learn the clean data distribution when you have many corrupted samples and few clean ones, assuming black-box access to the corruption process. They cast “recover clean from corrupted” as a KL Ambient Projection (KLAP) problem and show two key variational views: (i) the classic AmbientGAN min–max formulation, and (ii) a new formulation via the Donsker–Varadhan principle that turns KLAP into a one-sided entropic optimal transport objective with a tractable alternating minimization (posterior update + prior update). \n\nThis yields closed-form updates and naturally plugs into diffusion/bridge models to learn the posterior \n $u_\\theta(x∣y)$. To address non-identifiable corruption (e.g., grayscale or blur), they augment the objective with a KL regularizer toward a small set of clean samples $h$ and analyze the limit as the regularization weight vanishes, recovering the I-projection within the feasible set. They present SFBD-OMNI and an online variant (partial refresh of reconstructed samples) with convergence guarantees. Experiments on CIFAR-10 and CelebA show improved FID over baselines (including in non-identifiable cases) when a small number of clean samples is mixed in training.", "questions": "- Operator misspecification: How sensitive is SFBD-OMNI to errors in $r(y|x)$? Say you're doing pixel dropout and you have the wrong estimate of dropout probabilities?\n- Automatic hyper-selection: How exactly do you select the $\\lambda$ (clean-weight) and $\\gamma$ (online update ratio) parameters?\n- For my own understanding, what's the right way to think of $\\Phi(p)$ in Proposition 3? Is it a penalty for how different the pushforwards in y are for $p^*$ and $p$?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "- Unifying view + derivation: Recasting KLAP with Donsker–Varadhan to obtain a one-sided entropic OT objective is elegant and principled; it clarifies when/why alternating updates should work and connects to entropic OT literature (contrast with AmbientGAN’s min–max). \n- Identifiability analysis + “few clean” fix: Clear conditions for identifiability and a regularized KL that provably picks the best element in the feasible set, matching practice where a handful of clean samples are available.\n- Convergence guarantees: Both batch and online SFBD-OMNI have convergence results (including a rate bound when $\\lambda \\to 0$). That’s stronger theory than many EM-like or Ambient-style methods.\n- Empirical coverage: Results span identifiable (masking, Gaussian noise) and non-identifiable (grayscale, blur) regimes; OMNI competes with or beats specialized methods and EM-Diffusion. \n- Practical modeling choice: Using bridge/flow-matching to learn posteriors is a strong fit; the paper also addresses degeneracy of deterministic paths with noise injection.", "weaknesses": "- Assumption on black-box sampling from r(y|x): Many real sensors provide only forward passes on x, not easy sampling of y|x across conditions; measuring cost or calibration drift may complicate the assumed operator access. The paper could discuss robustness to operator mismatch (misspecified r). \n- Posterior learning scalability/details: Training $u_\\theta(x|y)$ as a conditional bridge can be heavy; stability hinges on path choices and endpoint noise. More ablations on bridge choices, noise level, and compute vs. FID trade-offs would help (training time is non-trivial).\n- Evaluation breadth: Only CIFAR-10 ($32^2$) and CelebA ($64^2$) with limited corruptions. Stronger evidence would include higher-res datasets, complex forward models (e.g., compressive sensing, Poisson noise), and domain tasks where distributional recovery helps downstream reconstructions.\n- FIDs are fine but you really do need to show empirical samples for the learned model.\n- Tuning $\\lambda, \\gamma$: While theory supports limits, practice shows a “sweet spot” for clean-weight; guidance for automatic selection (e.g., validation via corrupted likelihood) is missing.\n- Identifiability test practicality: The paper discusses conditions and a criterion qualitatively; a procedural test practitioners can run on a new corruption (and with finite data) would increase usability.\n- Minor nitpick, but Section 4.2 needs some cleaning up bc there's too much content in a small space and it's also a key contribution. Maybe reduce some content in Section 4.1 to give more space to 4.2? Also a small section with notation may be helpful, I had to search for the definition of $h$ which was last defined in Section 3.2 (maybe just say $p_data$?). Also, between Eqn (14) and (15), can you add a small phrase saying \"taking \\mathbb{E} over $y\\sim q$ and subtracting E[ log q(y) ], we get...\" It was a little confusing at first since there are many functions and it wasn't clear what's a function of y, x, etc.", "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": "z6HVWVY3Rv", "reviewer": "Reviewer_GiSu", "summary": "The authors propose SFBD-OMNI, a bridge model–based framework that maps corrupted sample distributions to the underlying clean data distribution for image restoration and generation recovery. The method extends the prior SFBD framework to handle arbitrary measurement models beyond the Gaussian corruption setting, providing a more general and theoretically grounded approach to learning from imperfect or partially observed data", "questions": "1.\tHow does the method scale with different amounts of clean data?\nThe paper fixes the number of clean samples to 50 (~0.1%), but it would be informative to see how performance changes when this ratio varies, or how it compares to a small supervised diffusion baseline trained on the same subset?\n\n2.\tDo the authors plan to evaluate SFBD-OMNI on real-world restoration datasets (e.g., RainDrop, AllWeather) to demonstrate its applicability beyond synthetic corruptions, and to compare its performance against recent fully supervised diffusion-based restoration methods?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Theoretical generalization capability for any unbound noise. The paper extends prior diffusion-based deconvolution frameworks SFBD to non-identifiable measurement processes, providing a principled formulation via one-sided entropic optimal transport. This generalization makes the approach applicable to a broader range of real-world degradations \n- Data-efficient restoration without extensive ground-truth supervision. The proposed framework learns to recover clean image distributions using only a handful of clean samples, making image restoration more practical and label-efficient compared to traditional fully supervised diffusion approach", "weaknesses": "- Limited novelty via SFBD framework extension. While the paper provides a principled generalization of SFBD to non-identifiable measurement models via one-sided entropic OT, the overall formulation and optimization pipeline remain close to the original SFBD framework. The methodological increment feels more like an extension or refinement rather than a fundamentally new paradigm.\n\n- Insufficient experimental evidence to substantiate the theoretical analysis. Despite a well-developed theoretical formulation, the experimental validation is narrow and mostly relies on simple synthetic corruption processes. The work lacks evaluation on real-world restoration datasets (e.g., **RainDrop (Qian et al., CVPR 2018)**, **AllWeather (Valanarasu et al., 2022)** ) that could better demonstrate generalization under uncontrolled degradations. Moreover, the notion of “limited clean data” (only 50 samples) is fixed and not systematically analyzed against traditional fully-supervised baselines, leaving unclear how much clean supervision is truly needed.\n\n- Sensitivity to hyperparameter settings and practical complexity:\nThe algorithm involves several critical hyperparameters, such as the clean-sample weight λ and the online update ratio γ, which the authors fix empirically (λ / (1 + λ) = 0.2, γ = 0.002). The paper does not provide an ablation or sensitivity study, and training remains complex due to alternating optimization of all hyperparameters. This raises concerns about reproducibility and the need for manual tuning to achieve stable performance.", "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": "V6XBmqCgAK", "reviewer": "Reviewer_6bRN", "summary": "This paper considers the problem of restoring clean signals from corrupted measurements in a setting with little to no access to clean samples and a black-box access to the corruption model. Taking inspiration from the Stochastic Forward-Backward Deconvolution (SFBD) algorithm for Gaussian noise corruption, the authors introduce SFBD-OMNI as its extension to arbitrary corruption processes. At the core of the method lies a reformulation of the problem using the Donsker-Varadhan variational principle, which leads to an explicit objective for the (augmented) Kullback-Leibler Ambient Projection (KLAP) problem. In addition to the resulting formulation of SFBD-OMNI, the authors also propose its online (flow) variant that performs only partial updates to the data buffer used during training. Both methods are empirically compared with a proper set of baselines on both identifiable and non-identifiable processes. In addition, an ablation regarding the influence of the clean sample weighting is presented.", "questions": "I am generally very sympathetic to this work and consider the approach, together with its theoretical derivations, very elegant. I was leaning toward a score of 8. However, I am very interested in the authors' answer to W1 above and consider it an important clarification before raising my score from 6 to 8. Please also refer to W2. Below are some additional minor questions.\n\nQ1. lines 046-048: Is this phrased correctly? Shouldn't it be *With only a limited number of corrupted samples but an abundance of clean ones (...)*?\n\nQ2. line 160: The authors probably meant $g=0$ for Eq. 6.\n\nQ3. I-projection (line 216) should be expanded to Information-projection for clarity.\n\nQ4. line 253: What if $r(\\mathbf{y}|\\mathbf{x})=0$ under the $\\log$?\n\nQ5. lines 289-298: The subscripts and superscripts for $k$ are probably mixed up.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "S1. The paper is written extremely well considering that the topic is not the easiest to describe formally. The flow of the text is great, with additional intuitive explanations following the introduction of more difficult concepts (like Figure 1).\n\nS2. The authors propose a novel reformulation of the considered objective using the Donsker-Varadhan variational principle, showing a connection to the entropic optimal transport objective. The theoretical results are interesting and relevant to the considered problem. Importantly, the convergence analysis applies to both the standard SFBD-OMNI and the online version.\n\nS3. The proposed methods outperform the considered baselines and the empirical results align with the theoretical considerations.", "weaknesses": "W1. My main concern relates to the accuracy of one of the claims. The authors suggest that, under suitable identifiability conditions, the ground-truth clean data distribution can be recovered without access to clean samples (as lines 065-069 seem to suggest). Proposition 4 guarantees convergence under the considered update rules, even with an arbitrary initialization and full replacement in the training buffer $\\Large \\varepsilon$. Does this imply that SFBD works when no clean samples are used at all, i.e., even when the initial pretraining phase is skipped? Is this scenario empirically verified in the paper, or do the results only cover the case where clean samples are used for pretraining but omitted from the updates in line 4 of Algorithm 1/2? If this is not verified, I would ask the authors to provide results for this case.\n\nW2. I understand that some methods in Table 1 are only applicable to specific corruption processes. Are there any instances where a baseline is theoretically suitable for one of the tested problems, but its results are not included in the table? If so, the paper would greatly benefit from evaluating these baselines in such cases, as this would provide a more complete comparison and support future work.", "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": "3lZ0srObNl", "reviewer": "Reviewer_Sh1d", "summary": "The paper introduces SFBD-OMNI, a new framework for generating from a clean data distribution while training from corrupted samples and a limited number of clean samples. The proposed method generalizes SFBD to arbitrary corruption models, providing both theoretical guarantees and practical algorithms, including an online variant that supports end-to-end training. The authors handle non-identifiable corruptions by introducing a regularization that acts as a \"projection\" (in the KL sense), enforcing the solution to be close to a given distribution constructed from a small set of clean samples.", "questions": "Why was flow matching selected as the primary instantiation? Would other stochastic bridges (e.g., Schrodinger) behave differently? Perhaps using another bridge could alleviate the need to perturb the endpoint samples? It would be interesting to assess several types of flows, especially since, from the beginning of the paper, the authors formulate the problem in the stochastic sense (using Brownian motion, eq. 8), but then implement it with flow matching (which does not involve Brownian motion).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Overall, the proposed approach is clean and elegant, relying on well-established principles.\n2. The paper is well-written and easy to follow. The paper is structured in a clear and logical manner, and the authors provide several intuitive examples throughout to help the reader understand the concepts presented.\n3. The experimental results seem good -- the proposed approach improves upon previous methods, and it is able to handle non-identifiable degradations such as de-colorization.", "weaknesses": "1. Only two datasets (CIFAR-10 and CelebA) are evaluated. Testing on more complex real-world domains (e.g., MRI, satellite data) could have been more appealing from a practical standpoint. In my opinion, this is a major weakness.\n2. While clean-sample weighting is explored (Figure 2), the effect of other hyperparameters such as $\\gamma$ (update ratio) remains unclear.\n3. The authors perturb the endpoints $y$ to \"prevent degeneration\", since flow matching is a deterministic map. But an explanation about the effects of such perturbations (both in terms of theory and practice) is missing.\n4. The proposed approach still uses a small number of clean samples to train the generative model, which is a limitation. In real-world scenarios and scientific applications (e.g., astrophysics), we often only have corrupted samples.", "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)