Generative Modeling from Black-Box Corruptions via Self-Consistent Stochastic Interpolants
Authors: Chirag Modi
Jiequn Han
Eric Vanden-Eijnden
Joan Bruna
Venue: ICLR 2026 Poster (2026)
Keywords: generative models
corrupted data
inverse problems
stochastic interpolants
Primary area: generative models
TL;DR: We develop a novel transport-based approach to learn a generative model of a data distribution using only access to corrupted observations from a black-box observation map

Abstract: Transport-based methods have emerged as a leading paradigm for building generative models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through a noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce a novel approach to this task based on Stochastic Interpolants: we iteratively update a transport map between corrupted and clean data samples using only access to the corrupted dataset as well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards a self-consistent transport map that effectively inverts the corruption channel, thus enabling a generative model for the clean data. We refer to the resulting method as the self-consistent stochastic interpolant (SCSI). It (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance on inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions.

Published as a conference paper at ICLR 2026
GENERATIVE MODELING FROM BLACK-BOX CORRUP- TIONS VIA SELF-CONSISTENT STOCHASTIC INTER- POLANTS
Chirag Modi" Jiequn Han"*, Eric Vanden- Manden-iijnden,,, Joan Bruna
New York University
2 Flatiron Institute
3 Machine Learning Lab, Capital Fund Management
modi modichiragnny..ce edu, jhan@flatironinstiii org, eveuna, nyu. edu
ABSTRACT
Transport-basse methods have emerged as leading paradigm for building genera- tive models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce novel approach this task based Stochastic Inter- polants: we iteratively update a transport map between corrupted and clean data samples using only access the corrupted dataset well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards self-consistent transport map that effectively inverts the cor- ruption channel, thus enabling a generative model for the clean data. We refer to the resulting method as the self-consistent stochastic interpolant (SCSI). It (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions.
INTRODUCTION
Generative modeling has become a central aspect of high-dimensional learning. Transport-based methods, including diffusson--bsed models (Ho al. 2020 Song et al. 2021] and flow-based models (Albergo & Sanden-Eijndee 2023 Lipman et al., 2022 Liu et al. 2023b), have emerged as leading frameworks with wide range applications from natural image synthesis (Rombach et al., 2022) to molecular design (Watson et al. (023). These methods rely access to clean samples 2 ι of the target distribution, which are plentiful in many machine learning tasks. However, in many scientific and engineering applications, such clean data of interest are unavailable. Instead, we only observe corrupted measurements y through a forward map y = F(z) that is typically noisy and ill-conditioned. Examples include medical imaging (tomographic projections of internal structures), astronomical observation (atmospheric distortion), and other measurement processes subject to noise and information loss Taaantola 2005). As result, the target data is never observed directly, rendering standard generative modeling inapplicable.
To further motivate the problem, consider experiment that produces iid measurements y1..... VN ), each arising from some unobserved signal of interest # .N E X', Scientists often have an accurate physical model, or simulator F, whose output y = F(x)) can be viewed sample from the conditional distribution P(dylr), and the goal is to infer the signal associated with a specific measurement y. i.e., sample P(dz|y - g). Computing this posterior distribution requires prior distribution л P(X)), which is often not explicitly known. However, we can still characterize it implicitly: denoting by μ E P(Y)) the law of the observations y,, and by K. P(x)) + P(Y)) the
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integral operator associated with the simulator F, i... MCT fx P())z)T(dz), our desired prior distribution is the solution of the inverse generative problem K = μ..
A classical framework to address this scenario is Empirical Bayes, dating back to Robbins 1992). The key idea is to consider an iterative procedure, based on the expectation expectationmmxximiaaatoo maximization (EM) algorithm:
given the current 'prior' 7(*), one considers the posterior p(h) (dxly) x P(dy x)) (1) (10 (dz), and (k)
performs (approximate) posterior sampling oi (-k))) The samples (ar NEN are then used
to estimate the new prior (((+11) for instance using diffusion models, in Rozet et al. 2024); Bai
jt al. (2024). Despite its conceptual appeal, this Empirical Bayes approach presents two important challenges: one must know and be able to evaluate the likelihood term P(dy(æ) associated with the forward model (having a simulator that merely produces samples from P(dy(ı) does not guarantee this), and ii) the EM algorithm relies on a posterior sampling which is known to be computationally hard and whose sampling errors are generally difficult to quantify. As a result, one often needs to impose further restrictions on the forward model, such as linearity with Gaussian noise enable Tweedie's 's formula for approximate posterior sampling (Daras et al. 2023 Rozet et al. 2024).
We introduce novel framework for this inverse generative modeling task, using only a black-bor bbo simulator of the forward process F. Our approach leverages stochastic interpolants (SI) (Albergo & Vanden-Hijnden 2023 Albergo et al. 2023a) with a self-consistent training procedure: we iteratively transport observations y to samples via learned velocity field, then enforce consistency by requiring that these generated samples, when passed through F, reproduce the original observation distribution. This scheme not only eliminates the need for clean data, but also avoids backpropagation or posterior sampling through F. and enjoys provable convergence guarantees. As a result, our framework is applicable to nonlinear forward models (e.g.. motion blur), doneeifferentiable operators (e.g., JPEG compression), and non-Gan non-Gaussian noise (e.g., Poisson noise), substantially broadening the applicability of inverse generative modeling. Conceptually, this makes our approach akin to model- free reinforcement learning, which optimizes policies through interaction with a simulator, whereas most prior methods resemble model-based control, relying on explicit knowledge and differentiability of the underlying physics.
Additional related works Several recent works have aimed to generate clean data x using only corrupted observations y. Most existing approaches, however, require the forward model to be explicitly specified and differentiable, often with additional structural assumptions. For example, Daras et al. (2023) ; Kawar et al. (2024); Chen et al. (025); Zhang et al. 2025) train diffusion models with corrupted data under explicit linear forward models and additional rank condition. [Akyildiz et al.) (2025) learns a generative prior by directly minimizing the sliced- Wasserstein-2 2 distance between observed data and model outputs. There is growing interest in generative models trained on mixtures of clean and noisy data which assume access to some clean samples (Daras et al. 2024, 2025 Lu eet al. 22255] Meanti al.) (025). While in this work we only work with corrupted observations, approach can straightforwardy be applicd to these mixed settings. On the theoretical side, [Li et al. (2024) 2025) study inverse problems over measure spaces, analyzing stability, variational structures, and gradient flows. Our work complements these by introducing a practical and scalable algorithmic framework while also establishing convergence guarantees under appropriate assumptions.
Finally, concurrently and independently from our work, Hosseintabar et al. (2025) proposes an EM- based approach to inverse generative modeling using conditional diffusion models, which overcomes the aforementioned limitations of the EM algorithm. Their method can be viewed as an instance of our proposed scheme, whereby the Stochastic Interpolation conditional, so that it directly models the posteriors P()) (dzly))
2 PRELIMINARIES
2.1 STOCHASTIC INTERPOLANTS
Let be the clean data distribution we wish to sample from and y be the distribution of the corrupted data that are available to us, both supported on Id.. Following Albergo et al. (202a)), a linear stochastic interpolant 4; between and μ is defined by
7 04330 3,01 7e2,
1 LE EE(0,1) [0,
(1)
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where (zo, 21) is sampled from joint distribution (or coupling) r(dro, dzı) that maintains the
marginals fRa p((;,zz) = т. Jg. v(dzo, -) = μ., and x ~ Ya is independent Gaussian noise. The
schedules at 3, "t satisfy boundary conditions ag Bi 1, a Po = 0, and "o = "n = 0. Define the velocity fields
b(1,z))= = E(i =] , g(4,2) == E| = x].
Then the solutions to the following reverse-time SDE
ax,² xt" (..xxtddtt )dt + aad gt X ( + V2
Here e E ( with 6t 2 0 is an arbitrary time-dependent diffusion coefficient, and W. = = We note is closely related to the SDE in based diffusion models: specifically, when "r > O. we have s(t, z) ;= Va log p(t, x)) = -2 g(t, x), where p(t, x) denotes the probability density of 4, From now on, for simplicity, we will assume a fixed (i.e., time-independent) diffusion coefficient c. Note that setting c = 0 recovers the probability flow ODE which only depends on the field b. The vector fields b and g can be learned efficiently in practice by solving least-squares regression problems:
have the property that if X [== μ is independent of W then X
b = arg min
F. (b)
with
E (b) = fo 1lbi(,, 1t) - ds]"dtt,
(3)
b
g arg min , (g) with E ( == So Bilg(t, I.) - dt.
(4)
where E denotes an expectation over the coupling (No, x1) ~ v and 2.
Learning the velocity field b and the denoiser g separately allows one to generate samples via the reverse-time SDE (2) with an arbitrary diffusion coefficient schedule e.. Alternatively, by fixing
specific ez, one can instead learn the combined drift required in ( directly through single least-squares regression problem:
v arg ming So E [ia(t,1) - - eo 2/] dt.
(5)
Notation To simplify and unify notation, we use θ to denote the required functions for generative modeling: θ = {b} in the ODE case, and θ = {b. g} in the SDE case. Let te denote the backward transport map induced by θ; that is, Te = Xu under backward probability flow ODE with terminal condition X, = y or (v(v)
X under reverse-time SDE 2 with terminal condition X = y. Accordingly, such a transport map induces a pushforward from the observation distribution Iu to the clean data distribution, denoted by me == ($e) #1.. Note that in the SDE case, t is a random map due to the Brownian motion, and the pushforward should be interpreted as the expected pushforward, i.c., averaging over the randomness of the Brownian motion. Finally, recall that in (3) we use E (b) to denote the loss associated with a candidate drift function b, defined with respect to the SI between π and μ.. Similarly, the objective E4 (g) for the denoiser is defined in (
2.2 PROBLEM SETUP
We consider a probability distribution of interest π E P(A') and a forward model F X' + y. which we allow to be stochastie, i.e., y F(x) defines conditional distribution of y given x. In many applications, Y embeds naturally into X' without information loss, allowing us to identify the two; all numerical experiments in this paper fall into this category. For the general case, we can always construct a common state space via a lifting argument: let Ω = X' x y, and define F(x, v) == F(r)), where w is drawn independently from an arbitrary base measure To E P(2). If KC denotes the integral operator associated with F, then for any v E P(2), we have K To &ux, where vx is the marginal of on X. Thus, if म E P(2) solves Küt jú with jı = no 3 μr, then मं solves the original inverse problem K = μ. In the embedding case, we identify ) with its image in X' and write F X X; in the general case, replace X' with Ω and proceed analogously. Either way, we henceforth assume F F + X.
Injectivity at distribution level (and therefore non- non-invertible) as a mapping in X, thus justifying the need to regularize the inverse problem of recovering from the observations y F(x)) However, the situation is different
The forward model F is often ill-conditioned. non-deteriinistic
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when viewed at the level of probability measures P(2):: soon K. is injective in P(X), i... KT = KT imples т = म. one can hope to recover т from by inverting the linear relationship Km. To illustrate this point, consider the white gaussian noise (AWGN) channel
of, with { ~ 7 = (0, [2):: while the optimum reconstruction at the level of the samples (in the MSE sense) is given by the posterior mean & = [x)] and generically
always have reconstruction error E | - ~|| > 0, the associated inverse problem at the level of distributions amounts to deconvolution, i.c., 14 πT * Ye, which is invertible for any noise level -- and thus и can in principle be reconstructed exactly. Here are few examples of injective channels:
Example 1. The AWGN Channel: F(æ) = x + -.. E ~ "Yd is injective for any a < cc.
Example 2 (Random Projection Channel). Let S(d, r) be the Stiefel manifold of rank r unitary matrices in R and consider distribution E P(S r)) such that span {A; A E supp(p)} = Rd Then the random projection channel F(x) = (A x, A) R' r(,,r) injective. This channel captures applications such as tomography or inpainting.
Remark 1 (Channel composition preserves injectivity). We immediately verify that F1 and Fz are two injective channels, their composition Fi F2 is also injective. Therefore, channels that combine random projections and additive noise are also injective, such Cryo-EM,
Restoration, Generation and Inference Assuming that we have sampling access to μu, our goal is to find a transport map that pushes μ to a distribution = 4 satisfying
Kf Kh μt,
(6)
i.e., á solves the linear inverse problem. Note that in practice our samples from μ come from dataset of observations }4 t. Ys ~ μ4, so our method pushes a dataset of "corrupted" observations into another dataset (d(y)) J of samples from न.. In that sense, our model is restoring dataset. In order to obtain a bona fide generative model for æ. we thus need the additional ability to generate further independent samples. This be easily achieved by combining
method with a standard generative model: either train a generative model for μ using the original corrupted samples {3/ } and then restore its outputs via the transport map é or train a generative model for ı directly on the restored dataset.
Further, the ultimate goal is often to perform posterior inference, requiring us to sample from the posterior distribution P(dx y)) rather than its marginal æ. Again, the natural solution to compose our transport map with a standard conditional generative modeling procedure. Once we have sample access to the marginal æ, we can produce samples from the joint distribution (a, F(x)), which can then be used to train conditional transport model, as in e.g.. Dax et al. (2023) 7hou et al. (0024); Chen et al. (2024).
To summarize, our focus will be on the inverse problem 6 at the distribution level, since it unlocks key applications when combined with standard downstream transport-based generative procedures.
SELF-CONSISTENT STOCHASTIC INTERPOLANTS
In the standard generative modeling setting with direct access to clean data samples 2o ~ π and corrupted samples 21 μ., one may use, for example, the independent coupling r/dzo, dz,
n ((dxr) to construct a Monte Carlo approximation of the expectation in the objective 3 However, in our inverse problem setting, clean samples from т. So it is a priori not obvious how to construct the SI (I. We now describe how construct and train self-consistent SI using only black-box access to the forward map F.
only observe corrupted data from μ and lack access to
3.1 ITERATIVE SCHEME FOR SELF-CONSISTENCY
Eq II provides natural transport between the observed and clean distribution, but is actionable only when one has sample access to both æ and μ.. Observe first that if we replace the sample access of la by oracle access to the simulator F, we could still build a transport from to by leveraging the fact that μ Kr. Indeed,
T a4T B.F(x) 762, LE 1010, n, 2 Vi
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defines valid interpolation between π and μ4, and can be directly sampled for training the optimal vector functions O For a generic measure replacing я in we denote by bp and 9p their associated velocity and denoiser fields.
Observe that the associated backward transport map 4- == & pushes observations from μ toward clean samples from æ, effectively defining local inverse, in the sense that K # words, Φ specifies a self-consistency condition in the space of observation measures; see Fig.
/ In other
However, there a crucial difference in our setup: rather than accessing FT, we instead have acccess to F). The key idea is to turn the self-consistency equation K 3 / = μu into procedure that adjusts θ to push KC # back js. We use SIs connect each of these two distributions to
common 'empirical prior m ( Consistency is then enforced by bringing the two SIs close to each other, leading to natural bi-level fixed-point iteration scheme; see Alg. The outer loop updates (44) to ((++1) by constructing, at each step k, the following SI
f(++1) ar ( (v) + F.F( + 7e=, t E [0, 14 Nal
(8)
This SI is directly sampleable given ( and samples from μl, and we train it using standard SI loss 4 via stochastic gradient descent as the inner loop to obtain (k++1);
minimiaces in 5 with 1((++1) (k++1)
(++1)
step
step
We remark that this bi-level scheme resembles the EM-type algorithm in Rozet et al. (2024); Bai
eet al. (2024). where the clean data is updated in the outer loop and the score function is retrained in the inner loop. However, their method requires an explicit linear forward model and relies on uncontrollable approximations to posterior sampling. In contrast, our data-driven backward transport map avoids these assumptions and enables learning the SI Eq. (8) with only black-box access F.
We easily verify that if the forward model is injective at the distribution level, π is the only admissible fixed point of our iterative scheme (see proof App. A.T similarly as the consistency guarantees in Daras et al. (2024). In Section 4 we will show that with additional assumptions beyond injectivity, one can further establish unconditional convergence guarantees for our scheme.
Proposition (π is the only admissible fixed point). Assume that K. is injective and that the iterative scheme (9) converges to a fixed point O Then øe-
defining Re)
Algorithm 1: Training of Self-Consisten SI
self consistency
θ (0)
// Initialize
12 (for updaring 9)
K
for in 1... K do for i in 1... Ter do
Data Probability Space
Observation Probability Space
Sample I, in (8) with (R)
Figure I: Schematic the SCSI: the fixed point 0* satisfics KrTe- μ,, which in turns implics Te- No samples from are required- the approach only uses corrupted samples from z and the map F.
SGD update of via losses
4
(k) < θ // Update transport map
return (K)
3.2 TRUNCATED INNER- LOOP OPTIMIZATION FOR EFFICIENCY
For the sake of efficiency, in practice, we do not solve the inner problem to convergence at each step k; instead, we initialize the parameters from (k) and update them for Ti, gradient steps. See Alg. T for description of the resulting algorithm and Alg. 2 in App. [ for more details. In the special case Tú 1, the algorithm equivalent to treating J in S as dependent on Ô but applying shop-gradient to Ie when computing the gradient of the corresponding loss function. Nevertheless, we retain the two-loop formulation Alg [ to emphasize the more general form and better match the bi-level scheme introduced earlier.
In contrast to a global inverse, which would require Ka = for all v E P(X), much stronger condition.
In the spirit of Empirical Bayes.
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4 THEORETICAL ANALYSIS
In this seciton we analyze the iterative scheme for SCSI from Sec. 3.1 We establish convergence in KL divergence the ground truth in the SDE setting e >> O, and convergence in Wasserstein in the general setting. We focus on the idealised continuous-time limit, and leave time discretization aspects for future work. In the following, we will often use reference L metric in the space of flows C 1] x X; x)), given by ||6 == E BB [lb(t, =)) where is the law of the oracle SI defined in 7 We define x(6) = (Deas) " the estimate of data distribution at (outer loop) iteration k. While initially introduced the denoiser g. will henceforth mainly use the score s in analysis for convenience, noting that the two are equivalent.
4.1 CONTRACTION IN WASSERSTHIN METRIC
We first give a simple contraction bound in the Wasserstein metric, that applies both to the SDE setting (where > 0) as well as the ODE setting (e = 0). With slight abuse of notation, given a measure से and its associated
KGt, denote by 4 the backward transport map that pushes ji to म. obtained by integrating the SDE/ODE associated with the solution of
We consider the following simple assumption:
Assumption 1 (Lipschitz Stability in W2). There exists R >> O such that for any ã E P()) we have
E 10 $= S (T,
(10)
where the expectation is with respect to the randomness of the reverse transport map.
In words, we are assuming that the map + 4; between (P(x), W2) and (2' u)) vR- Lipschitz at æ. In the ODE setting, the reverse transport is determinsitic diffeomorphism. so the expectation disappears. More generally, the Lipschitz constant = R(e, 7, a, 33 depends on the design of the SI, as well the choice of diffusion coefficient during inference.
Now, since n (k+1) = क (k)
# μ, we have that (v)) 700 (y)), y ~ μ4, a coupling between w and (1++1), and therefore by directly applying the Lipscthiz assumption to (k) we immediately obtain
W ,T, क. (k+1)) < (x) D2 a (y)ll < RW3 (T, (*),
(11)
indicating that have contracting scheme
R < 1. To summarize, we have
Fact 2 (Wasserstein Linear Convergence). Under Assumption L we have W2 ((TT,((())) (6)) < W2 (TT, xt (0)), and therefore we have exponential convergence soon as R < 1.
We first note that the injectivity of the channel is a necessary condition to have R < 1. Indeed. if is such that K Kr therefore (x, < E + - + 1 In general, it is not immediate how to verify the condition R < 1, since it has the flavor of 'reverse transport inequality', but in Appendix B we verify that it holds in the Gaussian AWGN setting under appropriate SNR regimes.
then ($), (v), 2x (v)), for μ,, is valid coupling between π and म. and
4.2 CONTRACTION IN KL DIVERGENCE
Our next result focuses on KL contraction. This contraction relies two structural assumptions, summarised here and further described in Appendix A.2 (i) KL. Lipschitz stability. This is the analogue of Assumption [ where
replace the Wasserstein metric with KL divergence: the
function π + fa := bæ + E is Lipschitz with respect to the KL divergence, with a Lipschitz (o.1) S L=))(= (ii) Condition Number. We restrict the constant of order e: Vn, T. [ Jf=-Jal l
hypothesis space of target distributions E S (see more details in Appendix A.2 and consider the ratio
X sup (L(T|lp)
(12)
PES KL [ITp)
This condition number quantifies the injectivity of the channel, requiring regularisation an appro- priate hypothesis class. The (regularised) inverse problem becomes non-singular whenever X < co.
We are now ready to state the main result of this section, with the proof deferred to Appendix A.3

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Theorem (Contraction in KL). Assume that c > is such that (17 holds for Le and let x be the
condition number of the regularized channel. Let 8(k) = max ((bx() ((0) 1a [|s g(k) (
be the error incurred at iteration k and assume that ((h) S 6 for all k. Then, if L.X < 4, defining
Le -x,, we have
n 1
8²
(I( S 21/ [[[(<=>>>>)>>>)>) + 0
(13)
(1- vT)²
Instrumental to the contraction the ability to relate errors measurement space back to data space precisely what is enabled by the condition number. An interesting interpretation of Theorem 3 is that it provides global convergence guarantees for a seemingly complex non-convex objective function, given in (16), by replacing the ubiquitous gradient descent strategy with a tailored 'Picard-type' iterative scheme. In that sense, our guarantees go beyond the qualitative results of the self-consistency loss in Daras et al. (2024). The upper bound (13) captures the typical tradeoff between approximation and estimation errors: a 'small' function class has a smaller condition number, which improves the contraction rate, but in turn causes the error d to increase (the proof provides explicit error dependencies in ö). That said, a quantitative analysis of this tradeoff in specific function classes is beyond the scope of this work, but an interesting question deserving further attention.
Remark (Stability). Theorem 3 shows that the scheme is stable to estimation errors of the drift and score. However, notice that the error is measured on path distribution (TT)te(o.1) different
[ee[..)), and we rely uniform guarantee across all iterations. In that sense, the quantity ö captures an dusto--distribuutinn error which is more stringent than the rypical Fisher stability bounds in generative diffusion literature (Chen al. 2022 Benton et al. 2023). Finally, we remark that if one has access to an estimate p rather than p. the scheme pays an additional O(KT(())))) additive term, following standard data- mata-processing inequality argument.
from the training distribution (TT (k))
Remark& 5 (Role of Diffusion Coefficient e). A key parameter controlling the strength of Theurem is the diffusson coefficient On one hand, the second term in the RHS cupturing learning errors is of order O(E On the other hand, the diffusion coefficients also affect the contraction rate through the Lipschitz constant L. In any case, the current KL contraction results do not cover the limiting regime of c - 0 where the transport is performed with the probability flow ODE.
Contraction in Fokker-Plank channels The Lipschitz assumption (17) is admittedly difficult to verify in general. However, there is class of channels where one can explicitly control this Lipschitz property, given by Fokker-Plank Channels (Wibisono et al. 2017). These are channels where the forward map F can be expressed as diffusion process itself; in other words, the law of F(x) given x agrees with the law of X1, where X, solves
dX,- = (1,, X.)dt + V2edW,, Xo = x
(14)
for some well-posed drift fü see also (Sobieski al. (025) for SDEs with matrix-valued diffusion coefficients for more general inverse problems. In this case, if the Fokker-Planck representation of the channel is known, we can replace the linear SI in 7 by (14). A prominent example of a Fokker-Plank channel is the AWGN channel, where f = O. Now, observe that in this case the drift of the forward process does not depend on the initial distribution, thus O. We thus obtain:
Corollary (Contraction for FP channels). Under the same hypothesis as in Theorem 3 using the Fokker-Plank interpolant (4] yields a KL exponential contraction with rate 1 X for any O.
EXPERIMENTS
We apply self-consistent SI a variety of forward models across three settings: (i) synthetic low- dimensional datasets, (ii) imaging tasks, and (iii) scientific application in quasar spectral recovery. In settings (ii) and (iii), we report restored samples in comparison with the clean samples as a byproduct of the learned transport map, though our primary goal is to learn generative model of the clean data distribution rather than an inverse solver operating on individual corrupted samples.
In some tasks, a latent variable M associated with F is observed, such as the random mask accompa- nying each observation in the masking task. In such cases, we additionally condition the vector fields on M. This procedure is fully compatible with our framework: it is equivalent to appending M to the observation, redefining the forward map as F(x) = (y. M). and keeping the corresponding channels of the SI constant with value M. More implementation details are provided in App. D
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5.1 Low DIMENSIONAL SAIIHALSYAA SYNTHETIC MODELS
We use this setting to compare between the ODE and SDE formalism. We take the two- moon dataset for the true data distribution and consider the AWGN channel, i.e., F the corruption with Gaussian noise of fixed variance on- In Fig. 2 we show the results for high noise (0,, noise, both formalisms give similar results. However, for the high noise case, ODE restora- tion collapses into artificially thin arms for the two moons while SDE results remain stable. While the SDE formalism can be more ro- bust for highly corrupting forward mod- els in synthetic examples, we find that for high dimensional experiments, it is
1.0) and low noise (o, 0.5) setup. For low intermediate
Data
Observations
ODE restored
SDE restored
de- 1.00 d. -0-55
also more sensitive to hyperparameters like noise schedule and number of transport steps. On the other hand, ODE approach works well for moderate corruptions, is largely robust to choices of hyperparam- eters and is less computationally expensive than SDE. Hence we only present ODE results for other experiments in the maxn-text and defer discussion on SDE results to Appendix E.2
Figure 2: AWGN channel: Comparing ODE and SDE restoration for different noise levels (o,,).
5.2 IMAGING TASKS
Setup We use CIFAR-10 10 dataset (and CelebA dataset for JPEG compression) as the clean data distribution and generate one observation y per image with the forward model. We model the velocity field in our SI with the U-net from Dhariwal & Nichol (202T), but using only channels, resulting in ~32 million parametery" We train all networks for 50,000 iterations, which required ~5 GPU hours on A100. In Appendix E.1 affects final performance.
we provide additional ablations on how network size
Comparison methods for sample restoration We assess restoration quality by evaluating the LPIPS metric (Zhang al. (0188) and compare SI approach with two reference methods. The first is DPS (Chung et al.) (222), a popular and strong inverse solution based on diffusion models. DPS requires a pre-trained diffusion model on the original dataset solve the inverse problem. Hence, we train a large diffusion model with similar U-net architecture but 96 channels instead of 64 (~ 70 million parameters), which achieved an FID of 5.16. For every inverse problem, we also did a grid search to select the best guidance strength hyperparameter we found the good values to be very different from the recommendations in the original paper. Hence, this baseline has four advantages over our approach: i) most importantly, it uses the clean data to train generative model, ii) it requires gradients the forward map unlike our black-box only access, iii) our implementation uses 2x larger neural network, and iv) benefits from task-specifie hyperparamter search.
The second reference is an oracle restoration model, obtained by training an SI model with paired dlean-corrupted data as in Liu et al. (2023a). It is not a competing baseline, but upper bound on the performance achievable by interpolant- based methods with comparable architecture and compute.
i) Random masking Following Daras et al. (2023) Rozet al. (2024), this map generates an observation by masking each pixel of an image x independently with probability p. and adding isotropic Gaussian noise (o,,). As in their setting, we assume access the mask M for each y and use it to condition our SI. We also pre-process the observations by adding independent standard Gaussian noise to masked pixels as it improves the final results. We show the restored images Fig. Bal and LPIPS metric for masking probability p = 0.5 and two noise levels (o,,) in Table [ Our restored samples are comparable to DPS in the low-noise case and outperform they remain reasonably close to the oracle SI model across most settings. Additional quantitative metries are provided in Appendix
higher-noise regimes, and
To compare with inverse generative models from prior work, we use the trained SI to restore the observations; that is, we transport all observations y to the data space via (y)y), and use these samples to train generative diffusion model. We use the same architecture above, but with 96 channels.
SSpecifically, the implementation here
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Table I: LPIPS for restoration quality of our SCSI, DPS, and the SI-Oracle. DPS requires clean pre-training data and forward-map gradients for restoration, and the oracle is trained with paired clean-corrupted data.
Table 2: FIDs for random masking with dif- ferent masking probabilities p. To account for differing architectures, 'baseline' FID for our model on the clean CIFAR-I0 data.
Forward Model
SCSI
DPS
SI-Oracle
Method
FID !
0.20
11.70
Random Mask (o,, 10
0.0051
0.0049
0.0044
Ambient Diffusion
0.40
18.85
Random Mask (a,, 0..1)
0.0064
0.0072
0.0055
0.25
5.88
Gaussian Blur (a,, -0.1)
0.005
0.009
0.0051
EM Posterior
0.50
6.76
Gaussian Blur (a,, 0..5)
0.015
0.025
0.0011
0.25
5.38
SCSI (generated)
Motion Blur (a. = 10"")
0.0069
0.0026
0.003
0.50
6.74
Motion Blur (o.,. 0.1) 0.011
0.012
0.003
Baseline
0.00
5.16
Table shows the FID scores for observations with two different masking probabilities and negligible added Gaussian noise (E 10 6). SCSI vastly outperforms Ambient Diffusion (Daras et al., (203).. is comparable with EM Posterior method (Rozet et al. 2024), but more computationally efficient: required combined 86 GPU hours (54 and 32 GPU hours to train SI and diffusion model respectively) compared to their 512 GPU hours.
Original Image
Original Image
Original
Random
Gaussion
Mcdon
Okur
abir
Restarad 10uns
Restord (Ouns)
Restored [Quns)
DPS
DPS
DPS
(a) F: Random mask
(b) F: Gaussian blur + noise (an 1, 0.1).
(c) F: Motion blur + noise (kernel size on 0.1)
(p 0.5, 10-9).
Figure 3: Restored samples for different forward maps from our interpolants and DPS.
ii) Gaussian blurring with noise The forward map is blurring with a Gaussian kernel with OR = 1 and adding noise. Here add Gaussian noises with different levels (o,, 0.10, 0.25) and show the results for Poisson noise case in Appendix E.6 This demonstrates that, unlike previous works, e.g., Daras et al. (2023), approach can handle non-negligible and oon-Gaussian noise.
iii) Motion blurring The previous two examples involve linear forward maps. We now consider a nonlinear one: motion blur. Fig. 3c shows restored samples for observations with 5-pixel motion kernel and small Gaussian noise E 10-6) The blur direction is randomly assigned per image and assumed known for conditioning the SI. While Daras al. (2023): Rozet et (2024) are limited to linear operators, SCSI handles nonlinear maps with only black-box access.
iv) JPEG compression This another common non-linear corruption operator with real-world applications. The forward map is JPEG compression with quality factor (q) and added Gaussian noise (o,, = 0.01). For training, we corrupt every image randomly with a different factor q ~ (( 1, implies compression) and assume this latent parameter y is known for each observation to condition the SI. Fig. 4 shows the restored image with our trained SI for different strengths of compression. The restored image gets closer to the original with lower compressions, and we restore a physically plausible image even for = 0.1. In Appendix E.5.3 we show that our trained SI is stable extrapolations of q outside the training regime.
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quality: 10
Qualityy 30
Quality :s0
Quality 70
Quality 90
Original
Figure 4: JPEG + noise: results for different compression level (Top: Corrupted; Bottom: Restored).
5.3 QUASAR SPECTRA
Quasars are observed through telescopes and hence the observed spectra differ from the underlying true spectra due to noise, finite spectral resolution and finite observation time. Recovering the true spectra from these observations of interest both study individual objects and understand the evolution of quasars as whole. For the true data distribution, we take the quasar spectra from Sloan Digital Sky Survey data release (Lyke et al., 2020). We isolate 30,000 quasars in redshift. [2.75, 3.25] and consider À E [400mm, 650mm resulting in spectra of length D 1024. We approximate the forward model with a combination smoothing, and added Gaussian noise. Unlike imaging examples, for every observation, we randomly vary the size of the smoothing kernel within 5% and add noise with a different magnitude depending on randomly chosen SNR. We assume do not have access these latent parameters for any observation. In Fig. 5 we show the restored spectra for observations in the two extreme regimes that different telescopes operate in: observations with high spectral resolution and low SNR (high noise), and ii) those with low spectral resolution and high SNR. For comparison, we also show Wiener filter (WF) restoration baseline. SCSI best able recover important features like peak heights and locations which are used to determine the distance and metallic composition of quasars.
flux calibration error (offset), a Gaussian
5.0
Ob. D33 242 0.92
10
7.5
Flux
5.0
52. 0.501
0355)
4000 5000 3200
5000
4000
3000
1500
5000
86:00
Figure 5: Restored quasar spectra for different scenarios. Legends show MSE over 1000 samples. Left: high spectral resolution, low SNR; A log À 5000 + 5%, SNR E [0.5, 2]. Right: low spectral resolution, high SNR; A log À 50 + 5%, SNR E [30, 50].
CONCLUSION
We presented a self-consistent SI framework for reconstructing the underlying data distribution using only corrupted observations and black-box forward model. The proposed bi-level iterative scheme computationally practical and enjoys provable convergence under suitable assumptions. Compared existing approaches, the proposed SCSI accommodates much broader class of nonlinear and non-Gaussian forward models. Experimentally, we demonstrated its effeectiveness across a range of inverse problems, achieving competitive performance even against methods that rely on additional access to the forward model (e.g., Ambient Diffusion) or even clean data (e.g., DPS). Interestingly. our theoretical analysis as well as our experiments point towards an advantage of the ODE-based transport over the SDE counterpart.
Looking ahead, with recent extensions of stochastic interpolants to discrete domains (Gat al. 2024; Kim et al.) 2025). our approach can also be adapted to corrupted discrete data. In addition, an important aspect to be further investigated is the comparison between marginal and posterior transport, as in the EM algorithm and its diffusion-based implementations (Rozet et al. 2024 Hosseintabar
et al. 2025)- both captured in our framework by adapting the SI; see Appendix C22 While the EM
scheme is convergent under weaker assumptions than the marginal transport
analyzed in Section
the analysis in Appendix demonstrasss (on simple settings such as Gaussian AWGN) an inherent advantage of marginal transport in convergence rates, particularly for the probability flow ODE.
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REFERENCES
O Deniz Akyildiz, Mark Girolami, Andrew M Stuart, and Arnaud Vadeboncoeur. Efficient prior calibration from indirect data. SIAM Journal on Scientific Computing, 47(4):C952-C955 2025.
Michael Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants. In ICLR 2023 Conference, 2023.
Michael Albergo, Nicholas Boffi, and Eric Vanden-Eijnden, Stochastic interpolants: unifying framework for flows and diffusions. arXiv preprint arXiiv23033088797 2023a.
Michael Albergo, Mark Goldstein, Nicholas M Boffi, Rajesh Ranganath, and Eric Vanden-Eijnden Stochastic interpolants with data-dependent couplings. arXiv preprint arXiv:2310.03725, 2023b.
Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their
Applications, 12(3):313336, 1982.
Weimin Bai, Yifei Wang, Wenzheng Chen, and He Sun. An expecation-maxxiizztion algorithm for training clean diffusion models from corrupted observations. Advances Neural Information Processing Systems, 17:19447-19471, 2024.
Joe Benton, Valentin De Bortoli, Arnaud Doucet, and George Deligiannidis. Nearly d-linear conver- gence bounds for diffusion models via stochastic localization. arXiv preprint arXiv:2308.03668 2023.
Nicholas M Boff and Eric Vanden Saiden-Eijnden. Probability flow solution of the fokker fokker-planck equation. Machine Learning: Science and Technology, 4(3):035012, 2023.
Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru R Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209.1121: 2022.
Tianyu Chen, Yasi Zhang, Zhendong Wang. Ying Nian Wu, Oscar Leong, and Mingyuan Zhou. Denoising score distillation: From noisy diffusion pretraining to one-step high-quality generation. arXiv preprint arXiv:2503.072 077.7777 2025.
Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S Albergo, Nicholas M Boffi, and Eric Vanden- Fijnden. Probabilistic forecasting with stochastic interpolants and Föllmer processes. arXiv preprint arXiv:2403.13724 2024.
Hyungjin Chung, Jeongsol Kim, Michael T Mccann, Mare L. Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. arXiv preprint arXiv:2209. 14687, 2022.
Giannis Daras, Kulin Shah, Yuval Dagan, Aravind Gollakota, Alex Dimakis, and Adam Klivans. Am- bient diffusion: Learning clean distributions from corrupted data. Advances in Neural Information Processing Systems, 36:288-311 313, 2023.
Giannis Daras, Alex Dimakis, and Constantinos Costis Daskalakis. Consistent diffusion meets Tweedie: Training exact ambient diffusion models with noisy data. In Forty-first International Conference on Machine Learning, 2024.
Giannis Daras, Adrian Rodriguez-Munoz, Adam Klivans, Antonio Torralba, and Constantinos Daskalakis. Ambient diffusion omni: Training good models with bad data. arXiv preprint 1XXiv2556610033 2025.
Maximilian Dax, Jonas Wildberger, Simon Buchholz, Stephen R Green, Jakob H Macke, and Bernhard Schölkopf. Flow matching for scalable simulation-basett inference. arXiv preprint vrXiv:2305.17161, 2023.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8794 2021.
11

---

Published as a conference paper at ICLR 2026
Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky TQ Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching. Advances in Neural Information Processing Systems, 37: 133345-133385, 2024.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840- -118800-8551, 2020.
Danial Hosseintabar, Fan Chen, Giannis Daras, Antonio Torralba, and Constantinos Daskalakis. Diffem: Learning from corrupted data with diffusion models via expectation maximization. arXiv preprint arXiv:2510.1200 2025.
Bahjat Kawar, Noam Elata, Tomer Michaeli, and Michael Elad. GSURE- diffusion model training with corrupted data. Transactions on Machine Learning Research, 2024. ISSN 2835-8856.
Jaeyeon Kim, Lee Cheuk- Kit, Carles Domingo-Enricn Enrichh Yilun Du, Sham Kakade, Timothy Ngotiaoco, Sitan Chen, and Michael Albergo. Any-order flexible length masked diffusion. arXiv preprint arXiv:2509.0102. 2025.
Frederik Kunstner, Raunak Kumar, and Mark Schmidt. Homcomorphic- Homcomorphi--nvaariaee invariineeee of em: Non- asymptotic convergence in kl divergence for exponential families via mirror descent. In Interna- tional Conference on Artificial Intelligence and Statistics, pp. 3295-3303, PMLR, 2021.
Qin Li, Maria Oprea, Li Wang, and Yunan Yang. Stochastic inverse problem: stability, regularization and wasserstein gradient flow. arXiv preprint arXiv:2410.00229, 2024.
Qin Li, Maria Oprea, Li Wang, and Yunan Yang. Inverse problems over probability measure space. arXiv preprint arXiv:2504. 18999, 2025.
Yaron Lipman, Ricky TQ Chen, Heli Ben- Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations, 2022.
Guan-Horng Liu, Arash Vahdat, De-An Huang, Evangelos Theodorou, Weili Nie, and Anima Anandkumar. 12SB: image-to-mmage Schrödinger bridge. In International Conference on Machine Learning, pp. 22042-22062. PMIRR 2023a.
Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In The Eleventh International Conference Learning Representations, 2023b.
Haoye Lu, Qifan Wu, and Yaoliang Yu. Stochastic diffusion models with finite noisy datasets. In Forty-second International Conference on Machine Learning, 2025.
deconvolution: Training
Brad W. Lyke, Alexandra N. Higley, J. N. McLane, Danielle P. Schurhammer, Adam D. Myers, Ashley J. Ross, Kyle Dawson, Solène Chabanier, Paul Martini, Nicolás G. Busca, Hélion du Mas des Bourboux, Mara Salvato, Alina Streblyanska, Pauline Zarrouk, Etienne Burtin, Scott F. Anderson, Julian Bautista, Dmitry Bizyaev, W. N. Brandt, Jonathan Brinkmann, Joel R. Brownstein, Johan Comparat, Paul Green, Axel de la Macorra, Andrea Muñoz Gutiérrez, Jiamin Hou, Jeffrey A. Delabrouille, Isabelle Pâris. Will J. Percival, Patrick Petitjean. James Rich, Graziano Rossi, Donald P. Schneider, Alexander Smith, M. Vivek, and Benjamin Alan Weaver. The Sloan Digital Sky Survey Quasar Catalog: Sixteenth Data Release. Astrophysical Journal Supplement Series, 250(1):8, 2020.
Newman, Nathalie Palanque-Delabrouille
Giacomo Meanti, Thomas Ryckeboer, Michael Arbel, and Julien Mairal. Unsupervised imaging inverse problems with diffusion distribution matching. arXiv preprint arXiv:2506. 14605, 2025.
Herbert Robbins. An empirical Bayes approach to statistics. In Breakthroughs in Statistics:
Foundations and basic theory, pp. 388-394. Springer, 1992.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF confer- ence on computer vision and pattern recognition, pp. 10684-10695, 2022.
12

---

Published as a conference paper at ICLR 2026
François Rozet, Gérôme Andry, François Lanusse, and Gilles Louppe. Learning diffusion priors from observations by expectation maximization. Advances in Neural Information Processing Systems, 37:87647-87682. 2024.
Bartlomiej Sobieski, Matthew Tivnan, Yuang Wang, Siyeop Yoon, Pengfei Jin, Dufan Wu, Quanzheng Li, and Przemyslaw Biecek. System-embedded diffusion bridge models. arXiv preprint TrXii:25006.3726. 2025.
Yang Song, Jascha Sohl-Die Dick-Diicssten,, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
Albert Tarantola. Inverse problem theory and methods for model parameter estimation. SIAM, 2005.
Joseph L Watson, David Juergens, Nathaniel R Bennett, Brian L Trippe, Jason Yim, Helen E Eisenach, Woody Ahern, Andrew Borst, Robert Ragotte, Lukas Milles, et al. De novo design of protein structure and function with ridiffusion. Nature, 620(7976); 1089-1100, 2023.
Andre Wibisono, Varun Jog, and Po-Ling Loh. Information and estimation in fokker-planck channels. In 2017 IEEE International Symposium on Information Theory (ISIT), pp. 2673-2677. IEFE, 2017.
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 computer vision and pattern recognition, pp. 586-595, 2018.
Yasi Zhang, Tianyu Chen, Zhendong Wang, Ying Nian Wu, Mingyuan Zhou, and Oscar Leong, Restoration score distillation: From corrupted diffusion pretraining one-step high-quality generation. arXiv preprint 2rXiv:2505.13377 2025.
Linqi Zhou, Aaron Lou, Samar Khanna, and Stefano Ermon. Denoising diffusion bridge models. In The Twelfth International Conference on Learning Representations, 2024.
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A PROOFS
For notational simplicity, we use K to denote K,=, the forward integral operator that pushes π to and we adopt this shorthand throughout this section without risk of ambiguity.
A.1 PROOF OF PROPOSITION
Proof. Recall our iterative scheme as
minimiaers in 4 with 1- (2+1)
1 (B+1)
(k+11)
(15)
step
'M' step
If the above iteration converges to a fixed point O', and the channel is injective at the level of distributions, i.e., KT K implies म म, then the corresponding transport map Pe. transports corrupted samples from into clean samples from To see this, consider Te = (Øe- / We prove only the SDE case, as the ODE case corresponds to the special case when e 0. By definition of Te- and the property of time-reversal SDE, O* transports Te- to under the forward SDE (Anderson) 1982, Song et al. 2021)
dx.- = b(t, X; dt - (t x. jtt + V2ndW.
On the other hand, since O" is the optimal solution trained from the SI between Te. and KTge, . the above forward SDE also transport samples from xe. Kree (Albergo et al. 2023a). As result must have Ke = 14, which means that Tg- = æ thanks to injectivity.
Loss function perspective Our iterative scheme can be viewed as a specific procedure to find fixed point O* satisfying self-consisteney. Alternatively, such fixed point can be characterized as a minimizer of loss function that penalizes discrepancies between two transport descriptions. Given a generic pair θ = {b, g) of drift and denoiser models, the corresponding backward transport defines a distribution Te == (De) # nl, and then the objectives associated with the SI between hee and KTo defines minimizers bise and Ine. We seek to align them with the original pair via the loss
((b,g) |b bxgn 1|2 + ||g 9m
(16)
where || here denotes an 7,2 with respect to an arbitrary base measure. The main challenge when analyzing gradient-based optimization of this loss the highly non-linear dependencies arising from the transport map.
A.2 ASSUMPTIONS FOR THE KL CONTRACTION
In this subsection we introduce our assumptions required to establish linear convergence rates for the KI. divergence.
KL Lipschitz stability Recall our definition of by, sn, the velocity and score associated with the SI in (7) To control the contraction along iterations, we will assume that the function + fx == by + CS., that maps a given data model π the drift of Fokker-Plank equation transporting x to KT. is Lipschitz with respect to the KL divergence, with Lipschitz constant of order e:
Assumption (Lipsethiz Stability in KL).
VT, ", Ilf ftlls S. LeKL(TT] (
(17)
In words, a SI builds a diffusion bridge between v and Kr, and Le measures the sensitivity of its drift initial conditions. To avoid confusion we shall write L L. to emphasize that this Lipsethiz constant does depend on the diffusion coefficient. Note that Il.fa f= < 2| b 2<21|sz - s 112 so the assumption balances the Lipschitz smoothness of the drift (of order (1)) and the score (of order ((2)) via the diffusion coefficient c. For general channels, this assumption indirectly imposes a macroscopic diffusion coefficient e = (1). Indeed, in the limit of infinitesimally small diffusion + 0 (capturing the ODE inference setting), this assumption imposes π + by to be constant (ie, by is independent of тT) and therefore that the channel F is deterministic diffeomorphism T X X, whose ODE representation is given by b. Notice that the Lipschitz constant depends on the SI design, and might be reduced by going beyond linear interpolants and more effectively 'preconditioning' to K', which leave for future work.
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Condition number An important aspect of the problem is that there are two distinct notions of enror, whether it is measured on the 'data' side, i.e., (T((=|[=()) or on the 'observation" side, i.e., ( = (L((TT [I(CT]][***)) Since the learner only has access to data from μl, a necessary condition to guarantee that we can recover the original data distribution is injectivity, i.e., that KL(KTr] 0 implies = f. However, this is not sufficient to provide quantitative estimate of KL(+](a) in terms
In other words, the inverse problem K - y is generally singular in ((X), even for the simplest channels, due to the ciiinit-dimensiona. nature of the domain.
To mitigate this issue, we need to regularize this inverse problem by restricting (or penalizing) the domain of possible velocities and scores arising from the SI objectives (3) so that the resulting
constrained optimization returns b,,,,, 8x(4) E Bx, where Xx is indexed by a complexity measure e.g., neural networks with O(x-1) parameters. In turn, these regularised objectives inject regularity in x x(*), i.e., for all l we have (*()) E Sx, the class of terminal densities obtained by running a Fokker-Plaat equation with drifts in Bx. We can now consider the condition number of K. around ..
KL(T|Ip)
X sup PES, KL(KT
(18)
Note that by the sata-processing inequality, we always have X 2 The (regularised) inverse problem becomes non-singular whenever < co. The purpose of regularisation, in this abstract context, is to restrict the range Sx as to make X small, while maintaining a small approximation error. Observe that, if p E Sx, by Girsanov's theorem we have (L(T]lp) S s <= 6 + ells* - 5pll2 oo, which shows that X is well-defined.
particularly simple form of regularisation is to consider a continuous parametric model {bw, s.) where D is in compact domain, which encompasses most practical setups. Combined with the injectivity of the channel, this allows us to have X < co. For technical reasons, we consider the following misspecified setting. Note that, under these general conditions, unsurprisingly, unable to quantify the condition number.
Proposition 2 (Finite condition number for compact hypothesis class), Assume that K. is injective, that D a compact parameter space, with continuous parametrization of the drift and score models, and that nt cannot be exactly represented by the model. Then X < co.
Proof. Let F + PX) be the funetion that maps model (he, se} to F(O) = Ti, where (æt)) is the marginal law of (Xi),, which solves the SDE
dX,- (t, + (ts dt + 2edW,,
(19)
Xo
(20)
Define G(@)= == [L(XT]|/F(()) We claim that G is positive for all @ E 'D and that G is lower semi-continuous. Indeed, since are assuming misspecified model, we have KI(+]|F(O)) 0 for all O E D, which implies G(@)= > O for all θ E 'D thanks to the injectivity of K.
Moreover, the mapping v + (L (v) is lower semi-continuous in the weak topology. This follows from the Donsker-Varade Varadhan variational representation of the KI. divergence:
KL(H||v> = sup { {f. log(ee .)))
fEC,
The map v ト - log(e!, ,V) is weakly continuous for all J E Cs, and the supremum of continuous functions is lower semicontinuous. Now, consider any sequence (0,), )n such that ||O, (0 OI1 - 0 as + oo. By Girsanov's 's theorem, observe that
KL(F(O))IP(E.)) S S "'Ibe - be., 1|? + el|se - se., 112
(21)
which shows that (L,) + 0 as n + oo thanks to the continuity of the mappings θ + (e. By Pinsker's incquality, we also have that (F(O) - F(O,)llTV + 0, which shows that F(O,)) converges weakly to F(O), and therefore
lim G(O.) 2 G( +
(22)
showing that G is LSC as claimed.
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Now, observe that
KL - - -01 =(0)).
and
KL(T|IF(O))
(l.(I]/FF()) G( - r()).
(23)
The function r is ratio between a continuous function and a positive, lower semicontinuous function. It follows that is upper semi-continuous, and therefore
X S sup r( <0,,
since USC functions attain maximum over compact sets.
A.3 PROOF OF THEOREM 3
Proof. The strategy of the proof establish comparison between (L(T|| and KL(T]| (L](|(]((+)) by exploiting the relationship between the diffusion bridges that relate them.
For that purpose, let Iī be the oracle SI, given by
F = a, + (x) + 702, X ~~T x².
(24)
Let ne be the law of 4, It solves the Fokker-Planck equation
0 = V ((-b" - es TT)) + eATT.
(25)
No ग M1 KTT = μ
where
(t, = E| 1ī = 2]
(26)
s* (t. == - El% | = x
well the reverse Bokker-Planck' equation
θ, = V ((-b" + Es (Te) E
(27)
1 Km
To
Consider also the SI iteration &o of our algorithm. Given ग. (8), we consider the interpolant
1 = axX + 3xF(()) + 722, X (A)
(28)
its associated (exact) drift and scores (t, = ( |1(b5 Ø|
(29)
( (1,x):= := : - ((( = x=]
as well as the estimated drifts and scores, that we recall are given by
(ik) = arg min gb
K00, (6) xR(6) (k)) arg arg min
(8) + AR(SS
(30)
b
They define respectively a forward Fokker-Planck equation 0. = V ((-b(4) - es (x)))) + edTT.
(31)
To (4) ni (k)(k) (b)
and a reverse Fokker-Plane Peaker-Planck equation (notice its initial condition is 1a):
0.. = V ((- + ( eAm,
(32)
T1 4 To -= (k+1))
also useful to define f == b + es be total drift of the forward (i.e., from data to measure- ments) diffusion; with the corresponding oracle f*, iiterate f(k) and estimated f(k) versions defined
analogously. From (55). (27). (31) and 32) we immediately verify that the reverse drift becomes
-f+ 2cs.
The following lemma relates the rate of KL along two SDEs. We reproduce the proof later for completeness, but is a known result, e.g., (Boffi & Vanden-Eijnden. 2023 Proposition 1) or (Albergo et al. 2023a Lemma 2.22):
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Lemma 1 (KL divergence along two diffusion processes). Let dX,, b(t, Xi)dt 2odW, and dY, a(t, Yj)dt + 2odW,, be two diffisiins, and pe, V denote the marginal law of X. and Y respectively. Then
d (L(ll) = -o1(pellps) + Bue (b V Vlog u4
V Vlogu)
(33)
dt where I(ullp) E V log μ V 1og is the Fisher divergence.
In the particular setting where b a, one obtains a de Bruijn identity: Lemma 2 (de Bruijn Identity).
KL [Te|| (x)) -ollollln |mm (k)
(34)
dt
Besides a control of the marginal KL, we will also use Girsanov's theorem to obtain control of the KL divergence between path measures of (Xt), and (Yi)::
Lemma 3 (Girsanov Theorem). Let dx, = b(t, Xt)dt + 2odW, and dY, = a(t, Yi)dt + 2odWe be two diffusions, and let brr and Mo,r be the path measures of X, and Ys, respectively. Assume the Novikov integrability condition. Then
(L( (L()oo)be) + E [ [la(t,z)=b(taa)) b(t,,))
1
(35)
4o
By the data processing inequality, direct consequence of Lemma 3 is Corollary 2.
Bygar L' [
(T(TT((UT) & (T((polPo)) + 4o
b((,
(36)
We first apply Corollary from = 1 to 1 0 to the two reverse Fokker-Planch equations (27) and 32). respectively sending back to æ,, and the current model sending back to (k++1), Since they share the same initial condition, we have
1
/ E.. (If"(t))) (t, - (t, - 3 (t,
(L(T||
(37)
4e
We now apply Lemma to the pair of forward Bekkerr-lancce equations 25) and 31
to obtain
(T)) (T.((ll((())) + L. V oe Vlog ( 11)
(38)
B,T / (5* ((()), log - log m. (2)
= (KI((ull((t)) + L/¹ || st (k) 11a
-E, ( (5" ((),,
06 (k) (
From 37 we have
(L(T||=(*)))
<
1 - f00 + =||ss 3 (f+ ((h) -
((()) )x
(39)
4e
Assuming a drift and score approximation error uniformly bounded by 6, plugging
38
we have
(T(x]|=((+1)) < ElJs - f()) + el|s*
1
(( 2 RBS - (() 5* - ((4)
(40)
4e
8²
+
2e 2e
11 - ((0) || + + e)|(8* - - -)115==
(41)
2c
<
(L(T (a))
(42)
- - + Cz(() ( - -600 + Cs(c)]ls* - -s(11
(43)
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Now, using the condition number and SI Lipschitz assumptions, denoting 17 1 + 4 -x= obtain
KL |L[T|| - 1)LL + C(()))))))+))))))))))))))))))))) + 28C2(e) - 11 + -1s - 1) (44)
with (c) 0 + ) and C2(c) = 0 +(1+
Observe that from 38 and using
we have
f e||s* 8 12 < (1
| (+)) + 15- - s(17))
(45)
s (1 - x )KL क7 (+)) + LEKI
- s(A0
(46)
S (T | (8)) + 2 TT T [|(+) - -(0
(47)
which implies ||** s 4e" -1 1/2 12 | (.) and therefore
| - /(4) | + ||s* - s |1 - Ca(c)) (TKL(T||T())
(48)
with Ca(e) = 0 + 1/2)). Thus, by redefining 8 = C.O for C. = =O(E (/2) we obtain
KL(T |L(T||T((+))) (1+ (KL((T| + 8 28 (KL(T]|T((1)
(49)
2
= (V 8
(50)
Setting ag = (+) we arrive at the linear recurrence
0k+1 S VToa +8.
(51)
Solving this linear recurrence yields
OAk 11 00 +
200
(52)
1 vn
hence
2
8
([(T||=(1)) S
11 1/2000 +
(53)
1 VT
28²
+ (1
(54)
52
= -2/ + +0(c-))
(55)
(1 TT)
as claimed.
Proof of Lemma 7 Let K, (L(mellu)) = J pc(x)) log
[x(x)) p((z) dx. By definition, the laws pe and
Vt solve the Fokker-Planck equations
Oepi V
(56)
dz4 = V A...v--00 (1-a log v))
(57)
We compute =
(x(x) (,p /
fLog log
[r((x) (x)r
K,
(58)
dt
(z)
(((z)
/ 1t V. (e--(((0 ((-a + o Voooooloo log logg))u))dz / log
14t
V ((-b ((-b+oVlogm)u))l) (59)
Vi
vi
/ v
/ V log
-a + r V log 2/
log utdr (60)
14
/
F
V Vlog
-a b ov log
1t
(61)
1/2
1i
-o0((e]lkz)) + En, (b - V log ut
V log vt)
(62)
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B CASE STUDY: AWGN WITH GAUSSIAN PRIOR
We illustrate our algorithm and the theoretical guarantees using arguably the simplest setting of
Gaussian prior (6, 2) and the AWGN channel. Since the only relevant quantity is the signal-to-noise ratio, we can assume without loss of generality that F(x) = r w, with w ~ (), (i.e., we renormalize the signal so that the noise variance is 1).
Linear Convergence and ODE advantage We suppose that Σ > 0, and we consider a Gaussian initialization To- (bo; Eo) with Eg > 0, and use the simple linear interpolant
I, = (1 Vtx + Viz(()) x + vtw.
In the AWGN channel, the transport equation between and Ki is straightforward, given by the heat equation 0, Ar with To
that is, for any Gaussian measure ă = [b, Σ), the law #, of
4 is explicitly given by मt = -N E + tz).
For any diffusion coefficient 2 0, the reverse transport equation is then
1
0.p: v running backwards from the terminal condition P = (6, 5 + 1). The corresponding reverse SDF
7 Seppt.
(63)
2
1 1+ Vooght - (Y;) V
dY; 2
VedW,,
(64)
where now = 0 corresponds to measurements and f to data. For Gaussian measure म, the score V log 7. is an affine field given by
V log (z) -(É ( b).
(65)
As result, the solution of the SDE (64) is an Ornstein-Ulhenben process that can be explicitly integrated. Denoting by
(t, s) ( + (1 ()) (1 -1+6)/2
(66)
the solution map from time to time t, we have
b) + Vc L. $11, s)d,,
Y1 = b +
(67)
and therefore that the Gaussian structure preserved throughout the iterations. Recalling that Tk+1 = Law(Yi), with bg. E = Σx above, given current parameters by. Ex, we verify from 66 that the parameters become
b by Bb bk) EA+1 E. B&(E BR with By = (Ex (2 + 111(400))/
(68)
We now establish that the means by and covariances Σ₁ converge respectively to b and Σ at linear rate.
Proposition 3 (Gaussian AWGN Linear Convergence). Let n min (Amin (E), Amin (En)) and
assume > 0. Then
m Σ+ $ (1 - - Coll |b b < (1 - 17 ( + 11) - bo 1
7)
(69)
(70)
Proof. Denote by Fkk b by and Rs Σ Σt. We have bb+1
B& Db and R&+1 Rs- Bg Ra.B Br. We verify that for any symmetric matrix X (not necessarily psd) and symmetre
matrix B such that < B I we have ||X BX B 5 (1 A Therefore
|| |Re41|| 11 S (1 - Amin (Bx)² |R
(71)
"Indeed, split X into X = Xp+ + X.. where X, > 0 and X.. 2 0. Then we verify that - X, =
X, BX 3 (1 Amin Amin(B)²) (B))) and > Xx = Nn BX B 2 (1 - Amin (B)))x,. It follows that
X BXB- X, X, with (1 Appia (B)2
)onia (X) 5 Amii (X,) Amin BX S Amax (X
BXB B) S >max (X,) 5 Amin )m))) (X).
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We claim that, for all k
min (Exs 2 min (Amin (2), Amin (Eo)) - 17
(72)
Indeed, this follows from simple induction argument: since BR and En commute, we have that As a result, we immediately obtain
Ex+1 = (I- ( B + BB = (2n)((2x)) - BB) + Amin (2
(73)
(74)
t min(Amin (Ei)), Amin (2))1
(75)
Amin (Bx) (Bx)
( Auiiu (Ex)
(1+r)/2
7
(1+6)/2
(76)
1 1 (mi)
1 1+11
establishing (69)
We thus verify that iterative scheme converges at linear rate, with factor 1 - (1 ()) 1 The setting that optimizes this upper bound is given by O, which corresponds to the ODE setting. Interestingly, this relative advantage of the ODE is also observed in our numerical experiments. In particular, in the low SNR regime where 1) < 1, the convergence rate is of order (1- -))* As may be expected, in the high SNR regime the convergence rate is much faster, since the effect of the channel is relatively minor. Additionally, and since in the fiite-dimensional representation of Gaussian measures all metrics are equivalent, from 69 one obtains linear convergence both in Wasserstein and in KL divergence.
Comparison with EM Another interesting aspect of the Gaussian AWGN example is that we can compare the performance of our scheme to the actual EM algorithm Rozet al. (2024). As in our scheme, the Gaussian structure is preserved throughout the iterations, and moreover it corresponds to mirror descent algorithm Kunstner et al. (2021).
Focusing on the centered Gaussian setting for simplicity, the EM algorithm updates a current Gaussian prior Te with covariance Σg by first considering the posterior
(x(3)) No (z)
(77)
Yo 2u.E++1 (v)
where 78.E denotes the Gaussian density with mean and covariance Σ. This posterior is then used update the prior via the 'M' step, defining the mixture
7x.,( dr.
d (I) = Ey = 7, (x) Byngo.Ee
(78)
%o Sy (1)
We can explicitly compute the RHS, since it Gaussian integral, resulting in Th+1 Yo. Ex4r with precision matrix 2-1 - +1 (( + 1)- (Ex + 1) 1)- As result, the covariances
Σx evolve under EM according to
Σx+1 Σg + Mr ( Σx) with Mj Ex (Ex 11
(79)
Notice that this EM update identical to the update (68) for the choice c 1. Crucially, the asymptotic rate of this choice is suboptimal in our class, as it is dominated by the choices < 1; in particular, the low SNR regime 7) & 1, EM has a rate of (1 - --)²²= of the ODE (c 0), indicating that the ODE marginal transport provides some form of acceleration. Beyond the asymptotic rates, the EM algorithm enjoys a non-asymptotic rate of (11kk) thanks to its mirror-descend structure. By analogy with the acceleration phenomenon in convex optimization, one could expect an accelerated non-as -onmstyptooicc rate (1/k²² for the ODE scheme: we verify that this indeed the case empirically; Figure o but this non-asymptotic analysis is out of the present scope.
contrast with the (1 17)
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10
000
10
10
10
10
Figure 6: Convergence of | Ex 1l using either ODE (c (e= 0, in red) or the EM algorithm (e = 1, green curve), for Σ drawn from the Wishart distribution. In dashed we plot the rates 1/ke and 1/k respectively.
Comparteon Metrica Tand
Smilden Toe
10g
110
417
700
412
Figure 7: Scatterplot of (A, B) (y-axis) versus T(A; B) (x-axis), for pairs of independent Wishart matrices A and B. Red line corresponds to R 1. Left: Low SNR, Center: Moderate SNR, Right: High SNR. We observe that R with high probability.
Verifying convergence assumptions Finally, the Gaussian AWGN example allows us verify our assumptions from Section Let us focus on the ODE setting, since it corresponds to the fastest regime. For a psd matrix A, we denote by Ta = (A(AA + I) 1)-1)/ 1)/2 its associated solution map from (66). Recall that the Lipschitz Wasserstein a compares, for pair of distributions (here both assumed to be Gaussian) (0, A) and (00 B), the transport cost
E (o>N -
TB( (T TB)(A T)(T
to the squared-Wassrestenn Wassrestein distance
(V) B)) Tr - -2
First, we notice that in the limit of the SNR going to zero, i.e., ||A|], || + O, we have T(A; B) Tr(A) () + Tr(rr))B200))) (000-2TT((A22 2T222A/22B1/22) O[] || (1]B))++1111 while W.- = =Tr(A)+T((B) 2 41/2 we thus verify
From the variational characterization |||| that R cannot be uniformly upper bounded by in the PSD cone. However, for macroscopic SNR, we simulate pairs (A, B) from the Wishart ensemble and verify numerically that TTA; B) < W3 B) with overwhelming probability, even for moderately small SNRs; see Figure
x (Y, X) -
Similarly, the condition number (88 needed in the KL. guarantees can be explicitly bounded this setting. Indeed, for ?>: we have
KL(T|1Ts) KL(KTTll7444
X sup 7i
S (1 + Amin (E)- -
(80)
By plugging the rate (1 - x 11* from Corollary corresponding to Fokker-Plank channel with we thus recover the previous rate.
note that in the Fokker-Plant channels, the diffusion coefficient e is part of the channel, and cannot therefore he tuned for inference in the general SI setting.
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C DETAILED ALGORITHM DESCRIPTION
C.I ALGORITHM PSEUDOCODE
The following pseudocode summarizes the full training procedure for our SCSI model in settings where the observation space y can be embedded into X' without loss of information.
Algorithm 2: Training of Self- Coffssonssttett Stochastic Interpolant (SCSI)
Input :Observation distribution μs, Forward mapping F. Interpolant schedule (a, 7,,, Initialization of drift and denoiser @ = (b(0) "g"") Total number of iterations K, Number of transport steps Tur
Dutput:Optiminn networks ( {b(K), ((K))
(10)
// Initialize transport map
for k in 1... K do for ... Ter do
y μ
I (g(k-1>(1)) (v)
// Backward transport to get a data sample // Map back to observations
ÿ F(x)
2 ~ (O, 1); ~~ (),
I, ax + B.ū 7+2
SGD update of via losses 3
10
(H) +- θ
// Update transport map
return (K)
C.2
CONDITIONAL GENERATION VIA LIFTING
In this subsection, we clarify how to proceed when embedding ) into X' is not straightforward. As noted the beginning of Section 2.2] one can always work a common state space through a lifting construction, which also makes explicit how the method naturally yields conditional generation. Throughout this subsection, we therefore do not assume =
We lift the problem to shared product space by defining Ω : X' y and introducing the map F(r,,) = (w, F(x)), where is drawn independently from an arbitrary base measure To P(Y). Let denote the joint distribution of (xr, ./. so that -mmarganal to generate. Under the lifted observation channel F, the observed distribution becomes jı = Ap @ To illustrate the resulting algorithm, consider the analogue of 7. in which do have access to
the target distribution we want
samples = (x, y) ~ n. this setting, we introduce a lifted interpolant I, = (4 (12) by
1 a&T + Brw, (
t [0, 1].
(81)
E [0, 1].
(82)
In the ODE setting, the velocity field satisfies b(1,z) = =Eii = (b(1, ar, y), 0). Operationally, this amounts to appending the observation as an additional input to the transport model with state space X, commonly done in conditional generation in score-based diffusion models Song al. (202I) and stochastic interpolants Albergo et al. (023b). When ß is not directly accessible, as in our inverse generative setting, the same iterative self consiisnssy scheme introduced in Section 3.1 can be applied to the lifted formulation without modification.
D IMPLEMENTATION DETAILS
D.1 ARCHITECTURE OF MODELS
We give the architecture details of our SI and diffusion model here. Both architectures are the U-net from Dhariwal & Nichol (022), specifically following the implementation here The main difference that we reduce the number of model channels in the first layer from default 192 to 96 for the diffusion model and 64 for the stochastic interpolant. This is primarily done for computational
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reasons. As result, the small model (64 channels) has ~32 million parameters while the large model has ~0 million parameters. Maximum positional embedding for the diffusion model and is taken to be 10,000 and respectively.
For 2-D latent parameters as used in random masking, we process them with small U-net consisting of 2 convolution blocks sandwiched between mode convolution layers and the number of channels given by channel multiplier. We concatenate this with the image along channel dimension. For 1-D latents as used in motion blur and JPEG compression, we process them with a three layer perceptron and then add them to the time embedding.
Table 3: Model configuration parameters
Parameter
Value
Model channels
96 (64)
Channel multiplier
[1, 2, 3, 4]
Channel multiplier for embeddings
4
Number of blocks
Attention on resolutions
[32, 16, 8]
Dropout Fraction
0.10
Max positional embedding
10000 (2)
Number of channels in latent U-Net
D.2 TRAINING PARAMETERS
We use the same hyperparameters for all the experiments. The backward transport map via ODE or SDE is performed in 64 steps. For experiments with ODE, we choose the schedule of SI as
a, = 1 - t, 3 t, Ye 0. For experiments with SDE, we keep the same schedule for a, Br, set Vt t(1 11--t t), and use 0.1.
We experimented with different choices of Ti, the number of backward transport steps in Alg. I and observed only minor differences across values, with T, already sufficient for all current experiments. To support this observation, we revisited the experiment in Section [.. (Fig. 2 and varied Tú while keeping the product Tú. K fixed so that the overall computational budget remained unchanged. The Wasserstein distance between the final solved distribution and the true distribution is summarized in the Table 4 below (three independent runs). The results show that the performance quite insensitive Tú unless it becomes so large that it forces too few outer iterations. We also did not observe notable differences in convergence speed. Given this robustness and avoid unnecessary hyperparameter tuning, use Tir throughout this work.
Table 4: Effect of Tú on final Wasserstein distance (three independent runs) for the synthetic example in Section 5.1 (Fig.)
Tir
Wasserstein Distance (Mean + Std)
1
0.0491 0.0038
10
0.0460 + 0.0038
100
0.0476 + 0.0047
1000
0.0593 + 0.0021
When (A) far from the optimal at the early stages of the outer iteration, the distribution of self-generated observations K.=($)()) 4 may differ significantly from μ,, and consequently slow down the convergence in practice. To mitigate this effect, we modify the interpolant (8) by replacing the generated
(y)) with mixture: with probability (set to 0.9 in our experiments), observation, and with probability - p. we use the original y. As long as p > 0, following the same argument in Prop. I know the fixed point still gives us the desired optimal parameters Or.
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Furthermore, to enhance computational efficiency, for every data mapped back with ODE integration, we (re)-sample the observations twice to generate two interpolated points. This amortizes the cost of ODE integration, which is the most expensive step in the training process. We fix the learning rate to be 0.0005 and use cosine schedule with warmup. Random masking, motion blur and JPEG experiments trained for 50,000 iterations while other experiments trained for 20,000 iterations.
E ADDITIONAL RESULTS
IMPACT OF NETWORK SIZE
We first show the results for training generative diffusion model using our network architecture and choices hyperparameters like learning rate, number iterations etc. This to provide simple intuition for the modeling capacity of our overall configuration. We train a big model (with 96 channels) and a small diffusion model (with 64 channels) on clean CIFAR-10 data. The big model corrssonnds to the model used for training generative model the restored samples, well for DPS experiments. Small model corresponds to the SI used in our experiments. The FID for samples generated with these models after training is 5.16 and 6.64 respectively. Note that these numbers are close to the FID reported in Table 2 for generative model trained on restored samples of random masking. In Fig. and we show some randomly drawn samples from these models.
Figure 8: Randomly drawn images from the large diffusion model trained on cleaned images.
Figure 9: Randomly drawn images from the smaller diffusion model trained on cleaned images.
To see the impact of network capacity our results, we repeat the random masking exercise with p 0.5 (i.e., 50% pixels masked) with networks of different sizes. We vary the number of channels in the SI model from 48 to 96 which varies the number of network parameters in a 3x range from 17 million to million. We train each network for 25k iterations and use ODE sampling.
Table 5 the FID between the restored samples and original CIFAR- 10 results and we find that using a larger network improves the quality of restoration. Figure I0 shows the training loss curves for these and even though all the networks converge to similar loss, the loss for larger networks begins to drop earlier than for smaller networks. However note that here we focus on the number of iterations, and in terms of wallclock time, smaller networks do train much faster. Given these, in principle it is possible to design an iterative scheme which optimizes the wallelock time by training a smaller network and using its restored samples to warm-start larger network.
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10a
Intorpaiant
Channels
FID
10
17M (4Blayers)
48
1.69
layers)
48M (80 layers)
64
1.40
70M (96 layers)
80
1.26
10a
10
10
96
1.17
Number of Iterations
Table 5: FID for restored samples.
Figure 10: Training loss curves.
E.2
SDE RESULTS
We present results for restoration using SDE in this section. We experiment with two different strategies: learning combined drift for the velocity drift and score terms using the objective (5). and ii) learning different networks for drift and score separately using losses 43. While the latter offers more flexibility in varying the noise schedule during inference, training two networks is least twice expensive, and we found no improvement over learning a single network for the combined drift. Hence we adopt the combined-drift formulation as our default for SDE.
In general, we find that, compared to ODE, SDE is more sensitive to the tuning of hyperparameters, specifically the noise schedule and number of transport steps. We have not thoroughly explored these choices for each experiment, and hence the results presented here generally comparable or inferior to the ODE results presented the main text. This merits more investigation in the future.
Results for CIFAR-10 and quasar spectra are shown in Figures and 12 respectively. All results use the default choices of noise schedule 7/2 = "n t) with To 0.05, c 7/s and T₁ = 1. For random mask and Gaussian blur, approach is able to restore the images but the quantitative performance is worse than ODE (e.g., for Gaussian blur, SDE samples have FID of 16.8 as compared 6.17 for ODE). However, for random motion, the restored images have visible artifacts. For quasar spectra, SDE-restored samples have MSE very close to ODE in the low-resolution/hig--SNN setting, but noticeably higher MSE in the high-resolution/lowww SNR setting.
Restered Coussien Oniginal
Original
(b) F: Gaussian blur + noise (aR 1, 1, 0.1). Restored Caussian blur Original
(a) F: Random mask (p 0.5, o. 10 ")
(C) F: Motion blur + noise
(kernel size 5, a,, 0.1)
Figure 11: Restored samples for different forward maps with SDE based SI.
E.3 ADDITIONAL. METRICS FOR RESTORED PERFORMANCE COMPARISON
Table reports additional quantitative metrics of imaging sample restoration for all corruption types, including comparisons with DPS and the SI-Oracle model. The SI-Oracle is trained using paired clean-corrupted data with the same SI architecture and serves performance upper bound for method. We find that our method approaches this oracle for Gaussian blurring and random masking, indicating that the lack of clean supervision incurs only modest penalty in these settings. The gap more pronounced for random motion blur, suggesting this corruption poses greater challenges for unsupervised recovery.
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True
FlUx
5.0
Dbs (0.87 0.05)
2.5
True
005: 114 80 1.36)
7.5
SI [0. 10:324 0.03)
Flxx
5.0
WF (0.55= 0.04)
51 10.53: 0.40)
18.20 0.061
4000
4500
5000
35000
6,00
6500
4000
45.00
5000
55.00
6000
6500
(a) F: High spectral resolution, low SNR 5%, SNR E [0.5, 2]
(b) F: Low spectral resolution. high SNR A log À = = 5%, SNR E [30, 50]
A log À 5000
Figure 12: Restored quasar spectra using SDE. The numbers in the legend represent the squared emror (MSE) averaged over 1000 true spectra.
Table 6:
Quantitative comparison of individual individual-sampll sample restoration quality across corruption types and noise levels with DPS and an oracle SI baseline.
Forward Model
Metric
Ours
DPS
SI-Oracle
LPIPS
0.00503
0.00497
0.00443
Random masking (p 0.5, 6)
PSNR
28.97
29.4
29.4
SSIM
0.931
0.935
0.935
FID
1.57
1.47
0.85
LPIPS
0.00650
0.00776
0.00551
Random masking = 0.5, on 0.1)
PSNR
27.8
26.7
28.4
SSIM
0.902
0.854
0,912
FID
2.78
17.54
1.07
LPIPS
0.00537
0.00949
0.00510
Gaussian blur (aR = 110, on = 0.1)
PSNR
29.2
26.8
28.8
SSIM
0.914
0.849
0.906
FID
6.74
29.9
1.33
LPIPS
0.0149
0.0249
0.0107
Gaussian blur (ØR = 1.0, 0n = 0.25)
PSNR
26.5
24.1
26.1
SSIM
0.852
0.744
0.841
FID
14.5
60.52
1.57
LPIPS
0.00692
0.00275
0.00297
Motion blur (k: = 5, o, 100-
PSNR
28.9
33.9
32.1
SSIM
0.904
0.970
0.947
FID
7.7
0.98
2.47
LPIPS
0.0108
0,0120
0,00297
Motion blur (A: 5, dn 0.1)
PSNR
24.9
25.3
32.1
SSIM
0.804
0.796
0.947
FID
21.9
50.1
2.70
E.4 SYNTHEIC MANIFOLD EXAMPLE
As noted the main text, when the original forward model produces observations space different from that of the input variable, we may introduce an equivalent formulation in which both the input and output of the forward map lie in a common ambient space. illustrate this procedure, we consider synthetic example of Rozet et al. (2024), where the original forward model maps X = R to y R In this example, the ground truth distribution ग is supported on a one-dimensional curve in 5 Observations whose rows are sampled uniformly from $4 e NO0 10 (1)), and a different realization of M is observed with each sample.
generated
Mr + R² where M E 2 is random matrix
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To construct observation in five-dimensional space where SCSI operates, consider two simple embeddings: (1) Direct padding. where y is augmented with three independent standard Gaussian coordinates; (2) Adjoint embedding, where y is mapped to M y R This choice still preserves all information in and is consistent with the general structure that quantities of the form (V.F) always have the same dimension
We follow the same setup in Rozet et al. (024), using a dataset of 65536 independent observations, and train the transport model parameterized by three-layer MLP of width 256. Figure 13 shows the reconstructed distributions through the 2D marginals ( 22), (I2, 23), and (23, 244. Both embeddings recover the underlying one-dimensional structure, with the adjoint embedding yielding closer reconstruction in this setting.
-2
-2
X2
X2
X;
(a) Ground Truth
(b) Direct Padding
(c) Adjoint Embedding
Figure 13: Reconstruction on the synthetic manifold example of Rozet et al. (024). Shown are three 2D marginals: (2), (I2, 23).. and (xz, x44 (from top to bottom) for the ground truth, the direct padding embedding, and the adjoint embedding. Both embeddings enable recovery of the one-dimensional curve, with the adjoint embedding producing the closer match.
E.5 VARYING LEVELS OF CORRUPTION
In this section, we present additional results for varying the levels of corruption for different forward models used in the main-text.
E.5.1 RANDOM MASKING
In Fig. 14
show additional results for random masking experiment with 25%, 50% and 75% pixels randomly masked. The quality of restored images declines with increasing corruption, but the restored samples close original image even for 75% corruption.
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fecoved
(a) Random masking with 25% pixels masked.
Necvee soodimn
(b) Random masking with 50% pixels masked,
Resbood Rovin MoL mnggra
(c) Random masking with 75% pixels masked.
Figure 14: Restoring images with for varying fractions masked pixels (levels of corruptions).
For generative modeling, we train new diffusion model on the samples restored with SI. Fig 15 shows samples from the model trained on the restored samples of the random masking experiment with 50% corruption and negligible noise. As reported in the main text, FID of this model is 6.74.
Figure 15: Samples from the diffusion model trained on the restored samples of random masking experiment with 50% corruption.
E.5.2
MOTION BLUR
show additional results for the motion blur experiment with increasing size of the motion blur kernel from 5 to 9 pixels.
In Fig. 16
E.5.3
JPEG COMPRESSION
In Fig. 17 we show restorations for JPEG corruption for additional images that have been compressed with randomly chosen ratios. The SI is able restore samples across broad range of corruptions.
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Published as a conference paper at ICLR 2026
beewed Suboetee
Motion blur with blur kernel of pixels.
(b) Motion blur with blur kernel of pixels.
Resbood Necobarr
(c) Motion blur with blur kernel of pixels.
Figure 16: Restoring images with SI for varying size motion blur kernel (levels of corruptions).
Figure 17: Additional images for JPEG restoration for the model trained samples with 1.).1.1.].
In addition, we consider another setting where we have training samples that are corrupted with q [4,01,,, 0.5]. i.e.. we never see high quality samples. The results for the trained SI in this setting are shown in Fig. 18 and 19 The restoration for low-quality samples poorer than when SI was trained
samples with compression ratio of more than 0.5. However, note that the SI remains stable in the extrapolation range, i.e., when restoring sample of q >> 0.5, the interpolant does indeed improve the restored image even though it has never samples in this regime.
E.6 NON-GAUSSIAN NOISE: GAUSSIAN BLURRING WITH POISSON NOISE
In the main text, we considered only Gaussian additive noise to different forward models. In this section, we present additional results for when the forward map blurring with a Gaussian kereel followed by adding Poisson noise. We add noise with two different levels, X, 0.1 and 0.5. The restored images here demonstrate that our approach also works the non-Gausaa noise setting.
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Published as a conference paper at ICLR 2026
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Figure 18: F: JPEG compression + noise (o, 0.01): results for different compression levels (Top: Corrupted: Bottom: Restored), Model is trained only on samples with 1/0. 0.5]. Results for higher qualities are in extrapolation regime.
Restove Conaree
Figure 19: Additional images for JPEG restoration for the model trained on samples with q 4[0.1,0.5, only.
conno
Neewed Grown Mar
(a) Gaussian blur (an 1) with Poisson noise X, (..1.
bocwnd Peblee wate
(b) Gaussian blur (a 1) with Poisson noise X., 0.25.
Figure 20: Restoring images with SI for Gaussian blurring with Poisson noise for different noise levels.
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Reviews:
[{"review_id": "3ozlmvn39H", "reviewer": "Reviewer_RjDz", "summary": "The authors propose a method to solve the inverse problems, formulated as a generative modeling problem, but in an unusual setup. While usually, when solving the inverse problem, one knows the *clean* data $x$ and corrupt operator $\\mathcal{K}$, the problem setup authors solve when one knows the *corrupted* data $y$ and corrupt operator $\\mathcal{K}$. Since this setup doesn't have clean data samples, it doesn't allow for the straight application of already developed generative modeling methodologies and requires the development of new ones. \n\nThe authors propose to utilize the Stochastic Interpolants (SI) framework and an iterative  (Expectation Maximization-like) scheme for learning it in a no-clean data setup. The key idea of the method is to build such an SI that the corrupted data distribution $Y$ mapped by SI into clean generated data would be mapped back by the corruption operator $\\mathcal{K}$ to the initial corrupted data distribution $Y$, i.e., self-consistency property, see lines 177-182. The practical algorithm consists of the iterative repetition of: 1) sampling data from the current SI ($\\Theta^{(k)}$), i.e., Expectation step 2) learning the next SI ($\\Theta^{(k+1)}$) on the data generated in the previous step, i.e., Minimization step.\n\nIn addition, the authors provide a theoretical analysis of their iterative procedure under different conditions and provide convergence guarantees of their procedure.\n\nIn practice, the authors evaluate their model on a range of tasks, including toy 2D problems, image inverse problems, and quasar spectrum restoration, and provide comparisons with competing approaches within the image inverse problem setting.", "questions": "See Weaknesses section", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The SI framework allows for a corruption operator $\\mathcal{K}$ with black box access, which is less restrictive than previous works  and allows for wider range of corruption operators $\\mathcal{K}$.\n- The SI is, in general, known to be a good generative modeling framework for the inverse problem solving [1, 2], so the application of SI to such an inverse problem is very well motivated even in $\\mathcal{K}$  cases, e.g. linear inverse problems, where one can utilize other diffusion methodologies [3, 4]\n- The theoretical analysis is rather comprehensive and convincing. Section 4 results deliver valuable theoretical analysis on the convergence of the method and the possible room for error, which can be helpful in practice and is good as the theoretical result by itself.\n- In the image inverse problems experiments method slightly outperforms DPS method that learns *using the clean data* and outperforms other methods that learn only from corrupted data.", "weaknesses": "- Some results in the paper are made under the assumption that the corruption operator is *injective*. That seems as not a mild assumption. Can authors comment on the restrictions of such an assumption on practice and show some examples of injective/non-injective mappings?\n- The learning procedure is non-simulation free, since one has to sample from learned SI during the \"Expectation\"-step. That leads to a computationally heavy algorithm. Can authors provide an ablation study on the parameters that control the computational demand of the procedure, i.e., the number of procedure iterations $K$ and the number of gradient updates  $T_{tr}$ inside of one iteration.\n- The authors explain the *trade-off* between the condition number $\\mathcal{X}$ and restriction of the SI approximation class $\\mathcal{S}_\\lambda$. This trade-off is shown in Theorem 1, but as far as I understand, authors do not test this trade-off in practice. So, can authors implement the restrictions on SI approximation class in practice and indeed show 1) faster convergence, 2) bigger error. That trade-off and the way to enforce it could be very useful in practice and seems like a useful hyperparameter that hasn't been explored.\n- The method's performance gap with EM posterior in Table 2 is marginal and the comparison of the proposed method with other methods that learn only on corrupted data has been carried out on the generation task. Which is strange, as far as I understand, at least some of the reference methods (Ambient Diffusion, EM Posterior or others) can solve inverse problems by starting from corrupted observations. It would be nice to compare the proposed method to competitors in particularly in the image restoration setup and not the generation setup, i.e., add methods that learn only on corrupted data in Table 1.\n- The applications of the proposed method are not described properly. Can authors describe the possible real-world scenarios where one has only corrupted observations and a black-box corrupt operator?\n\n\n[1] Liu, G. H., Vahdat, A., Huang, D. A., Theodorou, E., Nie, W., & Anandkumar, A. (2023, July). I $^ 2$ SB: Image-to-Image Schrödinger Bridge. In _International Conference on Machine Learning_ (pp. 22042-22062). PMLR.\n\n[2] Albergo, M. S., Goldstein, M., Boffi, N. M., Ranganath, R., & Vanden-Eijnden, E. (2024, July). Stochastic Interpolants with Data-Dependent Couplings. In _International Conference on Machine Learning_ (pp. 921-937). PMLR\n\n[3] Francois Rozet, Gerome Andry, Franc¸ois Lanusse, and Gilles Louppe. Learning diffusion priors from\nobservations by expectation maximization. Advances in Neural Information Processing Systems,\n37:87647–87682, 2024.\n\n[4] Giannis Daras, Kulin Shah, Yuval Dagan, Aravind Gollakota, Alex Dimakis, and Adam Klivans. Ambient diffusion: Learning clean distributions from corrupted data. Advances in Neural Information\nProcessing Systems, 36:288–313, 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": "YSFh4hFlVS", "reviewer": "Reviewer_L5cg", "summary": "The paper addresses the problem of inferring a probability density function from data that is obtained by a forward mapping F.\nThe topic is interesting and important. But the paper contains serious mistakes and therefore cannot be published.", "questions": "see above", "limitations": "", "rating": null, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "The main strength of the paper is in problem definition and casting. The rest has some serious errors", "weaknesses": "The main problem with this paper is that it has a significant error that propagates through the paper.\nThey define the path \nI = \\alpha x_t + \\beta F(x) + \\gamma z\n\nBut this path is wrong. In particular, F(x) is not even in the same space (functional space) of x and in the finite dim case often has a very different dimensions. Take the example you propose, tomography. F(x) is projection data and x is the image. The size of F(x) depends on the number of projection I take. This path does not make any sense!\n\nHint for next time - try the path I = \\alpha x_t + \\beta (\\grad F)^T F(x) + \\gamma z\nThis additional operation is crucial both in the finite and infinite setting", "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": "OjPAQlDSLS", "reviewer": "Reviewer_vQVs", "summary": "The authors consider a problem of inverse generative modelling in which they have access to samples of the distribution of corrupted samples and access to a black-box corruption map. The authors propose and theoretically justify a novel iterative algorithm based on iterative learning and inference of a stochastic interpolant model until it generates data such that the corruption map produces exactly the distribution of corrupted samples. The authors further evaluate their approach on image setups with different corruption maps and one scientific setup.", "questions": "Do you have any understanding of it it possible to extend this framework to discrete domains by considering, e.g., masking or uniform diffusions?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed approach is novel and theoretically justified.\n- The proposed approach does not require knowing or using the specific functional form of the corruption map used.\n- The authors evaluate their method on a wide range of image setups and one scientific setup, showing that the proposed method solve the problem.", "weaknesses": "- All the setups are more like synthetic, rather than real setups mentioned in the introduction: “... medical imaging, where we observe tomographic projections of internal structures, astronomical observations affected by atmospheric distortion, and other measurement processes that introduce noise and information loss”. The quasar spectra setup also assumes the usage of the synthetically provided process F. While the considered setups clearly show that the method works, the addition of one in the mentioned real-life examples would further strengthen the method. Specifically, they would show that the requirements of the method (like injectivity of F) could be satisfied in real-life problems and that there is a direct practical result.", "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": "t2wbcKpzjp", "reviewer": "Reviewer_hPBy", "summary": "This work addresses the problem of restoring clean samples from their corrupted measurements under two key constraints: 1) access is limited to the corrupted data, without corresponding clean samples, and 2) the forward map from the clean signal to the measurements is treated as avilable in a black-box manner. The authors approach this problem using Self-consistent Stochastic Interpolants, an extension of standard Stochastic Interpolants (SI) framework. This method adds a self-consistency mechanism, ensuring that the trained model's outputs map back to the true measurements. The method is introduced with rigorous mathematical formalism, solid theoretical results, and a broad analysis of convergence guarantees. The experimental section covers both standard benchmarks and a sophisticated application to quasar spectra recovery.", "questions": "I think that the method is very elegant and I am generally sympathetic to the paper. However, I think some important points should be addressed, which I mentioned in the Weaknesses above. I have one additional question related to the theoretical results.\n\nQ1. How do the considerations from lines 308-319 apply when the considered SDE has matrix-valued coefficients? Moreover, how do these results connect with the matrix-valued SDE proposed in [2], which embeds the Gaussian linear forward model into the coefficients? \n\n[1] Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, ICLR, 2021\n\n[2] Sobieski et al., System-embedded Diffusion Bridge Models, arXiv, 2025", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. The considered problem of obtaining clean samples when only the measurements and the black-box forward model are available seems important and challenging, yet the authors' approach based on self-consistency is both novel and effective.\n\nS2. The proposed approach has strong theoretical foundations and is accompanied by a detailed convergence analysis.\n\nS3. While the paper often utilizes more sophisticated mathematical tools, the authors took great care in writing the paper in a clear way with good flow.", "weaknesses": "W1. My primary concern relates to the lack of evaluation of the SDE version of the algorithm. The theoretical analysis focuses mainly on the SDE case, so it is natural to ask whether these results translate to practice. Moreover, as the authors also mention, the toy example (section 5.1) indicates the superiority of the SDE-based approach, and it seems counterintuitive that no other non-toy evaluations appeared in the paper. While I understand that it might be more computationally demanding and slower in terms of convergence, some kind of comparison in a standard experiment (section 5.2) should be provided.\n\nW2. Equally concerning is the lack of standard evaluation metrics used in typical inverse-problems-related papers.\n\n1. For the standard benchmarks (Table 1), only LPIPS is used. Methods for inverse problems are typically evaluated with the perception-distortion tradeoff in mind [1], so I would recommend adding PSNR, SSIM (distortion) and FID (perception).\n\n2. For the comparison with other inverse generative models, only FID is provided. This is not crucial, but some metric like the Inception Score [2] should also be provided for a more balanced comparison.\n\n3. For the quasar spectra recovery, no quantitative results are given. I'm not an expert in this subfield, but I expect the authors to provide proper metrics used in the evaluation scheme in such scenarios.\n\nW3. The scope of the chosen baselines is very small. DPS is a good starting point, but as the authors correctly point out, its assumptions largely differ from their work. It would be beneficial for the paper to also include other methods that better cover the spectrum. Examples include: unsupervised methods that make different assumptions about the forward model [3,4], supervised bridge methods that ignore the forward model [5,6,7,8] or assume access to it [9], unpaired-data-based bridges [10,11] and the approaches explicitly cited by the authors as the closest ones in terms of assumptions and recency (lines 044-053). I don't expect the authors to include all of the above, but the comparison should be more representative.\n\nW4. (Minor) I believe that the black-box assumption is lost when the method is additionally using either the mask in the random masking experiment or the compression magnitude in the JPEG one. The text should be slightly rephrased to emphasize that.\n\n[1] Blau and Michaeli, The Perception-Distortion Tradeoff, CVPR, 2018\n\n[2] Salimans et al., Improved Techniques for Training GANs, NeurIPS, 2016\n\n[3] Wang et al., Zero-shot image restoration using denoising diffusion null-space model, ICLR, 2023\n\n[4] Song et al., Pseudoinverse-guided diffusion models for inverse problems, ICLR, 2023\n\n[5] Liu et al., I2SB: Image-to-image Schrodinger Bridge, ICML, 2023\n\n[6] Zhou et al., Denoising Diffusion Bridge Models, ICLR, 2024\n\n[7] Luo et al., Image restoration with mean-reverting stochastic differential equations, ICML, 2023\n\n[8] Yue et al., Image restoration through generalized ornstein-uhlenbeck bridge, ICML, 2024\n\n[9] Sobieski et al., System-embedded Diffusion Bridge Models, arXiv, 2025\n\n[10] De Bortoli et al., Schrodinger bridge flow for unpaired data translation, NeurIPS 2024\n\n[11] Kim et al., Unpaired image-to-image translation via neural schrodinger bridge, ICLR, 2024", "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)