Inverse Flow and Consistency Models
Authors: Yuchen Zhang
Jian Zhou
Venue: ICML 2025 poster (2025)
Keywords: consistency models
flow matching
diffusion models
inverse problem
Primary area: deep_learning->generative_models_and_autoencoders

Abstract: Inverse generation problems, such as denoising without ground truth observations, is a critical challenge in many scientific inquiries and real-world applications. While recent advances in generative models like diffusion models, conditional flow matching, and consistency models achieved impressive results by casting generation as denoising problems, they cannot be directly used for inverse generation without access to clean data. Here we introduce Inverse Flow (IF), a novel framework that enables using these generative models for inverse generation problems including denoising without ground truth. Inverse Flow can be flexibly applied to nearly any continuous noise distribution and allows complex dependencies. We propose two algorithms for learning Inverse Flows, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM). Notably, to derive the computationally efficient, simulation-free inverse consistency model objective, we generalized consistency training to any forward diffusion processes or conditional flows, which have applications beyond denoising. We demonstrate the effectiveness of IF on synthetic and real datasets, outperforming prior approaches while enabling noise distributions that previous methods cannot support. Finally, we showcase applications of our techniques to fluorescence microscopy and single-cell genomics data, highlighting IF's utility in scientific problems. Overall, this work expands the applications of powerful generative models to inversion generation problems.

Inverse Flow and Consistency Models
Yuchen Zhang'
Jian Zhou
Abstract
& Ermon 2020 Song al. 2021 2022). conditional flow
Inverse generation problems, such denoising without ground truth observations, is critical challenge in many scientific inquiries and real- world applications. While recent advances gen- erative models like diffusion models, conditional flow matching, and consistency models achieved impressive results by casting generation as de- noising problems, they cannot be directly used for inverse generation without access to clean data. Here we introduce Inverse Flow (IF), a novel framework that enables using these generative models for inverse generation problems includ- ing denoising without ground truth. Inverse Flow can be flexibly applied to nearly any continuous noise distribution and allows complex dependen- cies. We propose two algorithms for learning Inverse Flows, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM). Notably, to de- rive the computationally efficient, simulation- free inverse consistency model objective, we general- ized consistency training to any forward diffusion processes conditional flows, which have appli- cations beyond denoising. We demonstrate the effectiveness of IF on synthetic and real datasets, outperforming prior approaches while enabling noise distributions that previous methods cannot support. Finally, we showcase applications of our techniques to fluorescence microscopy and single-cell genomies data, highlighting IF's utility in scientific problems. Overall, this work expands the applications of powerful generative models to inversion generation problems.
matching models (Lipman al.) 2023 Tong al. 2024]. and consistency models (Song et al. [023] Song & Dhari
wal 2023) have achieved great success by learning a map- ping from a simple prior distribution to the data distribu- tion through an Ordinary Differential Equation (ODE) or Stochastic Differential Equation (SDE). We refer to their models continuous-time generative models. These mod- els typically involve defining forward process, which trans- forms the data distribution to the prior distribution over time, and generation is achieved through learning a reverse pro- cess that can gradually transform the prior distribution to the data distribution (Figure D.
Despite that those generative models are powerful tools for modeling the data distribution, they not suitable for the inverse generation problems when the data distribution is not observed and only data transformed by forward pro- cess is given, which is typically true for noisy real-world data measurements. Mapping from noisy data to the latent ground truth is especially important in various scientific applications when pushing the limit of measurement capa- bilities. This limitation necessitates the exploration of novel methodologies that can bridge the gap between generative modeling and effective denoising in the absence of clean data.
Here we propose a new approach called Inverse Flow (IF) D that learns mapping from the observed noisy data distribu- tion to the unobserved, ground truth data distribution (Figure I. inverting the data requirement of generative models. An ODE or SDE is specified to reflect knowledge about the noise distribution. We further devised pair of algorithms, Inverse Flow Matching (IFM) and Inverse Consistency Model (ICM) for learning inverse flows. Specifically, ICM involves a computationally efficient simulation- free objec- tive that does not involve any ODE solver.
1. Introduction
Recent advances in generative modeling such as diffusion models (Sohl-Dickstein et al. 2015 Ho et al.) 2020
A main contribution of our approach is generalizing continuous-time generative models to inverse generation problems such as denoising without ground truth. In ad- dition, in order to develop ICM, we generalized the con- sistency training objective for consistency models to any forward diffusion process or conditional flow. This broad-
Song
LLyda Hill Department of Bioinformatics. University of Texas
Southwestem Medical Center, USA. Correspondence Jian Zhou <jian.zhou@ utsouthwesterm..du
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
'Code available at htttps://ggihuub.com////aaaaa
InverseFlon

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Inverse Flow and Consistency Models
ens the scope of consistency model applications and has implications beyond denoising.
(x.) ue (x, [xx]|]
Compared to prior approaches for denoising without ground truth, IF offers the most flexibility in noise distribution, al- lowing almost any continuous noise distributions including those with complex dependency and transformations. IF can be seamlessly integrated with generative modeling to gener- ate samples from the ground truth rather than the observed noisy distribution. More generally, IF models the past states of (stochastic) dynamical system before the observed time points using the knowledge of its dynamics, which can have applications beyond denoising.
where Xo is sampled from the data distribution, and X, is sampled from the conditional distribution p(x; Xo) given by the conditional ODE.
The conditional ODE vector field u,(x Xo) can also be stochastically approximated through sampling from both prior distribution and data distribution and using the con- ditional vector field u, (x Xg. X1) as the training target
(Lipman et al. 2023 Tong et al. 2024):
(x,) - uy (x+ Xo, x1)
(3)
2. Background
This formulation has the benefit that u,(x Xp. X1) can be easily chosen as any interpolation between Xy and X1 because this interpolation does not affect the probability density at time 0 or 1 (Lipman et al. 2023 Tong al. Albergo & Vanden-Eijnden 2023 Albergo et al.,
2.1. Continuous-time generative models
Our proposed inverse flow framework is built upon continuous- generative models such as diffusion mod- els, conditional flow matching, and consistency models. Here we present unified view of these methods that will help connect inverse flow with this entire family of models (Section 3
2024
2023
For example, linear interpolation corresponds to
X, 2024
X + t(x, Xx) (Lipman et al., 2023 Tong et al.,
Liu et al. 2022). Sampling is realized by simulating
the unconditional ODE with learned vector field vß (x) in the reverse direction.
These generative modeling methods are connected by their equivalence continuous normalizing flow neural ODE (Chen et al., (211)). They can all be considered as explicitly or implicitly learning the ODE that transforms between the prior distribution p(x1) and the data distribution p(xo)
2. 2.1.2. CONSISTENCY MODELS
In contrast, consistency models (Song et al. 2023 Song & Dhariwal 2023) learn consistency functions that can directly map a sample from the prior distribution to data distribution, equivalent to simulating the unconditional ODE in the reverse direction:
dx =x(x)dt.
c(x;, t) = ODE" (x:))
in which (x(x) represents the vector field of the ODE, We use the convention that = 0 corresponds to the data dis- tribution and f = 1 corresponds to the prior distribution. Generation is realized by reversing this ODE, which makes this family of methods a natural candidate for extension toward denoising problems.
where x; denotes x at time f, and we use ODFL (x,) to denote simulating the ODE with vector field ux(x) from time t to time 0 starting from x,. The consistency function is trained by minimizing the consistency loss (Song et al. 2023). which measures the difference between consistency function evaluations at two adjacent time points (C)
Continuous-time generative models typically involve defin- ing conditional ODE SDE that determines the p(x, that transforms the data distribution to the prior distribution. Training these models involves learning the unconditional ODE (Eq. T based on Xo and the conditional ODE or SDE Lipman et al., 2023; Tong et al. 2024 Song et al. 2021) (Figure l). The unconditional ODE can be used for generation from noise to data.
Exx+,
+++1) - stopgrad (co +II
(4)
with the boundary condition c(x, 0) x. Stopgrad indi- cates that the term within the operator does not get opti- mized.
There are two approaches to training consistency models: one is distillation, and the other is training from scratch. In the consistency distillation objective, a pretrained diffusion model is used obtain the unconditional ODE vector field u, and Xt.. and x,, differs by one ODE step
1.1.1. CONDITIONAL FLOW MATCHING
Conditional flow matching defines the transformation from data to prior distribution via a conditional ODE vector field
(x(x Xp). The unconditional ODE vector field v (x) is
learned by minimizing the objective (Lipman et al. 2023
Xtat1 ~ p(xt, X0),
(5)
Tong et al. 2024 Albergo & Vanden-Eijnnden 2023):
X Xx,, = Ut (Xe4,) (le41 - t.)
2

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Inverse Flow and Consistency Models
Conditional Flow Matching Conditional (forward) ODE
Inverse Flow Matching (IFM)
Consistency Model
Inverse Consistency Model (ICM)
Conditional (forward) ODE
Conditional (forward) ODE
Conditional (forward) ODE
p( )((x))
Pesse Xo)
((x))
p(xn)
((xu)
Proce (X) Xx)
ag(x))
p(X.)
p(x))
Unconditional (reverse) ODE
Unconditional (reverse) ODE
Consistency functiion
Consistency function
Observed / given Unsbserved inferred
Figure 1. Inverse flow enables adapting the family continuous-ting generative models for solving inverse generation problems. For inverse flow matching and inverse consistency model, X indicates unobserved data and x₃ indicates observed data. For conditional flow matching and consistency model, Xo indicates data and x indicates variable from the prior distribution. Inverse flow algorithms modify continuous-time generative models to solve the inverse generation problem of recovering unobserved Xo from by replacing the unobserved p(Xo) with generated g(Xo) within the training loop.
If the consistency model is trained from scratch, the consis- tency training objective samples Xtz11 and xt, in coupled manner from the forward diffusion process (Karras et al. 2022)
Tweedie (Daras al. 2024) improves Ambient Diffusion by allowing exact sampling from clean data distribution us- ing consistency loss with a double application of Tweedie's formula. Rozet et al.] explored the potential of expecta- tion maximization in training diffusion models on corrupted data. However, all these methods are restricted to training on linearly corrupted data, which still limit their applica- tions when the available data is affected by other types of noises.
241 Xo + 2/;-1; Xt; Xo 2/,,
o0,
(6) where a controls the maximum noise level at t = 1. Consis- tency models have the advantage of fast generation speed as they can generate samples without solving any ODE or SDE.
2.2. Denoising without ground truth
2.1.3. DIFFUSION MODELS
Denoising without access ground truth data requires as- sumptions about the noise or the signal. Most contemporary approaches are based on assumptions about the noise, as the noise distribution is generally much simpler and better understood. Because prior methods have been comprehen- sively reviewed (Kim & Ye. 202T Batson & Royer] 2019) Lehtinen al. 2018 Xie et al. 2020; Soltanayev & Chun 2018 Metzler et al. 2020 and our approach is not di-
In diffusion models, the transformation from data prior distribution is defined by forward diffusion process (con- ditional SDE). The diffusion model training learns the score function which determines the unconditional ODE, also known as the probability flow ODE (Song et al. 2021
Denoising applications of diffusion models Diffusion models are inherently connected to denoising problems as the generation process is essentially denoising process. However, existing denoising methods using diffusion mod- els require training on ground truth data (Yue et al. [2023] Xie et al., (023b), which is not available in inverse genera- tion problems.
rectly built upon these approaches, only present a brief overview and refer the readers to Appendix A.3 referenced literature for more detailed discussion.
Another class of unsupervised denoising methods leverages variational autoencoders (VAEs) trained on noisy data com- bined with an explicit noise model (Prakash et al.) 2021b
Iwamoto al. 2025 Salmon & Krull 2025) In these ap- proaches, neural network jointly models the clean data distribution, the noisy observation distribution, and the pos- terior, either by assuming a fixed noise model a priori or by adapting it on the fly during training. While such VAE- based methods can provide diverse posterior samples and a tractable latent representation of clean signals. they typically rely the design and optimization of the encoder/decoder pair and are most effective when the noise statistics conform to the chosen VAE likelihood model.
Ambient diffusion and GSURE-diffusion Ambient Dif- fusion (Daras et al. 2023) and GSURE-diffusion (Kawar jet al. 2024) address related problem of learning the distri- bution of clean data by training on only linearly corrupted (linear transformation followed by additive Gaussian noise) data. Although those methods are designed for generation, they can be applied to denoising. Ambient Diffusion Pos- terior Sampling (Aali et al.) 2024), further allowed using models trained with ambient diffusion on corrupted data to perform posterior sampling campling-based based denoising for different forward process (e.g., blurring). Consistent Diffusion Meets
None of these approaches are generally applicable to any noise types.
3

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Inverse Flow and Consistency Models
3. Inverse Flow and Consistency Models
conditional flow matching (Eq. B an alternative form of the conditional ODE u, (x Xo; ) can be used instead of (x xo). Since x is sampled from the noise distribu- tion p(X1 Xu), the above condition is automatically satis- fied. The conditional ODE vector field can be easily chosen as any smooth interpolation between Xo and x1. such as u, (x Xo; x1) = x Xo. We detailed the inverse flow matching training in Algorithm with the alternative form in Appendix [A.
In continuous-time generative models, usually the data Xo from the distribution of interest is given. In contrast, in inverse generation problems, only the transformed data x1 and the conditional distribution p(x1] (x)) are given, whereas Xo are unobserved. For example, are the noisy obser- vations and p(x1 is the conditional noise distribution. We define the Inverse Flow (IF) problem as finding map- ping from X1 to Xy which allows not only recovering the unobserved data distribution p(xo) but also providing an estimate of Xo from X (Figure
Next, we discuss the theoretical justifications of the IFM objective and the interpretation of the learned model. We show below that when the loss converges, the recovered data distribution q (Xx) matches the ground truth distribution p( xx))). The proof is provided in Appendix A.2.1 1
For denoising without ground truth applications, the inverse flow framework requires only the noisy data x1 and the abil- ity to sample from the noise distribution p(x, (x)). This is thus applicable to any continuous noise and allows complex dependencies on the noise distribution, including noise that can only be sampled through a diffusion process.
Theorem 1 Assume that the noise distribution p(x1 xo)
satisfies the condition that, for any noisy data distribution p(x1) there exists only one probability distribution p p(xo) that satisfies p(x1) = =f p(x1 xo)p(xo)dxxx then under the condition that LIFM 0, we have the recovered data distribution q(xo) p(xo).
Intuitively, without access to unobserved data Xg, inverse flow algorithms train continuous-time generative model using generated Xy from observed data X1 within the train- ing loop (Figure D We demonstrated that this approach effectively recovers the unobserved distribution p( p()p) and learns a mapping from X to Xo.
Moreover, we show that with IFM the learned ODE trajec- tory from X X can be intuitively interpreted as always pointing toward the direction of the estimated xo. More formally, the learned unconditional ODE vector field can be interpreted as an expectation of the conditional ODE vector field.
3.1. Inverse Flow Matching
To solve the inverse flow problem, we first consider learning a mapping from x to xo through an ODE with vector field (x). We propose to learn v²(xx with the inverse flow matching (IFM) objective
Lemma 1 Given a conditional ODE vector field u2 (x
Xo, x1) that generates a conditional probability path p(x: I
Xg, x1). the unconditional probability path p(xx) can be
generated by the unconditional ODE vector field (x(x).
(0)
which is defined as
EE (((x)) ur (ODEY=O(X)))
(x) = xx (x) [u:(x Xo, xx,x
(8)
where the expectation is taken over 1, p(xi), and p(x: Xo ODEY"a This objective differs from condi- tional flow matching (Eq.. [ in two key aspects: using only transformed data x₁ rather than unobserved data xo, and choosing the conditional ODE based on the conditional distribution pxx) xo). Specifically,
The proof is provided in Appendix A.2. Specifically, with
the choice of u, (x X1) = X1 Xo. Eq. 8 has an intuitively interpretable form
nn (x(x) = (x(x)))) lx)
Xo
which means that the unconditional ODE vector field at any time t points straight toward the expected ground truth xo.
1. Sampling from the data distribution p(xo) is replaced with sampling from p(x1) ) and simulating the uncondi- tional ODE backward in time based on the vector field v, denoted as oD We refer to this distribu- tion as the recovered data distribution g(xo).
3.2. Simulation-free Inverse Flow with Inverse Consistency Model
IFM can be computationally expensive during training and inference because it requires solving ODE in each update. We address this limitation by introducing inverse consis- tency model (ICM), which learns a consistency function to directly solve the inverse flow without involving an ODE solver.
2.
The conditional ODE vector field u, (x xn) is chosen to match the given conditional distribution p(x1 (xo) at time 1.
For easier and more flexible application of IFM, similar to
4

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Inverse Flow and Consistency Models
However, the original consistency training formulation is specific to one type of diffusion process (Karras et al. 2022). which is only applicable to independent Gaussian noise distribution for inverse generation application. Thus, to derive inverse consistency model that is applicable to any transformation, we first generalize consistency training so that it can be applied to arbitrary transformations and thus
3.2.2. INVERSE CONSISTENCY MODELS
With generalized consistency training, we can now provide the inverse consistency model (ICM) (Figure I Algorithm
2
LICM(0)
E (co (Xt; (1> li+1) stopgrad (co (Xe,s
(1))
flexible to model almost any noise distribution.
X X1; U (Xum1 xo) 44)
(11)
3.2.1. GENERALIZED CONSISTENCY TRAINING
which is the consistency model counterpart of IFM (Eq.
To recall from Section 2.1.2 consistency distillation is only applicable to distilling pretrained diffusion or conditional flow matching model. The consistency training objective allows training consistency models from scratch but only for a specific forward diffusion process, which limits its flexibility in applying to any inverse generation problem.
7 The expectation is taken over i, p(x1),
p (X X x 1)). Similar to IFM, convenient
alternative form is provided in Appendix A.
Since learning consistency model is equivalent to learning conditional flow matching model, ICM is equivalent to IFM following directly from our Theorem 2 and Theorem I from (Sog et al. 2023).
Generalized Consistency Training < Consistency Distillation
Lemma 2 Assuming the consistency function co(x, () is twice differentiable and 0co(x.t) t)//x is almost everywhere nonzerd when the inverse consistency loss LICM O, there exists corresponding ODE vector field vî (z:) that minimized the inverse flow matching loss LirM to 0.
Conditional Flow Matching
} Flow Matching
Here we introduce generalized consistency training (GCT), which extends consistency training to any conditional ODE or forward diffusion process (through the corresponding conditional ODE). Intuitively, generalized consistency train- ing modified consistency distillation (Fq. and Eq. 5 in the same manner as how conditional flow matching modi- fied the flow matching objective: instead of requiring the unconditional ODE vector field (x(x) which not available without a pretrained diffusion or conditional flow matching model, GCT only requires the user-specified conditional ODE vector field (x(x xo).
The proof is provided in Appendix A.2.2 As in IFM, when the loss converges, the data distribution (Xp) recovered by ICM matches the ground truth distribution p(Xo). but ICM is much more computationally efficient as it is a simulation- free objective.
4. Experiments
We first demonstrated the performance and properties of IFM and ICM on synthetic inverse generation datasets, which include a deterministic problem of inverting Naiver- Stokes simulation and a stochastic problem of denoising a synthetic noise dataset 8- 8-gaussians. Next, we demonstrated that our method outperforms prior methods (Mäkinen et al., 2020 Krull et al. 2019; Batson & Royer 2019) with the same neural network architecture
(f)
=B (co t(41) stopgrad (co(Xi,) ++)
Xt41 Xt: U (Xts41 (o)(12+1 tx)
(10)
semi-synthetic dataset of natural images with three synthetic noise types, and a real-world dataset of fluorescence microscopy images. Fi- nally, we demonstrated that our method can be applied to denoise single-cell genomics data.
Where the expectation is taken over i, p(x)). and
p(Xtzy1/x)) [x0). An alternative formulation where the condi-
tional flow is defined by n (x Xu, x1) is detailed in
Appendix A.I
We proved that the generalized consistency training (GCT) objective is equivalent to the consistency distillation (CD) objective (Eq. Eq. 5). The proof is provided in Appendix
4.1. Synthetic datasets
To test the capability of inverse flow in inverting complex transformations, we first attempted the deterministic in-
A.2.2
20co(x, t)/8x * 0 is required to ensure the existence of corresponding ODE, and it excludes trivial solution such as co (x, = constant. With identity initialization of co(x, f), do not find it to be necessary for enforcing this condition in practice.
Theorem 2 Assuming the consistency function ce(x, t) is twice differentiable with bounded second derivatives, and Epixo.xils xx) (x xy, x1]x1)ll < co, up to a constant inde- pendent of 0, LGCT and LCD are equal.
5

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Inverse Flow and Consistency Models
Algorithm 1 IFM Training
Algorithm 2 ICM Training
Input: dataset D. initial model parameter 8, and
Input: dataset D, initial model parameter B, learning rate 1)
learning rate !
and sequence of time points 0 = 1 < l2 IN
2: repeat
repeat
3: Sample D and 210,
3: Sample X1 D and i U1,,
4: Xo f stopgrad (ODEY-((X)))
4:
Xo f- stopgrad (ca( 1))
(x1,
5: Sample Xt+t1 ~ p(xtut1 x0)
5: Sample X₁ ~ p(x: xo)
X1, f- X(,11 uhn (Kle41 (p) - (2)
6: ((0) - (x,) ) - Ur (Xt xo)
[()) t [x( (411) stopgrad (co (Xta 43)]
0 +
0 n)V(((0)
8:
until convergence
unfil convergence
Ground Truth
IFM
ICM
turbulent flows. The details of the simulation described in Appendix A.4.2.
0.2
0.1
To test inverse flow algorithms on a denoising inverse gener- ation problem. we generated synthetic 8-gaussians dataset (Appendix [A.4.2] for details). Both IFM and ICM are ca- pable of noise removal (Figure 2. ICM achieved a similar denoising performance as IFM, even though it much more computationally efficient due to the iterative evaluation of ODE (NFE=10) by IFM.
0.0
-011
0.2
0.4
0.2
0.0
4.2. Semi-synthetic datasets
0.4
We evaluated the proposed method on images in the bench- mark dataset BSDS500 (Arbeláez et al. 201T). Kodak, and Set12 (hhag et al. 2017). To test the model's capability to deal with various types of conditional noise distribution, we generated synthetic noisy images for three different types of noise, including correlated noise and adding noise through diffusion process without a flosed-form transition density function (Appendix A..43 for details). All models were trained using the BSDS500 training set and evaluated the BSDS500 test set, Kodak, and Set12. We show additional qualitative results in Appendix A.6
Inverse Flow Matching
Inverse Consistency Model
Figure Demonstration of inverse flow algorithms on synthetic datasets. Top panel shows an application to inverting Navier- Stokes fluid dynamics simulation color indicating difference hetween the input state and the initial state. Bottom panel shows a denoising application on 8 gaussians dataset with input (black) and denoised data (blue) connected with lines.
1. Gaussian noise: we added independent Gaussian noise with fixed variance.
Correlated noise: we employed convolution kernels generate correlated Gaussian noise following the method in (Mäkinen al. 2020)
verse generation problem of inverting the transformation by Navier-Stokes fluid dynamics simulation We aim to re- cover the earlier state of the system without providing them for training (Figure 2 Navier-Stoker equations describe the motion of fluids by modeling the relationship between fluid velocity, pressure, viscosity, and external forces. These equations are fundamental in fluid dynamics and remain mathematically challenging, particularly in understanding
1) = v @
(12)
where (O, oT) and g convolution kemel.
Jacobi process: we transformed the data with Jacobi process (Wright-Fisher diffusion), as an example of SDE-based transform without closed-form conditional distribution
Inverse flow algorithms can be applied to deterministic trans- formations from even though the general forms consider stochastic transforms de- scribed by p(x: xo).
by using matching conditional ODE,
dx _10(1 - x) -x]dtt + Vsx(1 - x)dw. (13)
6

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Inverse Flow and Consistency Models
Table Quantitative benchmark of denoising performances in multiple datasets for various noise distributions measured by Peak signal- to-noise ratio (PSNR) in dB,
Noise type
Input
Supervised
BM3D
Noise2 Void
Noise2Self Ours (ICM)
BSDS500
20.17
28.00
27.49
26.54
27.79
28.16
Gaussian
Kodak
20.18
28.91
28.54
27.55
28.72
29.08
Set12
20.16
28.99
28.95
27.79
28.78
29.19
BSDS500
20.17
27.10
24.48
26.32
21.03
27.64
Correlated
Kodak
20.17
27.97
25.03
27.39
21.56
28.53
Set12
20.18
27.88
25.21
27.43
21.58
28.46
BSDS500
14.90
24.34
20.32
23.56
22.60
24.28
SDE (Jacobi process)
Kodak
14.76
25.34
20.42
23.99
23.70
25.07
Set12
14.80
25.01
20.51
24.43
23.26
24.74
We generated corresponding noise data by simulating the Jacobi process with s and the conditional noise distribution generated by the Ja- cobi process does not generally has expectation that equals the ground truth (i.e. non-centered noise), which violates the assumptions of Noise2X methods.
from (Liu et al., 2013) for separate channels of each noisy microscopic images (Appendix [A.4.4] for details). Then denoising flow models were trained with the conditional ODE specified be consistent with fitted noise model. Our method outperforms Noise2Self and Noise2 achieving superior denoising performance for mitochondria, F-actin, and nuclei in the microscopic images of BPAE cells (Figure
Notably,
Our approach outperformed alternative unsupervised meth- ods in all three noise types, especially in correlated noise and Jacobi process (Appendix[AA) T.2le This can attributed to the fact that both Noise2X methods assumes independence of noise across different feature dimensions as well centered-noiss which were violated in corrleated noise and Jacobi process respectively.
4.3.2. APPLICATION DENOISE SINGLE-CELL GENOMICS DATA
In recent years, the development of single-cell sequencing technologies has enabled researchers to obtain more fine- grained information on tissues and organs at the resolution of single cells. However, the low amount of sample ma- terials per-cell introduces considerable noise in single-cell genomics data. These noises may obscure real biological signals, thereby affecting subsequent analyses.
Moreover, Our approach outperformed the supervised method on both Gaussian noise and correlated noise. Further analysis revealed that the supervised method encountered overfitting during the training process, which led to subop- timal performance. contrast, our method did not exhibit such issues, highlighting the superiority of our approach.
Applying ICM an adult mouse brain single-cell RNA-seq dataset (Zeisel et al.) 2018) and mouse brain development single-cell RNA-seq dataset (Hochgerner et al. 8b) (Fig- ure 4 Appendix [A.4.5] for details), we observed that the denoised data better reflects the cell types and developmen- trajectories. We compared the original and denoised data by the accuracy of predicting the cell type identity of each cell based on its nearest neighbor in the top two principal components. Our methods improved the accuracy of the adult mouse brain dataset from 0.513 + 0.003 to 0.571 + 0.003, and the mouse brain development dataset from 0.647 = 0.006 to 0.736 + 0.006.
In addition, in Appendix A.5 we conducted series of experiments that demonstrate the reliability of our method under different intensities and types of noise. Furthermore, our method yielded satisfactory results even when there is bias in the estimation of noise intensity. also achieved excellent performance RGB images and small sample- size datasets.
4.3. Real-world datasets
4. FLUORESCENCE MICROSCOPY DATA (FMD)
5. Limitation and Conclusion
Fluorescence microscopy is an important scientific appli- cation of denoising without ground truth. Experimental constraints such as phototoxicity and frame rates often limit the capability to obtain clean data. We denoised confocal microscopy images from Fluorescence Microscopy Denois- ing (FMD) dataset (Zhang et al. 2019). We first fitted a signal-dependent Ebisson-Gaussian noise model adopted
We introduce Inverse Flow (IF), a generative modeling framework for inverse generation problems such as denois- ing without ground truth, and two methods Inverse Flow Match (IFM) and Inverse Consistency Model (ICM) to solve the inverse flow problem. Our framework connects the fam- ily of continuous-time generative models inverse gener-

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Inverse Flow and Consistency Models
Input
Ground Truth
Ours (ICM)
Noise2Self
Noise2Void
Mitochondria
34.02
41.21
39.41
38.93
F-actin
30.25
38.90
37.44
37.26
Nuclei
34.05
43.18
Merged
32.38
41.00
39.12
Input
Denoised
Astrocytes
Cerebellumn
Dentate gyrus granule Intertmidbrain excitatory glla Microglia
inhibitory neurons Non-glutametergiic
Olfactory ensheathing Olfactory inhibitory neurons
Oligadendrocytes
Peplidergic neurons Pericytes
macrophages
inhibitory interneurons
Cerebrum projecting excitatory Cerebrum projecting inhibitory
Vascular endethelial cells
CA3 neupons
Olgnndensrrccylee
GABAeriic nourons
Ficilal glua celos i8tom odis)
Internedate progentor cels joydlng)
Input
Denoised
Mature-Astro
Immature-Astro
RGL
Young-RGL
nIPC
Per-iiPC
OPC
MOL
NFOL
Cajal-Retzius
Nouroblest
Inmatur-GABA
Mature-GABA
Immatura-GG
Mature-GC
Immature-Pyr
CA3-Pyr
types and developmental trajectories. top two principal components are visualized. Top panel: results for (Zeisel al. 2018). Bottom panel: results for (Hochgerner al., 2018b), Astro: astrocytes, RGL: radial glial cells, IPC: intermediate progenitor cells, OPC: oligodendrocyte precursor cells, MOL: mature oligadendrocytes; NFOL: newly formed oligodendrocytes, GABA: GABAergic neurons. GC: granule cells, Pyr: pyramidal neurons.
Figure Denoising single-cell RNA-seq data with ICM improves resolution
8

---

Inverse Flow and Consistency Models
ation problems. Practically, we extended the applicability of denoising without ground truth to almost any continu- ous noise distributions. We demonstrated strong empirical results applying inverse flow. A limitation of inverse flow is assuming prior knowledge of the noise distribution, and future work is needed to relax this assumption. We expect inverse flow to open up possibilities to explore additional connections the expanding family of continuous-tin generative model methods, and the generalized consistency training objective will expand the application of consistency models.
Joshua Batson and Loie Royer. Noise2Self: Blind Denoising by Self-Supervision, June 2019.
Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural Ordinary Differential Equations, December 2019.
Giannis Daras, Kulin Shah, Yuval Dagan, Aravind Gol- lakota, Alexandros G. Dimakis, and Adam Klivans. Am- bient Diffusion: Learning Clean Distributions from Cor- rupted Data, May 2023. URL abs 19256, arXiv:2305. [cs, math].
Impact Statement
Giannis Daras, Alexandros G. Dimakis, and Constantinos Daskalakis. Consistent Diffusion Meets Tweedie: Train- ing Exact Ambient Diffusion Models with Noisy Data, July 2024. URL http: /ttp:///aaivv..// org/ absa 10177 arXiv:2404.10177
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising Diffusion Probabilistic Models, December 2020.
Acknowledgements
The authors gratefully acknowledge funding support from the National Institutes of Health under grants DP2GM146336 (J.Z.), well from the Cancer Preven- tion and Research Institute of Texas under grant RR190071 (J.Z.).
Hannah Hochgerner, Amit Zeisel, Peter Lönnerberg, and Sten Linnarsson. Conserved properties of dentate gyrus neurogenesis across postnatal development revealed by single-cell RNA sequeneing. Nature Neuroscience, 21 (2):290-299)) 2990 February 2018a. ISSN 1546-1726, doi: 10.10386441593-017-00566
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---

Inverse Flow and Consistency Models
A. Appendix
A.1. Alternative forms of IFM and ICM
Here we provide the details of alternative objectives and corresponding algorithms of IFM and ICM which are casier and flexible to use.
A. 1.1.. ALTERNATIVE OBJECTIVES OF IFM AND ICM
We define the alternative objective of IFM similar conditional flow matching (Eq.
3)
||| v[(x)) u, (x, ODEY ( x1
LIPM (0) = E
1.p(xx)) =x=00 ((x))) /xxx
(14)
where x sampled from the conditional noise distribution. As described in Section 2.1.1 i (x Xo, x1) chosen any smooth interpolation between x and x1. such as u, (x Xo, x()
can be easily
x
Xo.
Since ICM is based on generalized consistency training, we first provide the alternative objective of generalized consistency training
LOCT(O) (0) pxx. xi) (X stopgrad (Co(Xx,, 4])
(15)
X1st1 Xt, M4.43 (Xta+1 Xo, x1) - 4.)
where the conditional flow defined jointly by p(x1 xo) and Us (x X0,
Then the alterntive form of ICM can be defined
(f)
I6 s.pp))))) (x) , (x. 1x
[X)co(Xe,)1 (441) stopgrad (x))
(16)
Xt+t1 Xt; Ut (Xe,41
Xo, x (le41 -t,) 4))
where ux(x Xo, x() can be freely defined based on any interpolation between xo and x1,, which is more easily applicable to any conditional noise distribution:
A.1.2. ALTERNATIVE ALGORITIMMS OF IFM AND ICM
Here we show the algorithms of alternative objectives of IFM (Eq. 14] and ICM (Eq. 16
Algorithm 4 ICM Training v2.
Algorithm IFM Training v2.
1: Input: dataset D, initial model parameter 0, learning rate 7) and sequence of time points 0 = t1 < t2 iN repeat
1: Input: dataset D. initial model parameter 6, and learning rate
2: repeat
Sample X1 and U[1, 1]
3: Sample X1 D and ~ 4[,0
Xo t- stopgrad (ca(x1, (x1,1)) 1))
4: Xo + stopgrad (ODEY
5:
Sample x² p(x{ xo)
6: Sample X ~ p(Xem. x()
5: Sample x1 p(x xo)
6: Sample p(x: Xo, x()
X1;
[(0)=
Xt:11 u (X Xo, (1i 4:) L(A)
7:
|| (x,) Ui (xe|
Xo. xo,x
+ |co (C 14+1), stopgrad (co(xe,, [4)]
8:
8:
- 0 (B)
9:
until convergence
V)(() 0
10: until convergence
12

---

Inverse Flow and Consistency Models
A.2. Proofs
A.2.1. INVERSE FLOW MATCHING
Theorem 1: Assume that the conditional noise distribution p(x, xo) satisfies the condition that, for any noisy data distribution p(x1) there exists only one probability distribution p(Xo) that satisfies p(x1) J p(x1 xo)p(xo))xx, then under the condition that LIFM O, we have (xxu) p(xu).
Proof:
The inferred data distribution is given by the push-forward operator (Lipman et al. 2023):
(xx)) = [ODDY" -u
* p(x1)
(17)
which is defined for any continuous normalizing flow @ from x1 to Xo in the form of
T8
[+] * p(xi) = p (xx)) det
(xo)
(18)
0x
where = (xo) The inferred noisy data distribution ((x1) is given by
((x1) = / p(x1 xo xx)q(xx)dxx
(19)
Under the condition LIFM O, we have
(xxo) = [ODEY", i-0 ((x1)
(20)
Then find that
ODEY p(x1) (ODEY_0) g(())
(21)
By the definition of the push-forward operator, we have
① (ODEY )*
)
p ((DDET".) (xx)) det
(xo)
Ox
(22)
① (ODEY".)
=9 ((ODEY"2)) ) (xx)) det
(xo)
Õx
Since the solution of ODE is unique, ODEY a bijective function with (ODEY".)) ODEY".
and
X = ODEY (1 = (ODEY-0) (xo)
Also, the nontrivial solution ensures that the determinant is non-zero. By substitution, get
p(x1)= = =x11)
(23)
and combine with Eq. we find that
p(x1) = / p(x1 x xx)q
(24)
We close the proof by directly applying the uniqueness of p(xo) and find that
g( (x)) = p(xo)
(25)
13

---

Inverse Flow and Consistency Models
Remark 1:
Our proof utilizes the one-to-one mapping property of neural ODEs Kidger 2022 diffeomorphic map between the noisy data distribution and the inferred data distribution.
which guarantees
Remark 2: Many inverse problems are ill-po ill-posed and impossible to address perfectly. The assumptions in our Theorem I provides guidance on the conditions under which IFM can recover the ground truth distribution. These conditions are:
Knowledge of the noisy data distribution: Either directly or through access to sufficient noisy data.
Transformability of p(x1): The noisy data distribution p(x1) must be transformable from p(xo) via ODE, This condition accommodates nearly any continuous noise distribution but excludes ill-posed transformations that lack one-on-one mapping between p(xo) and p(x1) (e.g., transformations like converting color images to grayscale).
Lemma 1: Given conditional ODE vector field u, Xo: X1) that generates conditional probability path p(x; Xpx x1), the unconditional probability path p(x,) can be generated by the unconditional ODE vector field u, ux(x), which is defined as
n (x) = (xpxoxx1 [u,(x Xu, X1]]
(26)
Proof:
To verify this, we check that p(x;) and u, (x) satisfy the continuity equation:
d
di p(x)) + div (xx(x))(x;)) =
(27)
By definition,
p(xi) dt p(x)|x0, x1)p(xo, x1 Jdxodx1- dt
(28)
With Leibniz Rule we have
r(x)) = / px x1)p(xo. x1 x1dxxdd11
(29)
dt
Since u, (x x1) generates pl X No, x1), by the continuity equation we have
d. xx Xo. x1) + div (u, x (xx x1)) = 0.
(30)
Substitution in Eq. 29 gives
di p(x)) = =- f div (x/xxx x1)p(x//xos x1)) p(xo, x1 dxxddxx.
(31)
Exchanging the derivative and integral,
d di p(xi) -div / uu (x)xxxxxx))))xxx
X1 )dxxdxx1
(32)
The definition of (x) is
f ux(xx xo. xi) p(x, (xo. x1 (p())))xoo. x1)
u, [p(x [x,(x] xo. x1)]
dx
(33)
p(x:)
Combining Eq. 32 and Eq. 33 gives the continuity equation:
d
dt p(x.) + div (ux(x)((x)) = 0.
(34)
14

---

Inverse Flow and Consistency Models
A.2.2. GENERALIZED CONSISTENCY TRAINING
Without loss of generality, we provide the proof for the form of LGCT in Eq. 15 and the proof for the form Eq. I0 follows by assuming that the forward conditional probability path is independent of x1.
Theorem 2: Assuming the consistency function co L) is twice differentiable with bounded second derivatives, and xp(x))))))) (x Xu, X1 x1]ll] < oo, up to a constant independent of 0, Leer and LCD are equal.
Proof:
The proof is inspired by (Song al. 2023). We use the shorthand Cg- to denote the stopgrad version of the consisteney function c. Given multi-variate function h(x, y), the operator 0₁ h(x, and 0zh(x, y) denote the partial derivative with respect and y. Let At == max; 1441 t and we use of At) denote infinitesimal with respect to At.
Based on Eq. 55am and Eq. 4 the consistency distillation objective is
LCD(O) = (x.,((x,xx)))xxxxx xx),))) ) {dl [co (X. (x)((x,11) (441); Co (Xe, 4)]
(35)
where X1, = -X4411 (fs-1 - & U (Xt..) ) and d general distance function.
We assume and Cg- are twice continuously differentiable with bounded derivatives. With Taylor expansion, we have
(f) = Ex,p(xxx1) x1) ((xe44 xx.x,) {d [co X (X 1411), ca- (xe. [,)]}
Ex.p(xn p(xx) xxxx)) {d (X lit1), ca- (x--(xe4 (li+1 - (s) (xt=-1); 4,)]
Px, x1)) xxxx) {d (Xt; li+1), Co (444111141)
-gico (X (1; ta) 46.4 (Xt41)
-Ū2ca- (Xa (4 1.) + o(A()}}
(36)
-Ex.p(xnxx xxxx {d [co li+1), ca- (x: (441)) }]}
p(xx x1) p(x; x1) { O2d (co(x (411),4))), co (X4,)) k+1)]
cr-c-- (X).. 1+ (1:+1 - to u (Xter))
Ex x )( x1)) 02d co (x ( Co- (444))441))
[a2co- (X (441) - +)]} + [o(Att
Then, we apply Lemma 1 and use Taylor expansion in the reverse direction,
(O)
(x x;) { tx-1x) co (xturs
- -E p1 x { (O2d ca li41), ca- (Xtr (ir1)
O Ca- (Xees 44 (1441 1,) E,)x [Mece (Xemn Xx, x1)]
Ex.p(xxxx)))xx.... xx.x)) {82d (X (411), co (8t41st441) [ (X l4p (144 4.) [o(A))
(
E, x(xo xx)))x))))(( ( ) {d co (o t441), ca- (xi41 ((t2))
xx, ((x (x))))) { (2g [co (x tv41), cy- (xi, t(+1))
[0,co- (Xea l (1+41 ti) Ut (x xo, x1)]}
Ex,p(xo.x) ((5. xxxx)) [32d [co (Xteis t++1): Ca- t
[ (Xuun let1) (1 [4)] [((()))
plxo, piple (x)))) { (d(co (x(xxx_,) Co (141)
15

---

Inverse Flow and Consistency Models
O,co- (41)) ue) (X No: x1)
- (Xe lit1 (li+1 44) + o(Di)}}
= (1) (x)xx) {d [co (xesn 1i11); Cg. (Xt,- - (4++1 - 4 (Xtet) Xo, (1,
+((tt
CGCr (0) (Att
(37)
where (i) is due to the law of total expectation.
Remark 3: Generalized consistency training enables us to extend the application of consistency models to any forward diffusion processes or conditional ODE including those that introduce non-Gaussian noise. For example, in the Dirichlet Diffusion Score Model (Avdeyev et al. 2023). the forward diffusion process is multivariate Jacobi process which transforms one-hot encoding of discrete data (e.g.. DNA sequences) into Dirichlet stationary distribution. Such diffusion processes are not supported by the original consisteney training approach but feasible with generalized consistency training. We leave further applications of generalized consistency training for future work.
Assuming the consistency function co(x, t) is twice differentiable and 0cp(x, t)/Ox is almost everywhere nonzero, when the inverse consistency loss LICM = 0, there exists a corresponding ODE vector field vß (x) that minimized the inverse flow matching loss LIFM
Lemma 2:
Proof:
When the inverse consistency function is minimized O, have
0 Licas(0) = E ( (100
(cco (441) stopgrad (co(xe,: (4)))
(38)
X(++ Xt, Uuet (X4.41 xo) t4)
which is equivalent to
co (x (+11) co(xx,,11)
d1 cx 1(41) -xt,) + O2cs (Xt,4) t++1)(t4+1 t,) co t411) co(X Ōi (Xt+4) u (Xt++ (t)((t+1 - t,) dgca (Xi..) (++41(1+1 tt),
(39)
Then we have the connection between the learned consistency function co f) and the conditional ODE vector field (x(x Xo) where (x l) is substituted for Xo:
0ica(x, l) (x cx(x1, 1)) 32co(x =0
(40)
Inspired by the above result, construct ODE vector field as
(x) = (2c)(x.t)
(41)
(x
where 0, 0co/0xx f 0 almost everywhere and for all (x, 1) where O co (x, = O, we define vx()
Now we show that va (x) minimized the inverse flow matching loss LIFM to
(0) E, t.p((x)) (xxxxx))))xxxx (x(x)) |19((x)) ur (x, (ODEY-=((1))
(42)
Firstly, we argue that ODEY" o = co(x1, 1), which can be proven by noting that the consistency function co(x, () maps every point along the ODE trajectory to the same point, Consider an N-step ODE and two consecutive points along the trajectory, say Xt, ODEY (x1) and X ODEY (x1). We have
(x(,, 1.) = co(xx.) - vo
(4++)) 42)4))
co (xt441)
(43)
-0,cp (Xessss 1441) Vo (8em)))44 14) - -O2co (Xc (41)) - 4))
= -Co(Xtur,s
16

---

Inverse Flow and Consistency Models
where derivative terms eliminated by the definition of v.. Further applying the boundary condition of the consistency function,
co(X1, 1) = co [xxx (N)
x X0
(44)
ODEY (x1)
where fn O. Secondly, substituting v2 (x) into LIFM, we get
(xc,(x(,t) t) (xc,(xx,1)
(0) =B (xp(x)) x0
(x: OODE
(45)
Multiplying the terms inside the norm by 01cy (xcg(xx, (xx, t) gives Eq. 40
(2c,(x,,t) u xx ODEY d (xco(x,,tt
02co(xx, t) un (xe co(x1, 1) 0 (x,,,)
(46)
Thus, (0) = 0 given the constructed v.(x).
Remark 4: The assumption that (x,(x, /0 * almost everywhere guarantees that the division in the construction of (x) is valid, preventing singularities except set of measure zero, More importantly, in neural ODEs, the ability uniquely map states forward and backward in time is essential for defining continuous and invertible transformation. The assumption ensures that the flow remains invertible almost everywhere, preventing singularities where trajectories might merge or become non-invertible. This property is crucial for ensuring that the learned dynamics remain well-posed.
A.3. Introduction to denoising without ground truth
The most comparable approaches to our method are those that explicitly consider a noise distribution, including Stein's Unbiased Risk Estimate (SURE)-based denoising methods (Soltanayev & Chun 2018 Metzler et al. 2020) and Noise2Score (Kim & Ye 2021). SURE-based denoising is applicable to independent Gaussian noise and Noise2Score is more generally applicable to exponential family noise. SURE-based denoising directly optimizes a loss motivated by SURE which provides an unbiased estimate of the true risk, which a mean-squared error to the ground truth. Noise2Score uses Tweedie's 's formula for estimating the posterior mean of exponential family distribution with the score of the noisy distribution. The score estimated by an approximate score estimator using denoising autoencoder.
Another family of approaches often referred to as Noise2X is based on the assumptions of centered (zero-mean) and independent noise. Noise2Noise (Lehtinen et al. 2018) requires independent noisy observations of the same ground truth data. Noise2Self (Batson & Royer] 2019) is based on the statistical independence across different dimensions of the measurement, such as the independence between different pixels. Noise2Vois (Krull et al. 2019) leverages the concept of slind-spot networks, which predict the value of a pixel based solely on its surrounding context. Similarly, Noise2Same (Xie) al. 2020) employs self-supervised learning using selectively masked perturbed regions train the model predict unobserved values. Both of them assume independence of noise across dimensions.
A.4. Experimental details
All experiments were conducted server with 36 cores, 400 GB memory, and NVIDIA Tesla V100 GPUs. All models implemented with Py'oorch 2.1 (Paszke al. 2019) and trained with the AdamW (Loshchilov & Hutter 2019 optimizer. Hyperparameters for both our method and basclines were chosen via systematic grid searches and 5-fold cross-validation. Model architectures and training hyperparameters are listed in Table A.4
1. TRAINING DETAILS
To train IFM or ICM, we first consider a discretized time sequence == t < to < small positive value close to O. We follow Karras et al. 2022 to determine the time sequence with the formula
< ty = 1, where e is a
17

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Inverse Flow and Consistency Models
Table 2. Model architectures and hyperparameters
dataset
architecture
channels
embed umbed_dim embed snaled_scale epochs
Ir
Ir schedule
Navier-Stokes
2000
5 x 10
[256,256, 256,256]
8 8-gaussians
MLP
256
1.0
2000 5 x 10 -4
None
Single-cell
1000
1 x 10-4
Gaussian noise
3000
1 x 10-1
StepLR
Correlated noise Jacobi process
[128,128, 256,256,512]
1000
1 X 4
None
UNet
512
1.0
1000
1 X 4
None
FMD
3000
1 10-4
StepLR
1P +
1-1 N-I (T2/P - 2101)² where p= 7,7 1, and N = 11. We choose the conditional ODE vector field as
u, (xx, Xp. x1 ) = X Xo.
(47)
Further, the gradient of the inferred noise-free data X6 is stopped to stabilize the training process, which is
X = stopgrad (ODEY
(48)
for IFM and
X = stopgrad (c.( (x1, 1))
(49)
for ICM. For ICM, the loss is weighted by
1(1) = ti+1 - 4
(50)
in the same way as (Song Dhariwal 2023).
A.4.2. SYNTHETIC DATASETS
We adopted a simple form of StoiersStokes equations which only includes the viscosity term in the fluid mechanies
dv pl at Vv) V + 2
(51)
0
where p is the density of the fluid, v is the velocity, is the pressure and is the viscosity coefficient. For inverting the Navier-Stokes simulations, we simulated the fluid data within a 2D boundary of [0, 1] x [0. domain from ( = 0 to f. = 0.1 with the spectral method (Spalart et al. 1991). For the upper simple case shown in Figure 2 the initial flow vector field was chosen as:
V, sin sin(y
(52)
Y, sin 4Tw)
For the bottom complex case, the initial flow vector was constructed by creating a random stream function:
(1 = Σ A sin sin(kaa) cos(kgy)
(53)
where we choose = -00, A, ~ ~2(0, 2), Kg 400 10), and ky =[[0, 10). Then the flow vector field was defined as
v. Oy
(54)
2
Vy dr
We show the original prediction of flow fields in Figure 5
The 8- A-gaussians generated by adding independent gaussian noise (a 0..15) to 8 points whose coordinates are (0, 1), (0.
1), (1, 0), (-,,0), ( ( -,,,, v2 and 1600 points for testing.
v2
)
/2
v2
v22). The dataset is composed of 8000 points for training
2- 2
2
We used a simple MLP- blsed model architecture with Gaussian Fourier time embedding in Table A.4 All methods were trained with learning rate of x 10-4 for 2000 epochs. The model training took about 10 minutes.
18

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Inverse Flow and Consistency Models
Input
Ground Truth
IFM
ICM
1.0
0.5
000
-0.5
-1.0
Figure 5. Original Prediction inverting Navier-Stokes fluid dynamics simulation, color indicating horizontal velocity.
A.4.3. REAL-WORLD DATASETS
All models were trained using the BSDS500 training set with 200 images randomly cropped to the size of 256 x 256 and evaluated on the BSDS500 test set, Kodak, and Sct12 with images cropped to the same size at the center. We used the same UNet-based model architecture as (Lehtinen et al. 2018) with additional Gaussian Fourier time embedding listed in Table
A.4
The URL for each dataset given:
BSDS500 (Arbeláez et al. 2011 http:///wwww....e.e////////////a
bsds/
Kodak: sttps://EOt.us//rrpphico
Set12 (Zhang et al.
Gaussian noise is applied with
X = X +
(55)
where Xy the noise-free data, X a noisy observation, and r (0, o²1). We chose o 25 the experiments. All models were trained with the following setting. The total epoch was set to 3000. The learning rate was initialized to 10-4 for the first 1500 epochs and decayed 5 x 10-5 for the last 1500 epochs. The model training took about 1.5 hours.
Correlated noise is applied similarly to independent Gaussian noise. We adopt the method from (Mäkinen et al. 2020) with
7) = 3
g
(56)
where v ~ NO, TT and g a convolution kernel. We consider g in the form of
cos |r| exp
(57)
2
2a
in polar coordinates and determines the level of correlation. We generated the correlated noisy observation with o = 25
and a = 2. All models were trained with a learning rate x 10-4 for 1000 epochs. The model training took about 30 minutes.
Jacobi process takes the following form
dx [a x) bx]dt + sx(1 x)dw,
(58)
where 0 < x < 1, when x approaches 1, the diffusion coefficient converges to 0 and the drift coefficient converges or -b, keeping the
> 1>>>>
is the speed factor, and a > 0, b > 0 determines the stationary distribution Beta(a, (a, b). Note that
19

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Inverse Flow and Consistency Models
diffusion within [0, 1]. We used 1 and a b and generated the noisy observation X1 with Euler-Maruyama sampler to simulate the SDE from the initial value Xo- All models were trained with learning rate of 1 x 10 for 1000 epochs. The model training took about 1.5 hours.
A.4.4. DENOISING MICROSCOPIC DATA
The Fluorescence Microscopy Denoising (FMD) dataset published by (Zhang et al. 2019) was downloaded from https? //github. http://ihccco////dcciiiii....... iiw.eeeoiisssn--eeooeeeeeeeeeee We adopted the signal dependent noise model from (Liu
et al. 2013
g f+ uuu
(59)
to estimate the condition noise distribution where y is the noisy pixel value, f is the noise-free pixel value, 7 is the exponential parameter, and u and to are zero-mean random variables with variance of and o respectively. The variance of the noise model is
=²= ou o.
(60)
To estimate the parameters in the noise model, we split an image into 4 x 4 patches. We assume the variance within a patch constant and approximate the noise-free pixel values of the patches by the mean values. The parameters the noise model are estimated by the Maximuum-Lkkelhoon method.
We used the same UNet-based model architecture (Lehtinen et al. 2018) with additional Gaussian Fourier time embedding listed in Table A.4 The learning rate initialized 10 for the first 1500 epochs and was decayed to 10 for the last 1500 epochs.
A.4.5. DENOISING SINGLE-CELL GENOMICS DATA
The adult mouse brain dataset published by Zeisel et al. 2018) was downloaded from https: www ncbii n nih gov/ sraerr//a waw.a//RR1359666
The dentate gyrus neurogenesis dataset published by (Hochgerner et al. 2018a) was downloaded from https::// //t////wwww /www.wwwoooooooo ncbi........... mlm. nih. gww query/ace. hwtp.//////eeeeuee////..co.o cgi ?acc= GSE--4EE133333 and the neuron- and glia- related cells were kept for denoising. We preprocessed the datasets by the standard pipeline (Wolf et al. 2018) and then performed principal component analysis. We further normalized the datasets by scaling the standard deviation of the first principal component to 1. After that, we denoised the datasets using the top 6 principal components with 0.4. We used simple MLP-based model architecture with Gaussian Fourier time embedding in Table A.4 The model was trained with learning rate of 1 10 10-4 for 1000 epochs. The model training took about 5 minutes.
A.5. Additional experiments
We provide extensive experiments to measure how different levels of Gaussian noise, different noise level assumptions, and different combinations of noises affect performance. We adopted the same model architecture and training strategy as for FMD Table A.4
A. DIFFERENT LEVELS OF GAUSSIAN NOISE
We conducted experiments to evaluate the performance of our method under different intensities of Gaussian noise. We performed experiments from
50 and found that method is robust over all noise levels applied (Table
A.5.I.
Table Denoising performance for different levels of Gaussian noise measured by PSNR in dB
5
12.5
25
50
75
Input
Pred
Input
Pred
Input
Pred
Input
Pred
Input
Pred
BSDS500
34.15
37.56
26.19
31.85
20.17
28.16
14.15
24.98
10.63
23.33
Kodak
34.15
37.92
26.19
32.56
20.18
29.08
14.15
25.96
10.63
24.33
Set12
34.15
37.87
26.20
32.78
20.16
29.19
14.13
25.78
10.63
23.86
20

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Inverse Flow and Consistency Models
A.5.2. DIFFERENT COMBINATIONS NOISES
We considered additive Gaussian noise and multiplicative noise such as Gamma noise, Poisson noise, and Rayleigh noise, as well as their combinations and
RGB dataset. We followed the noise distributions introduced in Noise2Score (Kim& & Ye 2021| Xie al. 2023a). For combinations of multiplicative noise and Gaussian noise, we added Gaussian noises with o 10 to the individual multiplicative noise models. As shown in Table A.5.7 our method is robust over all noise type combinations we applied and superior to compared methods in most noise types.
A.5.3. DIFFERENT NOISE LEVEL ASSUMPTIONS
We conducted experiments on data with o Gaussian noise, but training and denoising with different noise level assumptions from 12.5 to 50. Shown Table [A.577 our method demonstrates stable performance within the range of = 25 35, indicating that overestimating the noise level has minimal impact on the effectiveness of the model.
A.5.4. DENOISING SMALL DATASETS
In scientific discovery, the amount of data available is often very limited. To evaluate the performance of our method on small datasets, we conducted experiments on the electron microscopy denoising dataset (Mohan et al., 2021). Since the original authors did not release the real experimental data, we used the simulated dataset they provided and added Poisson noise, which is the noise distribution in the real data according to their analysis. The dataset consists of 46 samples. The results indicate that our method is applicable small datasets and outperforms other approaches in this scenario (Table A.5.7). While diffusion model is known as being data hungry, our method is efficient on sample size because it does not involve training full generative model.
A.5.5. COMPARISONS TO SURE-RELATED METHODS
We evaluated our method against Stein's 's Unbiased Risk Estimator (SURE) (Metzler et al. 2020) and its generalized variant, G-SURE (Kawar et al. 2024), which are applicable to Gaussian noise with known variance. Specifically, we compared performance on the BSDS500 dataset using SURE and assessed G-SURE on CelebA (Tabel A.5.7), where the method was originally optimized. Across both settings, our approach consistently achieved higher PSNR, demonstrating superior denoising performance.
A.5.6. COMPARISONS BETWEEN IFM AND ICM
Based on our experiments, we observed that while IFM tends to yield a slightly higher denoising performance than ICM (Table A.5.7). with ICM 2x faster during training and 10x faster during inference. In our setup. IFM requires solving the ODE at every training step, which makes it slower, whereas ICM uses simulation-fri free consistency function that bypasses the need for iterative ODE evaluations. In other words, if computation time is not a major constraint, IFM might offer marginal performance improvements. However, for most practical applications where efficiency is important, ICM offers a nearly equivalent performance substantially lower cost.
A.5.7. INTEGRATION WITH GENERATIVE MODELING
Our methods can be seamlessly integrated with generative modeling to generate clean samples from the ground truth rather than the observed noisy distribution. Specifically, we extend the training timepoint to a higher faar so that Pt.... becomes pure noise. For timepoints below the noise level a, the training follows our original methods whereas for higher t, the training identical regular continuous-timg generative models like flow matching. We have conducted experiments on CelebA with Gaussian noise a 0.01 where training with this extended timepoint results in the generation of clean samples from pure noise (Figure b). These experimental results validate our claim that IF can be integrated with generative modeling.
A.6. Additional qualitative results
We provide additional denoising results of the real-world datasets. Since there is not explicit noise magnitude in the Jacobi process, we did not apply the SURE-based method (Metzler et al. 2020) to this task.
21

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Inverse Flow and Consistency Models
Table 4. Denoising performance for different noise distributions measured by PSNR in dB
Noise type
Input
Noise2 Void
Noise2Self
Noise2Score
Ours (ICM)
BSDS500
23.78
28.29
28.52
30.53
29.91
Poisson
Kodak
23.60
28.76
29.36
31.10
30.58
= 0.01
Set12
23.08
30.01
29.23
30.94
30.68
BSDS500
26.75
29.17
27.43
31.14
32,48
Gamma
Kodak
26.67
30.26
28.26
31.67
32.97
k 100
Set12
25.53
30.44
28.54
31.21
33.08
BSDS500
14.03
28.57
14.86
30.37
30.55
Rayleigh
Kodak
13.95
29.73
14.83
30.96
31.16
0 0.3
Set12
12.81
29.98
13.74
30.89
31.17
BSDS500
22.40
26.45
27.76
28.54
29.26
Goisson+Gaussian
Kodak
22.25
27.67
28.86
29.02
30.02
Set12
21,88
27.81
29.23
29.10
30.03
BSDS500
24.29
27.98
26.10
29.34
30.53
Gamma+Gaussia
Kodak
24.24
28.99
27.08
29.90
31.22
Set12
23.62
29.53
26.84
29.69
31.27
BSDS500
13.85
28.01
14.72
29.36
29.79
Rayseigh+Gaussian
Kodak
13.77
29.12
14.69
30.12
30.49
Set12
12.78
26.81
13.59
29.82
30.50
BSDS500
20.17
29.72
27.33
28.28
29.99
GBussianRGB
Kodak 20.17
30.65
28.21
28.66
30.73
25
ImageNet
20.17
28.95
26.34
29.65
Table 5. Performance for different noise level assumptions
== 12.5
0 = 15
20
0
25
0
30
35
0 = 50
BSDS500
21.59
22.43
24.78
28.16
28.09
27.55
25.71
Kodak
21.62
22.49
25.03
29.08
28.99
28.43
26.66
Sct12
21.67
22.56
25.14
29.19
29.20
28.65
26.86
Table 6. Performance on the electron microscopy denoising dataset
Input Noise2Void Noise2Self
Ours (ICM)
PSNR
23.70
38.67
41.42
43.78
Table Comparisons SURE-related methods
Dataset
SURE
G-SURE
Ours
BSDS500
27.58
28.16
Kodak 28.23
29.08
Set12
28.95
29.19
CelebA
36.40
38.86
Table Comparison between IFM ICM
Input
ICM
IFM
BSDS500
20.17
28.16
28.33
Kodak
20.18
29.08
29.25
Set12
20.16
29.19
29.34
22

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Inverse Flow and Consistency Models
Figure 6. Generated samples from IF trained the noisy CelebA datasct
Input
Ground Truth
Ours (ICM)
BM3D
SURE
Noise2Self
Gaussian
Correlated
Jacobi process
NA
Figure 7. Denoising results of BSDS500 for natural images corrupted with three types of noise distributions. Methods compared are BM3D, SURE loss, Noise2Self, and ICM.
23

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Inverse Flow and Consistency Models
Input
Ground Truth
Ours (ICM)
BM3D
SURE
Noise2Self
Gaussian
Correlated
Jacobi process
NA
Figure 8. Denoising results of BSDS500 for natural images corrupted with three types of noise distributions. Methods compared are BM3D, SURE loss, Noise2Self, and ICM.
Input
Ground Truth
Ours (ICM)
BM3D
SURE
Noise2Self
Gaussian
Correlated
Jacobi process
NA
Figure 9. Denoising results of Kodak for natural images corrupted with three types of noise distributions. Methods compared are BM3D, SURE loss, Noise2Self, and ICM.
24

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Inverse Flow and Consistency Models
Input
Ground Truth
Ours (ICM)
BM3D
SURE
Noise2Self
Gaussian
Jacobi process Correlated
NA
Figure 10. Denoising results of Set12 for natural images corrupted with three types of noise distributions. Methods compared BM3D, SURE loss, Noise2Self, and ICM.
25


Reviews:
[{"review_id": "606CQzPOuW", "reviewer": "Reviewer_b7Av", "summary": "The authors provide a method to do inverse sampling of clean data p(x0) with having access to a distribution (data) of corrupted data p(x1) and assuming a knowledge of p(x1 | x0). They do not assume access to p(x0) at training time.\n\nThey propose an approach similar to the consistency model to do the trick.\n\nThey provide theoretical results and empirical evidence showing that their approach works better than some baselines.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for raising these important points. Below we address each point in the same order and numbering as in the review.\n\n1.**Regarding “Since the solution of ODE is unique…”**: In our framework we assume that the learned ODE (typically a neural ODE) satisfies standard conditions (e.g. Lipschitz continuity) that guarantee both existence and uniqueness of solutions. In Appendix, we clarified this by explicitly stating the necessary conditions [1][2]. We note that these are standard assumptions of all neural ODE including flow matching literature [3][4].\n\n2.**Assumption in Theorem 1** is natural in denoising—since without it, if any dimension of $x_1$ contained no information about $x_0$, no method could recover the clean signal. For example, consider the Gaussian noise model: $p(x_1|x_0)=N(x_1;x_0,\\sigma^2)$. The observed distribution is given by: $p(x_1)=\\int N(x_1;x_0,\\sigma^2)p(x_0)dx_0$. In the Fourier domain it becomes $\\Phi_{(x_1)}(\\omega)=\\Phi_{(x_0)}(\\omega)\\cdot \\exp(-1/2 \\sigma^2||\\omega||^2)$, and since the noise kernel is strictly positive for all frequencies, the mapping is injective. Further, considering that errors in estimating the transform may be amplified by $\\exp(1/2 \\sigma^2||\\omega||^2)$, if we assume that the clean distribution is band-limited to frequencies $||\\omega||<=W$, then the worst-case error amplification factor is $\\exp(1/2 \\sigma^2||W||^2)$. This bound shows that for moderate noise levels and naturally smooth signals, the recovery remains stable. In practice, many real-world denoising problems involve noise distributions—such as Gaussian noise with finite variance or other well-behaved noise models—where the Fourier transform of the noise kernel does not vanish. Thus, the assumption required by Theorem 1 is not restrictive. \n\n3. **Experimental Validity**: We have now provided additional details on how all baselines were tuned. We use the same model architecture for all methods (from [5]). In our experiments, hyperparameters for both our method and baselines were chosen via systematic grid searches and 5-fold cross-validation on the BSDS500 dataset. Therefore, the performance comparison is fair.\n\n4. **Choice of Baselines**. We argue that the baselines were carefully chosen and indeed the most comparable methods to our settings. All Noise2X methods also make assumptions about $p(y|x)$.  Except for Noise2Score, all other methods assume independence across dimensions and centered (i.e., zero-mean)  noise distributions. Noise2Score further requires exact knowledge of $p(y|x)$ and supports certain families of distributions such as Gaussian and Poisson,  and we showed that we outperform Noise2Score, as shown in Appendix A.5.2. We originally put the comparison in Appendix because Noise2Score loss function is specific to the distribution and not available for SDE (Jacobi diffusion). We will move the comparison to the main text.\n\nIn response to your comment, we have also added comparisons to SURE (assuming Gaussian noise) and G-SURE-diffusion (assuming linearly corruption followed by additive Gaussian noise, the same as ambient diffusion). We outperform both SURE and GSURE-diffusion in terms of PSNR, as shown below.\n| Dataset | SURE | G-SURE | Ours|\n|:---:|:---:|:---:|:---:|\n| BSDS500 | 27.58 | - | 28.16 |\n| Kodak | 28.23 | - | 29.08 |\n| Set12 | 28.95 | - | 29.19 |\n| CelebA | - | 36.40 | 38.86 |\n\nAs there are currently no alternative methods that support arbitrary noise distributions with the same generality as our approach, and we have now compared with most methods that do not require ground truth data and rely on assumptions of $p(y|x)$. Therefore, we feel that our choice of baselines is now both fair and justified.\n\n5. **Computational Cost**. We use the same model architecture for all methods. In our experiments, the training and inference cost of ICM is equal to that of the baselines. IFM, which simulates an ODE, incurs higher computational cost. For a detailed comparison, please refer to our response to the reviewer t72D. This part will be included in the revised manuscript.\n\n6.We clarify that “supervised method encountered overfitting” was considered as the explanation of the inferior performance in our benchmark. We agree that there are alternative possibilities and will remove this statement. \n\n7.We thank you for the suggestion on improving the text, we will add an inverse problem example at the start of Section 3, and will improve the flow in the Experimental section.\n\nWe hope that these clarifications and additional experimental results address your concerns.\n\n[1] Kidger, On Neural Differential Equations, 2022\n\n[2] Song et al., Consistency models, 2023\n\n[3] Lipman et al., Flow matching for generative modeling, 2023\n\n[4] Tong et al., Improving and Generalizing Flow-Based Generative Models  with Minibatch Optimal Transport, 2024\n\n[5] Lehtinen et al., Noise2Noise, 2018", "claims_and_evidence": "I expand on claims in “Theoretical Claims” and in “Experimental Designs or Analyses”.\n\nOn a high level:\n\n* The authors claim that their approach is theoretically justified (which might not be true). The assumptions for their method seem to be too strong and one line in the proof might not be valid.\n\n* The authors claim that their method outperforms the baselines empirically, but the choice of baselines and validity of experiments raises some doubts.", "methods_and_evaluation_criteria": "The main evaluation criteria the authors chose is to run experiments on inverse problems (i.e., denoising) and compare the empirical results to baselines.\n\nI have significant doubts about the choice of baselines and about the validity of results (see below “Experimental Designs or Analyses”) – the chosen baselines are not built for the same setting, there is no information about how the baselines were optimized, there is no information about hyperparameters optimization.\n\nThe authors also have some theoretical claims, but I also doubt whether the assumptions are realistic and whether the theoretical claims are actually valid (see below).", "theoretical_claims": "The authors provide the proof for the theorem, in line 697 in Appendix, the authors say that “Since the solution of ODE is unique,....”, this is not obvious at all. In ODE literature, proving that the solution exists and is unique is the core of theoretical analysis. Moreover, the exact form of ODE is not specified.\n\nIn Theorem 1, the authors suggest that `p(x_1|x_0)` must satisfy the condition that any noisy data distribution `p(x_1)` there exists only one probability distribution `p(x_0)` that satisfies p(x_1) = \\int p(x_1 | x_0) p(x_0) dx_0. It seems that this assumption is very strong. The authors should at least discuss how restrictive this assumption is for the model class they try to study. For example, if I take a noise distribution p(x_1 | x_0) which is factorised p(x_1 | x_0)=\\prod_{i} p(x^i_1|x^i_0), such that for some i, we have p(x^i_1|x^i_0)=p(x^i_1), it would not uniquely determine p(x_0).\n\nThe other theoretical claims seem valid (I haven't checked the proofs in detail), but they are not as significant as Theorem 1.", "experimental_designs_or_analyses": "I have a few doubts about the experimental validity and chosen baselines.\n\nMajor:\n\n* Experimental validity:\nThe authors do not provide any experimental details on how the baselines were tuned. Moreover, The authors do not provide any information on how the hyperparameters for their method were chosen and how many experiments they run to make it work.\n\n* Choice of baselines:\nIn the conclusion, the authors write “A limitation of inverse flow is assuming prior knowledge of the noise distribution, and future work is needed to relax this assumption”. It bears the question whether the baselines were chosen appropriately.\n\nThe authors focus their comparison on  Noise2X approaches which as far as I understand, do not require the knowledge of p(y | x). The authors focus efforts on comparing performance to these approaches, which is fundamentally unfair, because the authors approach uses more information (i.e., p(y|x)).\n\nThe authors in A.3 claim that \"The most comparable approaches to our method are those that explicitly consider a noise distribution, including Stein’s Unbiased Risk Estimate (SURE)-based denoising methods (Soltanayev & Chun, 2018; Metzler et al., 2020) and Noise2Score (Kim & Ye, 2021).\". Moreover, they mention “Ambient diffusion and GSURE-diffusion” baselines but do not compare them. They could add a comparison to these even if the data is linearly corrupted, to see whether the proposed method behaves similarly as in other experiments.\n\nThe authors write:\n“Further analysis revealed that the supervised method encountered overfitting during the training process, which led to suboptimal performance. In contrast, our method did not exhibit such issues, highlighting the superiority of our approach.” – this claim is not supported by any empirical evidence. No links to the appendix.\n\nThe results of the proposed method are very close to “Supervised” and to Noise2X baselines depending on the noise process. It would be useful to have a more detailed explanation on why this method performs better than baselines.\n\nIt is not clear what the computational cost of the proposed method compared to baselines?", "relation_to_broader_scientific_literature": "I think the paper adequately discusses existing literature.", "essential_references_not_discussed": "No concerns.", "other_strengths_and_weaknesses": "No other comments.", "other_comments_or_suggestions": "For unfamiliar audience, it would be useful to add an example of an inverse problem in Page 3 when Section 3 begins, especially when it makes sense to have access to both, x1 and noising distribution p(x1|x0).\n\nThe list of noises (1-3) seems a bit out of place with the rest of the text (in Experimental section).", "questions_for_authors": "No further questions.", "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": {"ethical_review_concerns": "No concerns."}}, {"review_id": "ZTHtJcnMns", "reviewer": "Reviewer_t72D", "summary": "The authors present two novel methods, inverse flow matching (IFM) and inverse consistency models (ICM), for unsupervised denoising based on flow matching and consistency models.\nFor training, both methods require only noisy data as well as a statistical model for the measurement  noise.\nIFMs learn a vector field describing an ODM that can through iterative application move a noisy data point to the clean data manifold.\nIFMs are trained in a bootrapped fashion by (i) using the current vector field model to derive a clean datapoint form a noisy one, (ii) applying noise according to the noise model, and (iii) using interpolation to derive a vector field whichis then regressed by the learned vector field.\nIn ICMs, a consistency function mapping noise and intermediate datapaoint to clean targets is trained bootstrapped in s similar way, using the current model to produce clean data and the noise model to create noisy versions and intermediate steps, which then serve ar training data for the consistency loss.\nBoth methods are evaualated on a number of different denoising problems and perform favourably, compared to baselines.\n\n\n## update after rebuttal\n\nI appreciate the insightful response of the authors.\nI will increase my rating to accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your encouraging feedback on our approach. We address each of your points in detail below.\n\n1.**For the RNA-seq experiment**, we performed denoising in a linear latent space transformed by PCA. We expect the noise in the latent space to be very close to Gaussian distribution due to central limit theorem, and have empirically verified it by bootstrap resampling of the scRNA-seq reads. We will add more details about the noise model to the revised manuscript.\n\n2.**“seamlessly integrate with generative modeling”**: we have now backed up this statement with new experimental results. We have conducted experiments on CelebA where training with this extended timepoint results in the generation of clean samples from pure noise. Specifically, we extend the training timepoint to a higher $t_{max}$ so that $x_{t_{max}}$ becomes pure noise. For timepoints $t$ below the noise level $\\sigma$, the training follows inverse flow whereas for higher $t$, the training is identical to regular continuous-time generative models like flow matching. These experimental results validate our claim that IF can be integrated with generative modeling. We will include this additional experiment and visualization of results in the revised manuscript to further support this point.\n\n3.**Interpretation of the denoising result**: Indeed our method is distinct from methods regressing an MMSE prediction of clean image like N2V/S. Specifically, we learned an ODE that simultaneously performs denoising and serves as a mapping between $p(x_0)$ and $p(x_1)$, therefore the denoising output has to be a valid sample from $p(x_0)$ and a clean image rather than a blurry image. This property is similar to generative sampling from flow matching or probability flow ODE models.\n\nFor the interpretation of the denoising result which is not MMSE, we can provide two perspectives: the first perspective is proven in our Section 3.1 Lemma 1, where we showed that the learned ODE vector field at any time $t$ points straight toward the expectation of the ground truth $x_0$. At high $t$ the expectation is similar to MMSE prediction which is a blurry image, but as $t$ reduces, the expectation gradually becomes a clean image.\n\nThe second perspective is by transport cost.  Our method leverages a linear interpolation to define the conditional ODE vector field, which has been proven [1] to induce an ODE-based coupling between $p(x_0)$ and $p(x_1)$ that reduces the transport cost between the noisy and the clean data distribution (compared to the training coupling based on the conditional noise distribution). Therefore, our method reduces the transport cost in the mapping from noisy to clean data, thus we obtain a sample from the clean image distribution that is close to the noise image. \n\n4.**Comparison between IFM and ICM**: Based on our experiments, we observed that while IFM tends to yield a slightly higher denoising performance than ICM (PSNR in the table below), with ICM 2x faster during training and 10x faster during inference. In our setup, IFM requires solving the ODE at every training step, which makes it slower, whereas ICM uses a simulation‐free consistency function that bypasses the need for iterative ODE evaluations. In other words, if computation time is not a major constraint, IFM might offer marginal performance improvements. However, for most practical applications where efficiency is important, ICM offers a nearly equivalent performance at a substantially lower cost.\n| Dataset | Input |  ICM  |  IFM  |\n|:-------:|:-----:|:-----:|:-----:|\n| BSDS500 | 20.17 | 28.16 | 28.33 |\n|  Kodak  | 20.18 | 29.08 | 29.25 |\n|  Set12  | 20.16 | 29.19 | 29.34 |\n\n5.**Apply to discrete noise**: While our theoretical derivations assume continuous noise distributions for analytical clarity, our method can be extended to discrete noise models. In practice, one can apply dequantization techniques to convert discrete measurements into a continuous representation. In fact, even without an explicit dequantization step, treating the discrete data as if it were continuous already yields strong denoising performance. As evidenced in Appendix A.5.2 (Table 4), our experiments on Poisson noise demonstrate that the method performs robustly under these conditions.\n\n6.We thank you for highlighting the VAE-based approaches which is another direction for utilizing generative models for denoising. We will discuss the distinctions of our approach in the final manuscript.\n\n7.We thank you for bringing up applications to deconvolution and denoising. Our method is indeed well-suited for deconvolution and deblurring, with and without additional correlated noise, as these operations can be easily represented via conditional ODE. \n\nWe genuinely appreciate the reviewer’s time and insights. We welcome any additional feedback or questions. Thank you again for your thoughtful evaluation.\n\n[1] Liu et al., Flow Straight and Fast:  Learning to Generate and Transfer Data with Rectified Flow, 2022", "claims_and_evidence": "I will discuss the most important claims below:\n\n**\"A main contribution of our approach is generalizing continuous-time generative models to inverse generation problems such as denoising without ground truth\"**\nThe authors provide the theoretical justification and derive two such methods and validate them experimentally.\n\n\n**\"IF can be seamlessly integrated with generative modeling to generate samples from the ground truth rather than the observed noisy distribution.\"**\nUnfortunately, this claim is not validated. It is not clear how exactly this would work.\nIn order to function as true generative models, the presented methods would have to map pure noise at x_1 to clean data points at x_0.\nWhile it is conceivable that this works it is not obvious and not discussed or experimentally validated.", "methods_and_evaluation_criteria": "I think the selected datasets make sense.\nI especially appreciate the fact that the authors show the applicability of the methods in so many different denoising tasks.\nWith respect to the denoising task, the selected evaluation metric (PSNR) makes sense.", "theoretical_claims": "I checked the background, derivations of IFM and ICM as well as the algorithms. I don't see any problems on the theoretical side.", "experimental_designs_or_analyses": "I believe the experiments are valid and sound. However, there are some open points and questions:\n\n* In the experiment regarding denoising of RNA-sequence data, it is not clear what noise model is used. How would could this be determined? Maybe I overlooked this?\n\n* I am missing a quantitative comparison of the two presented methods (IFM and ICM).\nMost experiments are only done using ICM. Is this due to the computational cost of IFM?\nI am missing a clearer analysis and discussion of pros and cons. It seems IFM is generally for expensive, but does it perform better?\nHow much more computation does it require? Does it perform better than ICM if time is not an issue?", "relation_to_broader_scientific_literature": "The methods are based on Flow Matching and Consistency models. This connection is clearly identified and described.\n\nThe presented methods allow denoising without clean GT, this is related to self-supervised methods like N2V and N2S, however, I think the relationship is not completely clear.\nN2V/S are regressing an MMSE prediction for the clean image given the noisy input that is they try to find the center of mass of the posterior of clean images given the noisy input.Even though this should be optimal wrt to PSNR, this can lead to blurry results for images that are very noisy, as the prediction is essentially a compromise between possible solutions.\n\nI believe this approach is quite different from the presented methods. I would love to see a theoretical discussion of this.\nHow should we interpret the denoising result of IFM or ICM? As MAP solution, or as a sample form the posterior or as MMSE? \n\nI believe the authors missed a line of work ([1,2,3] in next section) on unsupervised denoising. These methods allow sampling from an approximate posterior instead of producing a single output.", "essential_references_not_discussed": "The authors missed a VAE-based work [1,2,3,4]. \nThese works are trained using only noise data and a model of imaging noise (or learn is on the fly) and then provide a generative model for the clean data distribution, the noisy data distribution, and the posterior of clean images given a noisy image.\nI hope the relationship to this line of work can be discussed in the final version of the paper.\n\n[1]: Prakash, Mangal, Alexander Krull, and Florian Jug. \"Fully Unsupervised Diversity Denoising with Convolutional Variational Autoencoders.\" International Conference on Learning Representations.\n[2]: Prakash, Mangal, et al. \"Interpretable Unsupervised Diversity Denoising and Artefact Removal.\" International Conference on Learning Representations.\n[3]: Iwamoto, Yuichiro, et al. \"High throughput analysis of rare nanoparticles with deep-enhanced sensitivity via unsupervised denoising.\" Nature Communications 16.1 (2025): 1728.\n[4]: Salmon, Benjamin, and Alexander Krull. \"Unsupervised Denoising for Signal-Dependent and Row-Correlated Imaging Noise.\" WACV 2024", "other_strengths_and_weaknesses": "**Other Strengths:**\n\n* The problem of unsupervised denoising without clean ground truth is highly important especially for scientific imaging problems.\n\n* I think the approached are an elegant extension of the existing methods, turning a generative model into a denoisier.\n\n* I think the paper is well written. All concepts are introduced to the point in the background chapter.\n\n**Other Weaknesses:**\n\n* I think the computation time for IFM seems to be an issue. A comparision of the computation time would be good.", "other_comments_or_suggestions": "* The authors claim that there method is only applicable to \"continuous noise models\", but could it not for example be applied as well to discrete noise such as pure Poisson shot noise as well?\n\n* Typo after Eq. 2: should have a ',' at the end not '.'", "questions_for_authors": "* I am missing a direct comparison of the two presented methods.\n\n* How does the method relate to the references mentioned above?\n\n* Would it make sense to use the method e.g. for deconvolution/debluring?", "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": "MgpOltuC1d", "reviewer": "Reviewer_wSgT", "summary": "This work addresses the challenge of denoising corrupted data in the absence of ground truth observations with 2 proposed methods: Inverse Flow Matching and Inverse Consistency Model. \n\nIFM leverages a reverse flow process modeled by an ODE to transform noisy data towards a cleaner state, while ICM offers a more efficient solution by using a consistency function.  the key idea is train a denoiser $\\theta$ for both denoising and noising process, e.g., in IFM, KLD between $p_\\theta(x_t)$  and $q(x_t | g_\\theta (x_t) )$ while $g_\\theta$ provide a estimated $x_0$. \n\nHowever, I have some minor reservations with respect to ICM.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's detailed feedback. We address these questions one by one below:\n\n1.**Regarding recovering the distribution of $x_0$**, our theoretical analysis (see Theorem 1 and Appendix A.2.1) establishes that under appropriate assumptions our ODE-based inverse flow does indeed recover the full distribution $p(x_0)$ rather than merely its mean, which are well supported by empirical results including additional experiments which we will describe below.\n\n• By “the mean of all the observed noisy data”, if the reviewer means the overall dataset mean $E(x_1)$—averaging all noisy observations into one single point—this would indeed fail to capture any structure in the data. In our experiments (we are not allowed to upload figures, so we will describe the results in text and include them in the final revised manuscript), with a toy model where the clean data form two distinct circles, our method successfully recovers the two separate circles rather than converging to one average point.\n\n• If the reviewer refers to the conditional expectation $E(x_1|x_0)$, under centered noise distribution this equals $x_0$ and will perfectly recover clean data and its distribution, but when the noise is non-centered (as in a Jacobi process), the conditional expectation does not recover the clean distribution. In our toy experiment with a 1D distribution perturbed by a Jacobi process, our method was able to recover the clean data distribution rather than simply output the mean of the noisy observations which will produce a shifted distribution.\n\nIt is also important to emphasize that in our setting each sample has only one noisy observation, so averaging multiple observations to denoise is not an option.\n\n2.**Regarding the novelty of our method**: While our method indeed builds on the rich framework of continuous-time generative models, its key contributions lie in the formulation of the Inverse Flow framework and the development of the Inverse Flow Matching and Inverse Consistency Model algorithms. These contributions extend the scope of existing methods by enabling denoising without ground truth, accommodating any continuous noise distribution, and reframing the denoising task as an inversion of the forward noise process. It was not obvious that flow matching or diffusion can recover data from the earlier timepoint without requiring data for that timepoint and we showed to our knowledge the first general solution to this problem applicable to the entire family of continuous-time generative models.  Other works have attempted to address related questions without acheiving a general solution for all noise distributions. Thus, this novel perspective provides both theoretical insights and practical benefits that are not achieved by simply refining estimates of $x_0$.\n\n3.**Regarding validation on RGB datasets**. We would like to clarify that our evaluation spans a wide range of datasets including RGB images. In the Appendix A.5.2 (Table 4), we report RGB denoising results on the BSDS500 dataset, which clearly demonstrate the method’s capability in handling natural images. Moreover, we have now extended our experiments by conducting denoising on the ImageNet validation set. In these experiments, our method consistently outperformed the compared approaches in terms of PSNR, further underscoring both its effectiveness and scalability to larger, more challenging RGB datasets. We will include the results in the revised manuscript. We believe these additional results provided more robust evidence for the broad applicability of our approach.\n| Dataset  | Input | Noise2Void | Noise2Self | Ours  |\n|----|----|----|----|----|\n| ImageNet | 20.17 | 28.95 | 26.34 | 29.65 |\n\nFinally, we sincerely invite the reviewer to share any further comments or questions. We greatly value constructive feedback and are more than willing to provide additional clarifications or revisions in the manuscript. Thanks!", "claims_and_evidence": "In the author’s proof, the ODE appears to fulfill the claim made in the paper, namely, that it recovers the distribution of x_0. However, I question whether this is truly the case. In the context of unconditional generation, the prediction for x_0 could simply be the mean of all the observed noisy data, which would fail to recover an effective distribution. Would it be possible for the authors to conduct a set of control experiments, where in Algorithm 1, Line 4, x_0 is not derived from sg(ODE), but is instead the mean of all noisy observations? I believe this experiment could be crucial in determining the broader effectiveness of the proposed approach.", "methods_and_evaluation_criteria": "This work has been tested on toy datasets as well as some real-world datasets related to biology. However, I believe these tests fail to convincingly demonstrate the effectiveness of the proposed approach. The scale of these datasets is minimal compared to those of existing datasets, and the baseline comparisons are based on relatively old work.", "theoretical_claims": "please check my concerns about Claims and Evidence", "experimental_designs_or_analyses": "As previously mentioned, the experimental design in this paper fails to convincingly demonstrate its scalability.", "relation_to_broader_scientific_literature": "This paper primarily addresses the denoising of simple images, achieving acceptable high PSNR without relying on ground truth data. This approach could potentially influence a broad spectrum of image processing applications.", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "The novelty of the proposed approach is somewhat limited. Both the IFM and ICM are essentially predictions based on the existing frameworks, focused on refining the estimate of x_0.\n\nalso could I also expect the authors to conduct denoising experiments on RGB datasets, such as ImageNet, to further demonstrate the effectiveness of the proposed approach?\n\nthe last concern is about the effective of generated x_0 as I mentioned in previous section.", "other_comments_or_suggestions": "n/a", "questions_for_authors": "in this part I repeat my major concern:\n\n1. Would it be possible for the authors to demonstrate the validity of predicting x_0 by replacing the prediction with the mean of the noisy images, thereby supporting the claim that the approach recovers an effective data distribution rather than a shifted one?\n\n2. Would it be possible for the authors to present denoising results on RGB images?", "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:

Meta-review: The paper presents an method for unsupervised denoising based on flow matching and consistency models. The empirical results are broad and support the claims. There are number of issues raised by the reviewers that were mostly answered during the rebuttal.
Decision: Accept (poster)