Divergence-Free Neural Networks with Application to Image Denoising
Authors: Sébastien Herbreteau
Etienne Meunier
Venue: ICLR 2026 Poster (2026)
Keywords: image denoising
divergence
Stein's unbiased risk estimate
self-supervised learning
incompressible vector fields
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: We introduce a resource-efficient neural network architecture with zero divergence by design, adapted for high-dimensional problems. Our method is directly applicable to image denoising, for which divergence-free estimators are particularly well-suited for self-supervised learning, in accordance with Stein's unbiased risk estimation theory. Comparisons of our parameterization on popular denoising datasets demonstrate that it retains sufficient expressivity to remain competitive with other divergence-based approaches, while outperforming its counterparts when the noise level is unknown and varies across the training data.

Published as a conference paper at ICLR 2026
DIVERGENCE-FREE NEURAL NETWORKS WITH APPLI- CATION TO IMAGE DENOISING
Sébastien Herbreteau
Etienne Meunier
Univ Rennes, Ensai, CNRS, CREST- UMR 9194 F-35000 Rennes, France sebastien. weeennherrriee herbrrteeaueensal fr
Centre Inria de Paris Paris, France
et ieiieene... menne.mmunier@inrraa fr
ABSTRACT
We introduce resource-efficient neural network architecture with zero diver- gence by design, adapted for high-dimensional problems. Our method is directly applicable image denoising, for which divergence-free estimators are particu- larly well-suited for self-supervised learning, in accordance with Stein's 's unbiased risk estimation theory. Comparisons of our parameterization on popular denois- ing datasets demonstrate that it retains sufficient expressivity to remain competi- tive with other divergence-bel based approaches, while outperforming its counterparts when the noise level is unknown and varies across the training data.
INTRODUCTION
The divergence a scalar quantity that measures the rate at which a vector field "flows out" of an infinitesimal region of space. Formally, for weakly differentiable function f I - R", the divergence at point y E R" is defined as the trace the Jacobian matrix Jy(y):
0f: Oy: (y)
div f(y) & tr(J)())) Σ
In the special case where the divergence is zero everywhere, the vector field is said to be divergence- free solenoidal, indicating incompressible flow. One of the most famous example is without doubt the magnetic field, which, according to Maxwell's equations, has zero divergence (Maxwell 1873). Learning divergence-free vector fields is of particular interest at the interface of physics and machine learning (Richter-Power et al.) 2022 Raissi et al.) 2017a). as such fields naturally emerge systems governed by fundamental conservation laws. Parameterizations for learning often exploit the fact that, in R the curl of any vector field is divergence
(Morita 2001). or, more generally, draw on multidimensional extension via differential forms (Cartan] 1899] Richter- Rrichter-Powell et al., 2022). As long as the target functions remain low-dimensional, training can be performed efficiently with the help of an automatic differentiation engine (Paszke et al. (209)) that powers the computation of partial derivatives. However, scaling challenges arise quickly the dimensionality increases Miicheer-Powelle et al. 2022).
In this paper, establish representer theorem for divergence-free vector fields, based struc- tured combinations of conservative fields. Building on this result, and incorporating sparsity con- straints, show how this representation can inform neural network parameterizations that remain resource-efficient, thereby ensuring computational tractability in high dimension. With applica- tion to image denoising, for which divergence-free estimators are particularly well-suited for self- supervised learning, in accordance with Stein's 's unbiased risk estimation theory Stein (1981), we propose methodology to construct low-overhead network architectures that have zero divergence by design and which are adapted to image processing tasks. We demonstrate their competitiveness in comparison to other divergence-baseee approaches (Batson Royer (2019) Tachella et al.) 2025a Soltanayev & Chun 2018) for the removal of Gaussian noise without clean data.
In summary, the contributions of our work are as follows:
The establishment of a representer theorem for divergence- fields, on which we build construct neural network architectures that have zero divergence by design.

---

Published as a conference paper at ICLR 2026
2. A theoretical framework for analyzing self self--sueervied image denoising methods grounded in the principle of constant divergence.
3. The demonstration of the competitiveness of our approach in comparison with other constraint-based approaches, particularly when the noise level is unknown and varies across the training data.
2 RELATED WORK
Divergence-free networks are particularly studied within physics-informed machine learning and related scientific modeling tasks. which integrate physical laws into the training of neural networks to solve partial differential equations. Notably, enforcing incompressibility constraints is often im- especialyy in fluid dynamics, where velocity fields are required be divergence-free.
A common approach employs soft constraints by adding penalty terms to the loss fuucctin that encourage the predicted fields to be divergence-free (Raissi et al.) 201 Mao al., 2020, Jin et al. 2021). Although such penalty-based methods are straightforward to implement, they do not guarantee strict satisfaction of the incompressibilitty condition, and residual divergence can remain in some cases, particularly for complex or high-dimensional problems.
To overcome these limitations, recent works have explored hard constraints that enforce divergence- free properties by construction through network architecture parameterization. For example, in Raissi et al. (2017a), a stream function formulation is used in 2D to represent the velocity field as derivatives of scalar network output, which is analytically divergence-fres free. Extending this idea to the multidimensional case, |Richter- et al. (2022) designed networks that directly encode con- cesstraints-therele allowing modeling of flow fields
servation laws-including divergence-free and advected quantities without explicit divergence penalties. Nonetheless, scaling these models proves challenging due to their heavy reliance automatic differentiation. For example, the vector- field parameterization proposed by Brchter-Poweld et al. (2022) requires computing a Jacobian ma- trix, which becomes intractable as the dimension grows.
CONSTRAINT-BASBR APPROACHES FOR SELF- SULE-SUPERVISED DENOISING
We focus on denoising problems under the assumption of additive white Gaussian noise (AWGN):
y aTC,
(2)
where E R" is the noisy observation, x E IR" is the underlying noise-free signal drawn from distribution Pz, ~ N(00, I,.) is a standard Gaussian noise vector, and > the noise level, drawn from distribution Pa. Provided that sufficiently large dataset of noise-free signals æ is available, problem (2) is traditionally tackled a supervised manner by solving:
arg min y (y) x=122
that is, by finding the minimum mean square error (MMSE) estimator, which we denote fMMSE, Interestingly, MMSS admits closed-fome form expression whenever the noise level a is the same for all noisy observations y (i.e., Po is Dirac) which is given by Tweedie's formula (Efron 2011): (MMSE( = y + o²V log Pg()) In this latter expression, the optimal estimator SMMSE depends solely on the score of the distribution of the noisy data V log (g)) Accordingly, this formulation indicates that Gaussian denoising may be effectively addressed even in the absence of ground-truth data for training.
3.1 STEIN'S 's UNBIASED RISK ESTIMATE
Among all the methods proposed in the literature for tackling self-supervised denoising, the ap- proaches grounded in Stein's 's Unbiased Risk Estimate (SURE) (Stein) 1981] hold prominent place. This latter establishes a remarkable identity involving the divergence operator:
Ez.y ( 2 Ey.- [-no² + f 2 20 div f(y)]
(4)
provided that belongs L) the space of weakly differentiable functions. and under the assump- tion that (x)) is bounded. This result is particularly powerful, as it reveals that the mean square error can be reformulated depend solely on noisy observations and noise levels.
2

---

Published as a conference paper at ICLR 2026
When the noise level σ is known: Assuming that is known for every noisy training signal y, equation (4) allows to be rewritten in the following equivalent self-supe form:
arg min Ey. [1lf(y) 211 + 20 div f(y))
(5)
where the divergence term acts as a form of regularization, preventing the model from simply leamn ing the identity. Many traditional image denoisers-whin divergence admits closed-form expres-
sion-are are in fact rooted in SURE (Blu & Luisier, [0077] Van De Ville & Kocher 2009) Wang &
Morel 2013), even if this connection is not made explicit in some cases (Dabov et al. 2007 Le- brun et al. 2.13), as shown by Herbreteau & Kervrann (2025). However, when the estimator f is considerably more complex, such deep neural network, its divergence is generally intractable to compute exactly. A popular method is then to employ a Monte Carlo approximation of the diver-
gence, grounded in the following result (Ramani et al. 2008):
h f(y Th) f(y)
div (ly
lim E T-+0
T
provided that admits well-defined second-order Taylor expansion (if not, this is still valid in the weak sense provided that f is tempered. which is the case for networks with piecewise differentiable activation functions shown by Soltanayev Chun (2018)). practice, a single realization h from the standard normal distribution (0, is used for approximating the divergence and r is chosen as small constant. total, only two evaluations of the function f are necessary to estimate its divergence with this method. In deep learning setting, [Soltanayev & Chun (2018): Chen et al.) (2022) leveraged this Monte Carlo approximation in combination to the SURE loss ( to train neural networks datasets composed only of noisy observations y. leading to the MC-SURE approach. While they achieved performance close to that of the MMSE estimator, a slight gap remains, partly due to approximation errors in the divergence term.
When the noise level is unknown: In more realistic settings, one typically has access only to dataset composed of noisy signals, with no information about the underlying noise level a for each observation y. Although several methods have been proposed for ad hoc noise estimation (Chen ct al. 2013 Pyatykh et al. 2013 Foi et al. 2008), a recent line of research (Tachella et al. 2025a Batson & Royer 2019 Krull et al.) 2019) seeks to circumvent the need for estimating o altogether. This is achieved by restricting the estimator f' to belong to a constrained set S such that
YE R.
[y. div f(v) =
thereby effectively removing this term from the optimization objective ()) The final objective then consists in minimizing the measurement consistency under constraint:
arg
min min in fyllf(y) y s.t. S
In what follows, we describe two distinct types of constraints proposed in the literature for tackling self-supervised denoising with unknown noise levels, propose an alternative constraint set and then study their shared properties.
Remark In addition to the constraint based approaches studied in this paper, we also mention, for completeness, the approaches that directly utilize the score function log Py (b) in Tweedie's formula, as proposed in Kim & Ye (2021): Kim et al. (2022): Xie et al. (002), 3) all of which depend on the estimation technique introduced by Lim et al. (2020). Moreover, Noise2Noise-like (Lehtinen) et al. 2018) data augmentation techniques were also proposed (Pang et al. 202T Huang et al. 2021
Wang et al. 2022
Mansour & Heckel 2023) as an alternative to SURE.
3.2 BLIND-SPOT ESTIMATORS
A radical way to achieve (7) is to impose that each component function f; does not depend on y Under this constraint, f is trivially divergence-free by construction since By back to Efron (2004) and lies at the core of the Noise2Self approach (Batson & Royer 2019) and its
= 0. This idea dates
variants (Krull et al. 2019 Laine et 2019), in which so-called "blind-spot" network architecture
3

---

Published as a conference paper at ICLR 2026
SURE
UNSURE
CENSURE (ours)
Noise2Self, Blind-spot nets
0f. Dy
Less expressive estimators
Unconstrained
Ey div f'w) ne
div ((y) ne vy
(1)
Vy
known
unknown constant
unknown non-constant
unknown non-constant
(and applicable beyond Gaussian noise)
Balancing expressive power in constraint-based self-supervised Gaussian denoising. When the noise model is fully specified, SURE yields the most expressive estimators, whose perfor- mance matches theoretically that of supervised methods. As assumptions on the noise are gradually relaxed, however, the learned estimators must reduce their expressivity avoid overfitting the noise. In this work, we introduce divergence-constan estimators, family that preserves much of the ex- pressivity lost in bpin designs, yet still allows training even when the noise level a unknown and varies across the data.
Figure 1:
is employed. From a broader perspective, this constraint can be generalized by restricting f to the space
Süs = {f E L (RR, : Vy E R", 01 (y) c}. Oy:
(9)
which still satisfies as long as e E R is a constant fixed in advance. the case c = 0, the solution of (8) under blind-spot constraints is given by fP8( = (u [(alyy1) where
y-; refers to the vector obtained by excluding the ith entry.
The strength of blind-spot constraints lies actually in their versatility: they can handle wide range of noise types, specifically those that are zero-mean and spatially independent, which ( is prime example, without preciscly knowing the noise distribution. However, this flexibility comes a sigmificant performance cost. A blind-spot architecture is indeed inherently less expressive than classic one, especially since y: is usually highly informative about HG, and tends to introduce checkerboard artifacts (Höck et al. 2022).
3.3 UNSURE ESTIMATORS
In the particular case where every noisy observation y is corrupted by the same noise level a (i.e,, Po is a Dirac). Tachella et al. (2025a) proposed softened version of the constraint (9) imposing only that the estimator has zero expected divergence, that is, Ey div f(y O. This relaxation retains the validity of 7 while offering increased expressivity, since any blind-spot estimator automatically satisfies the zero cero-expected expected-divergenee divergence condition. Extending it to the constant case, this alternative constraint forces to belong the space
SOED (f E C R") Ey div fly) nc)-
(10)
In the case c 0. closed-form solution of the optimization problem (8 was established, namely fZED (y) + V log (y)), with n =
log Ps, (1)
An important difference with the blind-spot approach lies in how the optimization is carried out in practice. Indeed, unlike blind-spot approaches-where the constraint is enforced directly through the design of f-the so-called UNSURE approach seeks saddle point of the Lagrangian by formu- lating the problem as min-max (Arrow et al. 1958 Platt Barr 1987). However, such an optimization method comes with several caveats. First, constraint satisfaction is not guaranteed in practice; only the penalty term associated with violations minimized. Second, the outcome is highly sensitive to the choice of learning-rate pair for gradient-descent--an straint, and an inappropriate choice can lead to instabilities or oscillatory dynamics during training (Platt & Barr 1987 Gallego-Posare et al. 2022). Finally, in this setting, the divergence term is estimated via a Monte Carlo approximation based on (6) using a limited number of samples, which can further degrade the accuracy of the optimization.
max optimization which is solved by alternating
which controls the trade-off between the objective and the con-
4

---

Published as a conference paper at ICLR 2026
3.4 PROPOSED ALTERNATIVE: CENSURE ESTIMATORS
In this work, we propose study the set of weakly differentiable vector fields on R" with constant (normalized) divergence E R, denoted by
Sic = (f E L (R",R") ky E R" div fly) nc).
(11)
de facto introducing an intermediate constraint lying between the blind-spot constraint set and the much looser expected divergence constraint one: Sis C She C SEED We emphasize that all inclusions are strict, with in particular the possibility for E Shc to have each of its component function f; to depend on Ys. which is excluded for function in SBs (see Fig. O Appendix). We postpone the description of the way we construct in practice such divergence-constant estimators to the next section. Let us simply note that, similarly to existing alternatives, divergence-conntamt mappings are of particular interest self-supervised denoising since the condition (7) always satisfied, regardless of whether the noise level a is constant across observations y or varies from one observation to another. This is major difference with the UNSURE approach which cannot, priori, be used when the noise level varies across the noisy observations. Indeed, in full generality, Ey.olo? div [z)
] * Eg [o" [div fzy) because o² not independent div f(y). since y itself Figure
depends
| illustratss the expressivity trade-off that emerges in self-supervised denois- ing. Our proposed approach, termed CENSURE (Concealed and Erratic Noise level with Stein's Unbiased Risk Estimate), bridges the gap between constant expected divergence estimators and the more restrictive blind-spot ones.
3.5 PROPERTIES SHARED BY ALL CONSTRAINT SETS
For conciseness, S denotes one of the following sets: Sis: SED or Spc. This paragraph should be read by selecting one of these three sets consistently, without mixing them. As a preliminarily observation, we notice that the constraint set S admits an affine space structure, based e id, where id refers to the identity map on R... This statement is formalized in the following lemma (all the proof' of this paper are given in Appendix(C)
Lemma 1. S is a linear space and 8 is an affine space with S = cid +
direct consequence (see Proposition T is that the optimal denoiser within each class can be writ- affine combination of the identity function and the minimizer in S of the measurement consistency term. Thus, it is sufficient to restrict the search to estimators in S since any optimal denoiser in S can be recovered from an optimal denoiser in S
Proposition The solution of for the constraint set S can expressed as
arg min E. (I - y11 = cid +(1 c) arg min (o)l( JE8=
(12)
feso
An immediate question that arises at this point is: How to choose the constant e to achieve the best denoising? Proposition provides a theoretical characterization of the optimal constant c².
Proposition 2 (Optimal constant). the AWGN setting (see equation 2 Sf (Shs. Sho) or S == =SED and Po is a Dirac then
nE,,, |o
e" = arg min min (ly()) - =0123 seso E. ( cER
E [0,
(13)
JES=
y
Interestingly, the optimal constant o* lies in [0, 1]. As consequence, the affine combination in Prop. is in fact a convex combination in the optimal case. Naturally, c* depends on the knowledge of E, [o which may be unknown in some settings. This explains why the arbitrary choice is made in practice (Batson & Royer 2019 Tachella et al.) 2025a). Unless otherwise stated, we adopt
the same choice for CENSURE.
Finally, using the fact that Si C S we can derive a lower bound on the reconstruction error for divergence-constant estimators in the where constant.
5

---

Published as a conference paper at ICLR 2026
Proposition 3 (A lower bound). In the AWGN setting with constant noise level a (see equation
min SESBc 1l.y)
==12 2
min SESEED
(14)
σ²
MMSE
MMSE
MMSE,
MMSE
where MMSE = E | 01ل is the minimum mean square error.
DESIGN OF DIVERGENCE-FREB NEURAL NETWORKS
now present our proposed methodology for constructing divergence-free network architectures.
4.1 REPRESENTING DIVERGENCE-FREE VECTOR FIELDS
Lemma 2 Joffers straightforward method for generating divergence- free vector fields and highlights the key role played by skew-sy see-syymettiic matrices in ensuring zero divergence.
Lemma (A simple divergence-free vector field). Let R" + R be a smooth scalar field and let A E _axx be skew-symmetric matrix, i.e. A A. The vector field AV is divergence-free.
Nevertheless, this construction does not capture all divergence-free vector fields, except for the case 2. In fact, fully representing such fields typically requires combining multiple expressions of this form, as formalized in the following representer theorem.
Theorem (A universal representation of divergence- free fields). Let : RR" be a smooth vector field and let {A1..... Ak] E RaxR be basis of the space of real skew-symmelric matrices, with n 2 2. f is divergence-free if and only if there exist smooth scalar fields 1.,,..., VbK I - TR such that
K
A Vix..
(15)
The proof is an extension of the work of -oweeeer-Powell al. (022) which builds on the classical
Hodge decomposition theorem (Morita 2001 Berger 2003). Note that the space of real skew-
symmetric n n matrices is of dimension K (5) hence the above construction uses number of
scalar fields that scales quadratically with the dimension
Application for n = 3 Consider the following basis of real skew-symmetric 3 matrices:
0
0
- 0 0 ) )
0000
(A,. A2. As)
1
(16)
and let / be smooth divergence-free vector field. According Theorem I there exist thi, 1/2, tha 3 + R, such that
Be's
3
dya
dys Dats
11
A V
001
U2
(17)
D7g
8822
Oy1
Oyz
In other words, there exists vector field 2 known result in the literature (Morita 2001
(v1, 1he, v/s) such that f is its curl. This is a well-
4.2 PROPOSED ARCHITECTURE
Our goal is to construct a parameterized function f. under the form a neural network, that is divergence-free free by design and whose architecture is tailored for image processing tasks, in particular denoising. To this end, we build on the representer Theorem[[] which suggests defining f as struc- tured combination of conservative fields Ving. However, previously noted, the number of terms

---

Published as a conference paper at ICLR 2026
in this representation, namely K in Theorem | is on the order of n², which becomes prohibitive as soon as we work with images. This scalability issue was previously highlighted by Richter-Powaa et al. (2022). To keep computations tractable, we propose to constraint f to be represented using a sparse combination of conservative fields, which we assume retains sufficient representational fi- delity. This deliberate simplification ultimately consists in substituting K with K" < K (typically K' 8). More precisely, we build f under the form
f Σ V&VER
(18)
where $1,--.., DKK : R TR parameterized via single shared neural network and (AI,..., } E R=x* are (sparse) skew-symmetric matrices, also parameterized. Note that even though we retain fewer terms in (18) than required, the resulting function remains exactly divergence ervereence-free..o as ensured by Lemma] I 2 (divergencee functions are closed under addition). The parameter K' only affects expressivity: when K" 0. f = 0 which trivially divergence- free but not expressive at all; conversely. K' = K yields maximal expressivity, as stated in Theorem | but becomes computationally intractable. We now detail the construction of both types of parame- terization.
4.2.1 DESIGN OF THE SKEW-S SYMETRIC MATRICES
For the sake of computational efficiency, the skew-symmetric matrices A₁ in 18 are chosen to be sparse matrices with shared parameters as follows:
A P - Ps.
(19)
where θ is a shared parameterized repeated-block diagonal matrix and where each P3 E Raxx is a different and fixed permutation matrix (typically rotation or shift matrix). Note that the matrices A& are guaranteed to be skew-symmetrie by design thanks to the following equality of sets, valid for any permutation matrix Pa: {A E Rexa AT
A) P A Ps A Ruxn).
4.2.2 DESIGN OF THE SCALAR FIELDS
The idea of designing neural networks to represent exact conservative fields, i.e. of the form Vú, has already been explored in works targeting energy based models or plug-and-pla methods (Salimans & Ho 2021: Hurault et al.) 2022a). They all point out the critical choice of the architecture for the scalar potential function & order to achieve good performance in practice. In particular, as experimentally observed, modeling vi as a standard feedforward network, such as the ones used for classification, severely degrades performance. Instead, is recommended to incorporate an architecture tailored to the target application directly into the design of v. This is why, propose to consider parameterized scalar fields of the form
Be y E Rd
BI BAY (1())
(20)
where Bg E R and Dg E R" is a neural network specific to image processing, typically a U-Net (Ronneberger et al. 2015). Please note that the neural network parameters 0 are shared for all scalar fields. The specific form of the scalar fields (20) is strongly inspired by |Hrrull et al.)
2022a). with the addition of the B; matrices introduced in our formulation. It is justified by the fact that
V By (v) - B Do(y) + (z)
(21)
for which the first term is known to be effective for learning denoising functions. Beyond introduc- ing diversity for the scalar potential functions, the inclusion of the matrix B, in (20) has the effect of replacing the term Do(y) by B Do))), with the hope that this (transposed) matrix could counter- balance the potentially negative effect of multiplication by skew-symmetric matrix A& afterwards. In practice, expression (21) is computed by differentiating (20) with respect to the input y using an automatic differentiation engine (Paszke et al. 2019) which avoids computing the full Jacobian.
Finally, the matrices B& in 20 are parameterized analogously to (19 via a shared repeated- diagonal matrix O' E 2 in accordance with
B Py O'Pr.
(22)
7

---

Published as a conference paper at ICLR 2026
Table 1: The PSNR (dB) results of self-supervised methods on color datasets cornupted by synthetic white Gaussian noise. Training was conducted with unknown non-constant E [0, 75] (a single model handle all noise levels). The best and second best results in each category are highlighted in red and blue colors, respectively. Non-constraint- based methods are denoted with *
Dataset
Kodak24
CBSD68
Noise level
15
25
50
75
15 / 25
50
75
supervised
DRUNet light
35.18/32.78
29.77
28.14
34.22 / 31.61 28.42 26.74
Noise2Score* Neighbor2Neighbor*
34.69 /..9/32.022 28.17 22.91
33,82 31.00/ 27.16 22.49
pasyuaanss Spas
34.72/ 32.37 / 29.42 27.82
13.22/31.2 / 28.16 26.53
UNSURE
10
29.48 / 22.03 15.58 12.56
29.32 / 22.15 15.71 12.68
UNSURE (r = 10 1)
26.89 / 22.01 15.93 12.82
26.83 / 22.10 1000/11600 16.04 12.94
Noise2Self
34.08 31.90 / 29.07 / 27.49
33.06 / 30.70 27.78 26.21
CENSURE (ours)
34.10 / 31.95 / 29.16
33.10 / 30.81 / 27.89 26.32
Please note that the fixed permutation matrices Ps. are the same as in (9).. Ultimately, the learnable parameters for the proposed parametrization of (18) are (0, O 0' and their number is only slightly greater than that of Dg since @ and O' are sparse. which supports the practicality of proposed parameterization.
5 EXPERIMENTAL RESULTS
We demonstrate the effectiveness of our proposed methodology to construct divergence-free net- works, termed CENSURE (Concealed and Erratic Noise level with Stein's 's Unbiased Risk Estimate), in the case of self-supervised image denoising under the assumption of additive white Gaussian noise and compare its competitiveness with related state-of-thee -oce---r-tth--ttt-- approaches that rely on (4) such as MC- SURE (Soltanayev & Chun 2018) and its constraint- based variants, namely Noise2Self (Batson & Royer 2019) and UNSURE (Tachella et al. 2025a). We also compare with Neighbor2Neighbor (Huang et al. (021) and 2oooe2 (Kim & Ye (02T) (in the unknown- wowoowww-noi----e noise- setting, its ex-
tension (Kim al. 2022) was considered, still referred to Noise2Score) as non-constraintt-aased baselines. For a fair comparison, we used common backbone architecture for all methods and trained all models ourselves. Note that approaches that rely on Monte Carlo approximation (6) of the divergence involve an additional hyperparameter т. In our experiments, we considered two values: 7 10-2, as recommended for vectors with entries in [o, 1] by Tachella et al.) (2025b), and 10-4, value we selected based on test-set performance and which may be viewed as an oracle hyperparameter. Performance of CENSURE and other methods are assessed in terms PSNR val- ues. Details about training. datasets and implementations can be found in Appendix B Source code available at the following repository: https//ggthbbboomsshrrrrvvvvvggg
5.1 UNKNOWN NON-CONSTANT NOISE LEVEL
We first evaluate the self-supervised constraint-based methods for denoising under the assumption that the noise level a is unknown and varies across the training data. This scenario is the most re- alistic in practice because real-world sensors rarely operate fixed signal-to-noise ratio, and the corruption level can change over time, across devices, or even between consecutive acquisitions. According Figure [] our approach is, in principle, the most suitable in this setting. For each method, we learn a single model to handle all noise levels. Specifically, training is conducted using only noisy images synthetically corrupted with Gaussian noise and random ~ Z1((0, 75)). Please note that the values of a (assumed unknown) are not used in the loss function or during inference/test time. Table reports the PSNR results on two benchmark datasets for color image denoising, with representative qualitative examples shown in Figure 2 Results for grayscale images are presented in Table Appendix, along with corresponding qualitative comparisons in Figure 7 We can observe that CENSURE outperforms its constraint based counterparts, which is in accordance with the the- ory. In particular, UNSURE fails when a varies, mainly because it tends to overfit the noise. Indeed, contrary the case where is constant, E.y div ( f(y)) * [o E, /div f(y) in full gener-
8

---

Published as a conference paper at ICLR 2026
PSNRS20
PANRA
PSNRS &O 90
25
Noisy
Supervised
Seighbor2Neighbee
Noise2Score
F dB
12.98 32.98
Ground Truth
Ground Truth
2Sel
UNSURE
CENSURE (ours)
TSNN TAMSKB
ESNK
28001B
PSNK So
mmm
Noisy
Supervised
Neighbar 2Neighbo
2oire2Score
BSNR es ooutiny
USNR 23 M3
DSNR T8OTE
USNR GB
mim
mim
m
Ground Truth
Ground Truth
2ois
UNSURE
CENSURE (ours)
PSNR PSNR3106630 10.63
PSNR 27
PSNR-24377B
Noisy
Supervised
Neighbo
Noise2Score PSNR5277:44dl
PSNR oi
PSNK 26.92
PS NSNRR122 2 2:94
Ground Truth
Ground Truth
Noise2Sell
UNSURE
CENSURE (ours)
Qualitative blind image denoising results. All models were trained on dataset with unknown random E 75) (a single model to handle all noise levels). Best viewed by zooming.
Figure 2:
ality since o² and div /(y) are statistically dependent. Therefore, condition 7 is satisfied for UNSURE. Finally, although Neighbor appears to be the best- best-performing self-supervised method overall, is inherently limited natural images, as relies on the core assumption that two noise-free neighboring pixels share similar values most of the time. In contrast, constraint-basee approaches are more general for denoising problems.
5.2 CONSTANT NOISE LEVEL
For the sake of completeness, also assess the performance of self-supervised constraint based methods in the less realistic setting where the noise level a is constant (whether known or unknown) across the training data. As shown in Figure this scenario favors the use of estimators that are more expressive than CENSURE such as UNSURE or MC-SURE. Table ii Appendix reports quan- titative comparison on two datasets for grayscale images. Contrary the previous scenario, separate model is learned for each noise level. The results are organized into two categories; methods that require the noise level a during training or inference (known setting), and those that are com- pletely agnostic to it (unknown a setting), including CENSURE, UNSURE and Noise2Self. Impor- tantly, P and show that divergence frre denoisers, either everywhere in expectation, can be converted into noise-level-awaa ones (marked with symbol in Table 4 without additional y in
training. Specifically, for a divergence- Proposition be approximated using a single realization [} - y12.. where y denotes the noisy input image (averaging this term over more image realizations did not lead to further improvements).
estimator f. the quantity min fes" E,, (zy)
As expected, UNSURE achieves the best performance among divergence- based methods when the noise level is constant and unknown, followed by CENSURE and then Noise2Self. This ordering consistent with the fact that Shs C SDc C S the fewer constraints imposed on the search space of the estimator, the more expressive it becomes, shown by Figure I Note that slight performance gap is observed for UNSURE depending on the choice of the hyperparameter т.
When the noise level is assumed known, all divergence- frreee methods naturally benefit from this ad- ditional information, showing PSNR gains in line with theoretical expectations (see Subsection[3.5) except for UNSURE with 10-2 This hyperparameter choice is also suboptimal for MC-SURE,

---

Published as a conference paper at ICLR 2026
whose performance falls short of expectations, allowing CENSURE outperform it in most Indeed, according to (4), denoiser trained with the SURE loss should achieve performance com- parable to its supervised counterpart. The observed underperformance stability issues during training. As illustrated Figure 4 in Appendix, the training curves of MC- SURE and UNSURE exhibit pronounced fluctuations, in contrast the much smoother trajectories of the other methods, including ours. This phenomenon was already described in Snltannay & Chun (2188) and considering the oracle hyperparameter 7 10-4 (chosen based on test-set performance) instead solved this issue in our case. To the best of our knowledge, there is no systematic way to this value; it likely depends on the Lipschitz constant of the network. which itself difficult to estimate in Divance/Pintores & Després (2024) This highlights the strength of our approach, as CEN- SURE remains completely independent of this choice by naturally enforcing strict zero divergence. Finally, when the oracle value = 10-1 is used, MC-SURE becomes the best-performing method, achieving results close to the supervised model, in agreement with theoretical analysis. Qualitative examples illustrating the performance of self-supervised methods in the constant-noise regime presented in Figure 8 Appendix.
largely be attributed to
6 CONCLUSION
We presented an original approach for constraining neural networks to be divergence-free by de- sign. Our proposed parameterization is grounded in a representer theorem for divergence-free vec- tor fields, which characterizes them structured combinations of conservative fields. Leverag- ing this thcoretical foundation and incorporating sparsity constraints, we derived parameterizations for neural networks that both resource-efficient and scalable to high-dimensional settings. The practical relevance of our approach is illustrated in the context of self-supervised image denoising, where demonstrated that that these models achieve competitive performance compared to isting constraint-baser methods, especially when the noise level is unknown and varies across the training data. Beyond denoising, our results suggest that divergence- parameterization may hold promise for a wider range of high-dimensional learning tasks, in particular in physics-informed machine learning, opening new avenues for future research.
ACKNOWLEDGEMENTS
The authors acknowledge the support of the Research Group IASIS of the CNRS Informatics (Project DIVIN). Etienne Meunier was partially supported by the Research Programme "PPR Ocean & Climate" through a postdoctoral scholarship. This work was granted access to the HPC resources of IDRIS under the allocation 2025-AD01 1015932R1 made by GENCI. We thank Cédric Herzet for fruitful discussions.
REFERENCES
Eirikur Agustsson and Radu Timofte. NTIRE 2017 Challenge single image super-resolution: Dataset and study. In Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 1122-1131, 2017.
Kenneth Joseph Arrow, Leonid Hurwicz, Hirofumi Uzawa, Hollis Burnley Chenery, Selmer John- son, and Samuel Karlin. Studies in linear and non-linear programming, volume 2. Stanford University Press Stanford, 1958.
Joshua Batson and Loic Royer. Noise2Self: Blind denoising by self-supervision. In International Conference on Machine Learning (ICML), volume 97, pp. 524-533, 2019.
Marcel Berger. A panoramic view of Riemannian geometry. Springer, 2003.
D.P. Bertsekas. Nonlinear Programming. Athena Scientific optimization and computation series. Athena Scientific, 1995. ISBN 9781886529144
Thierry Blu and Florian Luisier. The SURE-LET approach to image denoising. IEEE Transactions Image Processing, 16(1 1):2778-2777 2007.
10

---

Published as a conference paper at ICLR 2026
Elie Cartan. On certain differential expressions and the Pfaff problem. In Annales Scientifiques de l'École Normale Supérieure, volume 3, pp. 239-332, 1899.
Dongdong Chen, Julián Tachella, and Mike E Davies. Robust equivariant imaging: fully unsuper- vised framework for learning to image from noisy and partial measurements. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pattern Recognition, pp. 5647-5656, 2022.
Guangyong Chen, Fengyuan Zhu, and Pheng Ann Heng. An efficient statistical method for image noise level estimation. In International Conference Computer Vision (ICCV), pp. 477-485, 2015.
Kostadin Dabov, Alessandro Foi, Vladimir Katkovnik, and Karen Egiazarian. Image denoising by sparse 3-D transform-doman collaborative filtering. IEEE Transactions on Image Processing, 16 (8):2080-2095, 2007.
Bradley Efron. The estimation of prediction error: covariance penalties and cross-validation, Journal of the American Statistical Association, 99(467):619-632 2004.
Bradley Efron. Tweedie's formula and selection bias. Journal of the American Statistical
Association, 106(496):1002-6611) 2011.
Alessandro Foi, Mejdi Trimeche, Vladimir Katkovnik, and Karen Egiazarian. Practical Poissonian- Gaussian noise modeling and fitting for single-image raw-data. IEEE Transactions Image Processing, 17(10): 2008.
Jose Gallego-Posada, Juan Ramirez, Akram Erraqabi, Yoshua Bengio, and Simon Lacoste-Julien. Controlled sparsity via constrained optimization or: How learned to stop tuning penalties and love constraints. In Advances in Neural Information Processing Systems (NeurIPS), volume 35, pp.
2022.
Sébastien Herbreteau and Charles Kervrann. A unified framework of nonlocal parametric methods for image denoising. SIAM Journal on Imaging Sciences, 18(1):89-119, 2025.
Sébastien Herbreteau and Michael Unser. Self-calibrated variance-stabiline stabiizig transformations for real-world image denoising. In International Conference Computer Vision (ICCV), pp. 10496- 10506, 2025.
Eva Höck, Tim-Oliver Buchholz, Anselm Brachmann, Florian Jug, and Alexander Freytag. N2v2- fixing noise2void checkerboard artifacts with modified sampling strategies and a tweaked network architecture. In European Conference on Computer Vision (ECCV), pp. 503-518. Springer, 2022.
Tao Huang, Songjiang Li, Xu Jia, Huchuan Lu, and Jianzhuang Liu. Neighbor2Neighhbr: Self- supervised denoising from single noisy images. In Conference on Computer Vision and Pattern Recognition (CVPR), pp. 14781-14790, 2021.
Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Gradient step denoiser for convergent plug-and-play In International Conference Learning Representations (ICLR), 2022a.
Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Proximal denoiser for convergent plug- and-play optimization with nonconvex regularization. In International Conference on Machine Learning (ICML), pp. 9483-9505, PMLR, 2022b.
Xiaowei Jin, Shengze Cai, Hui Li, and George Em Karniadakis. Nsfnets (navier-stokes flow nets): Physics-informed neural networks for the incompressible navier-stokes equations. Journal of Computational Physics, 109951, 2021.
Kwanyoung Kim and Jong Chul Ye. Noise2Score: Tweedie's 's approach to self-supervised image de- noising without clean images. In Advances in Neural Information Processing Systems (NeurIPS), volume 34, pp. 874, 2021.
Kwanyoung Kim, Taesung Kwon, and Jong Chul Ye. Noise distribution adaptive self-supervised image denoising using tweedie distribution and score matching. Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2008-2016, 2022.
11

---

Published as a conference paper at ICLR 2026
Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR), 2015.
Alexander Krull, Tim-Oliver Buchholz, and Florian Jug. Noise2 Learning denoising from single noisy images. In Conference on Computer Vision and Pattern Recognition (CVPR), pp- 2124-2132, 2019.
Samuli Laine, Tero Karras, Jaakko Lehtinen, and Timo Aila. High-quality self-supervised deep image denoising. In Advances in Neural Information Processing Systems (NeurIPS), volume 32, 2019.
M. Lebrun, A. Buades, and J. M. Morel. A nonlocal Bayesian image denoising algorithm. STAM Journal on Imaging Sciences, 6(3): 1665-1688, 2013.
Jaakko Lehtinen, Jacob Munkberg, Jon Hasselgren, Samuli Laine, Tero Karras, Miika Aittala, and Timo Aila. Noise2Noise: Learning image restoration without clean data. International Conference Machine Learning (ICML), volume 80, pp. 2965-2974, 2018.
Bee Lim, Sanghyun Son, Heewon Kim, Seungjun Nah, and Kyoung Mu Lee. Enhanced deep resid- ual networks for single image super-resolution. In Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 1132-1140, 2017.
Jac Hyun Lim, Aaron Courville, Christopher Pal, and Chin-Wei Huang. AR-DAE: Towards unbiased neural entropy gradient estimation. In International Conference on Machine Learning (ICML), volume 119, pp. 6061-6071, 2020.
Kede Ma, Zhengfang Duanmu, Qingbo Wu, Zhou Wang, Hongwei Yong, Hongliang Li, and Lei Zhang. Waterloo exploration database: New challenges for image quality assessment models. IEEE Transactions on Image Processing, 26(2):1004-1016, 2017.
Youssef Mansour and Reinhard Heckel. Zero-Shot Noise2Noise:
Efficient image denoising without any data. In Conference on Computer Vision and Pattern Recognition (CVPR), pp. 14018-14023 2023.
Zhiping Mao, Ameya D Jagtap, and George Em Karniadakis. Physics-infomee neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering, 360: 12789, 2020.
D. Martin, C. Fowlkes, D. Tal, and J. Malik. A database of human segmented natural images and its application evaluating segmentation algorithms and measuring ecological statisties. In International Conference on Computer Vision (ICCV), volume 2, pp. 416-423 vol.2, 2001.
James Clerk Maxwell. A treatise on electricity and magnetism, volume Clarendon press, 1873,
Shigeyuki Morita. Geometry of differential forms, volume 201. American Mathematical Society, 2001.
Tongyao Pang, Huan Zheng, Yuhui Quan, and Hui Ji. -ecorrupted-t--Reoorupp Unsupervised deep learning for image denoising. In Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2043-2052, 2021.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang. Zachary Veti,o, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Py PyTorch: An imperative style, perfforrfrreec deep learning library. In Advances in Neural Information Processing Systems (NeurIPS), volume 32, 2019.
M. Pintore and B. Després. Computable lipschitz bounds for deep neural networks. arXiv preprint arXiv:2410.21053, 2024.
John Platt and Alan Barr. Constrained differential optimization. Neural information processing systems, 1987.
12

---

Published as a conference paper at ICLR 2026
Stanislav Pyatykh, Jürgen Hesser, and Lei Zheng. Image noise level estimation by principal compo- nent analysis. IEEE Transactions on Image Processing, 22(2):687-66 2013.
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learn- ing (part II): Data-driven solutions of nonlinear partial differential equations. arXiv preprint 1IXiv:1711.100 2017a.
Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learn- ing (part D): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561 2017b.
Sathish Ramani, Thierry Blu, and Michael Unser. Monte-Carlo SURE: A black-box optimiza- tion of regularization parameters for general denoising algorithms. IEEE Transactions on Image Processing, 14095:55401155, 2008.
Jack Richter-Powell, Yaron Lipman, and Ricky TQ Chen. Neural conservation laws: divergence- free perspective. In Advances in Neural Information Processing Systems (NeurIPS), volume 35, pp. 38075-38088 2022.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomed- ical image segmentation. In Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 234-241, 2015.
Tim Salimans and Jonathan Ho. Should ebms model the energy or the score? In Energy based models workshop-ICLR, 2021.
Shakarim Soltanayev and Se Young Chun. Training deep learning based denoisers without ground truth data. In Advances in Neural Information Processing Systems (NeurIPS), volume 31, 2018.
Charles M. Stein. Estimation of the mean of multivariate normal distribution. The Annals of Statistics, 9(6):1135-1151 1981.
Julián Tachella, Mike Davies, and Laurent Jacques. UNSURE: Self-supervised learning with Un- known Noise level and Stein's Unbiased Risk Estimate. In International Conference Learning Representation (ICLR), 2025a.
Julián Tachella, Matthieu Terris, Samuel Hurault, Andrew Wang, Dongdong Chen, Minh-Hai Nguyen, Maxime Song, Thomas Davies, Leo Davy, Jonathan Dong, Paul Escande, Johannes Hertrich, Zhiyuan Hu, Tobías I. Liaudat, Nils Laurent, Brett Levac, Mathurin Massias, Thomas Moreau, Thibaut Modrzyk, Brayan Monroy, Sebastian Neumayer, Jérémy Scanvic, Florian Sarron, Victor Sechaud, Georg Schramm, Romain Vo, and Pierre Weiss. Deepinverse: A python package for solving imaging inverse problems with deep learning. arXiv preprint arXiv:2505.20160 2025b.
Dimitri Van De Ville and Michel Kocher. SURE-based Non-Local Means. IEEE Signal Processing Letters, 16(1 (6973-9773 2009.
Yi-Qing Wang and Jean-Michel Morel. SURE guided Gaussian mixture image denoising. Journal on Imaging Sciences, 6(2):999-1034, 2013.
SIAM
Zejin Wang, Jiazheng Liu, Guoqing Li, and Hua Han. Blind2Unblind: Self-supervised image de- noising with visible blind spots. In Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2027-2036, 2022.
Yutong Xie, Mingze Yuan, Bin Dong, and Quanzheng Li. Unsupervised image denoising with score function. In Advances in Neural Information Processing Systems (NeurIPS), volume 36, pp. 69752-69763, 2023.
Kai Zhang, Yawei Li, Wangmeng Zuo, Lei Zhang, Luc Van Gool, and Radu Timofte. Plug-and-Pil image restoration with deep denoiser prior. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):6300-3377 2022.
13

---

Published as a conference paper at ICLR 2026
Yide Zhang, Yinhao Zhu, Evan Nichols, Qingfei Wang, Siyuan Zhang, Cody Smith, and Scott denoising dataset with real fluorescence microscopy images. In
Howard. A Conference on Computer Vision and Pattern Recognition (CVPR), pp. 11710-11718, 2019.
Ruofan Zhou, Majed El Helou, Daniel Sage, Thierry Laroche, Ame Seitz, and Sabine Süsstrunk. W2S: Microscopy data with joint denoising and super-resolution for widefield to SIM mapping. In European Conference on Computer Vision Workshops (ECCVW), 2020.
14

---

Published as a conference paper at ICLR 2026
A LIMITATIONS
We would like to mention that the proposed parameterization for enforcing zero divergence en- tails higher computational cost during both training and inference compared its supervised and constraint-basee self-supervised counterparts (Tachella et al. al.] [00255|/Sotaaaaa
& Chun 2018 Bat-
ssnn & Royer 2019). In particular, our method requires one feedforward pass through the backbone network along with K' gradient computations at inference (K' = 8 in our implementation). Since each backpropagation has a cost comparable to a forward pass (Hurault al. 2022a), the overall inference cost amounts to roughly K' + times that of a single feedforward evaluation, which may limit its applicability in time-sensitive settings. Moreover, due scalability constraints, we deliber- ately restricted the number of terms in the sum (15) from Theorem [ While this sparsity constraint may not fully capture the underlying optimal solution, we demonstrated that it still yields strong performance in image denoising. Finally, our application in image denoising targets only additive white Gaussian noise, and its effectiveness under alternative noise models remains unexplored. We nevertheless demonstrate in Figure(Jthe that the Gaussian denoiser learned with our approach can be di- rectly applied to real-world noisy fluorescence images, when combined with a variance-stabilizing transformation learned using the Noise2VST (Herbreteau & Unser 2025) framework. Extending our parameterization to handle other types of noise, such as Poisson-Gaussian Gaussian corruption, presents promising direction for future research.
Table 2: Execution time (in seconds) comparison batch of size 16 x x 64 x 64 for different approaches with shared backbone architecture (GPU: Tesla V100). Model
Training 4 Inference +
# parameters
Supervised 0.036
0.000 17, 007, 744
MC-SURE
0.061
0.007
17, 007, 744
UNSURE
0.062
0.007
17,007, 007, 744
Noise2Self
0.006
0.120
17, 17,007, 744
CENSURE
K"
0.356
0..04
17, 008, 256
K"
0.679
0.039
17, 008, 256
K' =
1.002
0.055
17, 008, 256
K'
1.323
0.072
17, 00,008 256
B IMPLEMENTATION DETAILS
Common bane architecture For fair comparison, we adopt a variant of the DRUNet archi- tecture (Zhang et al. (2022) as the shared backbone across all approaches. In the original formulation, each scale is composed of four residual blocks of the form "3 x 3 conv + ReLU + 3 conv". To reduce computational cost, limit this to two residual blocks per scale. Moreover, as in Hurault et al. 2022b),, we replace ReLU by Softplus activations with sharpness parameter 3 = 100, which acts smooth surrogate for ReLU, easing training for conservative field networks (Hurault al., 2022a). Note that the noise level map was removed from the original architecture.
Datasets All models were trained using the same large-scale dataset proposed in Zhang et al. 2022)), which contains a total of 8,694 images. This includes 400 images from the Berkeley Seg- mentation Dataset (BSD400) (Martin et al. 200T), 4,744 images from the Waterloo Exploration Database (Ma et al. 2017). 900 images from DIV2K (Agustsson & Timofte
and 2,750 im- ages from Flickr2K Lim et al. 2017). The training set is augmented through random horizontal and vertical flips as well as random rotations of 90" For validation, we use the BSD32 dataset (Martin et al. 200T). consisting of 32 images, to monitor training progress and select the herf-peeforming model. Finally, the evaluation for color images is carried out on two test sets, Kodak24 and BSD68 (Martin et al. (00I), which are completely separate from both the training and validation data. For grayscale images, the test set Kodak24 is replaced by Set
Training details All models are trained for 600,000 iterations, where each training iteration in- volves gradient-based pass on batch of patches of size 128 128 that are randomly cropped from
15

---

Published as a conference paper at ICLR 2026
training images (except for CENSURE where patches are taken of size 64 x 64 in order accelerate training). We use a batch size of 16 and employ the Adam optimizer Kingmma & Ba 2015) with an initial learning rate of 10 Approaches that rely on a Monte Carlo approximation 66 of the divergence involve an additional hyperparameter 7. In our experiments,
in Zhang et al. (2022). which is halved every 150,000 iterations.
considered two values: 7 10-2, as recommended for vectors with entries in [0, 1] by Tacheela et al. (2025b), and T = 10-4, a value we selected based on test-set performance and which may be viewed as an oracle hyperparameter. For the UNSURE TTchella et al., [025a), Noise2Score (Kim & Ye 2021). and Neighbor2Neighbor (Huang et al., 2021] losses, we followed the default settings of the DeepInverse library (Tachella et al. 2025b); in particular, the momentum parameter for the gradient ascent on the noise level was fixed at 0.9 for UNSURE and the minimum and maximum noise level for Noise2Score, which are annealed during training, are set to 0.001 and 0.1, respectively.
Implementation choices for CENSURE Our proposed parameterization (88) of divergence-free estimators requires several additional hyperparameters that must be specified, including the number of terms K' in the sum, the size k x x of the blocks in the two repeated- block diagonal matrices O and O', and the selection of the permutation matrices Ps. First of all, in order drastically reduce the computational burden, we set K' = 8. The block size x is chosen as x 4 x C, where C' is the number of channels (C = 1 for grayscale images and C 3 for color ones), resulting in total of 512 x C' additional learnable parameters. which is negligible compared to the 17M parameters of the network backbone. Finally, the first four permutation matrices P& were selected to perform circular horizontal shifts of the input image by 0 to 3 pixels, while the remaining four are obtained by composing these shifts with a 90" rotation.
Our implementation is written in Python using the PyTorch framework (Paszke et al. 2019) and with additional support from the DeepInverse library (Tachella et al.) 2025b). Training was conducted on a Tesla V100 GPU.
C PROOFS
Proof of Lemmal 7 S is a linear space due to the linearity of the partial derivative operator and the
linearity of expectation. Let f E cid, +59, For all y E IR". ajeid..l nc. Therefore, f E S Reciprocally, let f E S Then, f = cid, +(ʃ - eid,,) E cid, +50 by
Jy. (y) cand so div div(cid.)(g)) =
linearity of the partial derivative operator and the linearity of expectation.
Proof of Proposition L According to Lemma 1 cid +(1 e) and we have
Sc = cid +50 In particular, if c * 1, S =
arg min (z)) y112 JES-
arg min fEcid +11-)80
(y) - 2
(23)
cid +(1 c)
arg min Ey (ciidy)) (1) + - c) - =112 fEsD
(24)
cid +(1 c) arg min(1 e)² [[v) -y1i fes
(25)
= cid +(1 c) arg min Ey By)((())= yll.. fEs
(26)
This still holds for since we have trivially arg min By y1l2 = id.
U
fest
Proof of Proposition
2 According to (4) (Stein 1981).
Be.y)((()) - 2012 = Eyy | 2 + [lf - yy12 + 20² div f(y))
(27)
Moreover, S E (Sis: } or if S SEED and Po is a Dirac then
Aʃ E 5" Eye 1o2 div f(y)] = ncFE. (o
(28)
hence,
min E. (l - 20 = [n2 + 2ncE. (02) + min fyll(() -
(29)
fES-
Jes
16

---

Published as a conference paper at ICLR 2026
According to Lemma
S cid +S5 In particular, if + 1, cid +(1 c)S and have
min fES=
Byls(v) v1l2 = min
fecid (11-c)80 fy)l(yy yl1
(30)
min By cy + (1 c)f(y) vl fES
(31)
(1 c)² min By f y y1l.
(32)
JESt
This still holds for as min fESt (l)
U 2 since id E S
Finally, for all c E R.
min Ee v (lf(z) - 2012 -nB3 2ncB, [o ] + (1 c)² min Byllf(y) - JES-
(33)
JES"
As consequence,
n.B, [
arg min min fl)) ==12 =1 min Ey [f v12
(34)
OER SE8'
But this latter quantity lies in [0, 1] since, in particular for e = 0 in 29
min SESo
Byllf(y) y = nEE + min Ev,y (1)) - x112 0, jES
(35)
and so,
n2 [o
n.E 10"
E [0, 1].
(36)
ses E (
nR, [o2) + min SESO E. Exy - 201 -x
Proof of Proposition 3
According to 22 for all R,
min
Ex,y
1 n fzy - x
12 -o2 200c c + (1 c)² min
fl yy11.
(37)
fESERD
JES
But since ZED = arg mig feSe bl - E1ا has a closed-from expression
Tachella et al.
2025a), namely FZED (y) = + V log (y) (3), with n = (EE V (3) - we have
min SESEED
E (5y)) - y = Ey - ZED (y) - y = E = V f1) = n
(38)
Moreover, f
SIRRE 2 2² Tachella et al. 2025a). Finally, since Sbc C SEED
min En 2012 SESic
min SESEED
Em.gy 1. [1) =01.
(39)
x²
+ 20 c + (1 c2
(40)
1 MMSE
02
1
MMSE o²
2 2c 1 1+(1+-00 (-0² +2000)
(41)
MMSE
d2
0
2 2c +1 1+ (2c-1) () (1- MMSE G²
(42)
MMSE 02
2
MMSE
2
MMSE o2
MMSE
(43)
1
MMSE
0²
o²
MMSE r²
MMSE
MMSE
(44)
MMSE 22
17

---

Published as a conference paper at ICLR 2026
Proof of Lemma 2 Let f x + (V). We want to compute div f(x) tr tr(V)())) for all a IR.
We have f = i22 (p1, with for all a E IR",
Pi a V (x)
( = (x) V ( = A
P2 a
Ax
where H (x) denotes the Hessian matrix of v evaluated at æ. According to the chain rule (Bert-
sekas 1995),
Vf(²) (=1(=) = H²)x)
(45)
hence div f() = ( skew-symmetric and symmetric matrix is zero:
(r ( (z)) 0. Indeed, the trace of the product of
tr(A = tr((AB) tr(B' T A = tr( = - tr(AB).
(46)
where B E R denotes a symmetric matrix.
L
Proof of Theorem 7 Assume that there exist smooth scalar fields , Ux R' - R such that f Σx Ag V. According Lemma 2
we have that each A& Vih is divergence-free. Finally, since the set of divergence-frre functions is a linear space from Lemma ! f is divergence-free.
Reciprocally, assume that RR R. is smooth divergence-fres vector field with n > 2. According to Richter-Powelly et al. (2022) (a proof is provided below which does not involve the formalism of differential forms), there exists a smooth nlew-symmetri matrix field F R" [ax= such that
" aF.
Vi ul. f
(47)
Ox,
By denoting E(d.) the standard basis matrix of Ruxn having a in the (i.j)-th entry and zeros elsewhere, we can rewrite as
E
VFF
E B
+
E (1)) VFF +
E V...
(48)
ja
EUJ +
(1i FiJs (since F = 0)
(49)
I(E E(id) VFF
(since F2, - FFa))
(50)
(E - E ) (kk)
(51)
B VF (()) F()):
(52)
where şo is a bijection from (1,.... () to (i.)) E (1,..., < j} and B A Ev(() (
We can notice that (Bı,... BR) E Raxn is nothing else than the canonical basis of the space of real skew-symmetric n X n matrices. Therefore, V1 < k S K, {x(²) AK
(k)
K
B
Σ
(k)
A.
(53)
1=1
Hence,
K
K
K
K
A;
V V(P(H) = Σ
A
Σ
x
Fe(t)
(54)
1=1
1=1
We conclude by setting v/2
Ex= k=1
K
(6)
FP(K)-
i.e. map F F: RR" m such that Vin R².
F(x)T FF()).
18

---

Published as a conference paper at ICLR 2026
Proof of 47 Let F" - I" be smooth divergence-free vector field with n 2 2.
Define, for all E (2....,72) ,n).
v :: + - 1 JclE, x2, F3,... .n) + 8,2
(3,,,,,,,...................,... ,3,..., ,.. kn)de, (55)
dy.,
where 6yy denotes the Kronecker delta. By construction, fi dr Moreover, for all x E R"
for all i m2.
(x) = θ
12
0 21 filE, 22, Zz. xn)df ,,,...,,,,d.
Σ
Σ
(56)
Dz,
dιz 3
dx;
Ju
2
2=
fi (2,......... .n)
fi(E, D2, Iz..... In dE
(57)
Oz;
1=2
fi(0, I2 ,2,,..,,In)
Inn
f1 22 33
In de
(58)
fi 32 ................................ Inn fi(x)
(0, 22 .... Inn
(59)
= fi()),
(60)
where, in 58
we used the fact that f is divergence-free.
Define
0
a
(x)
1/2 x
F: RR
Val x
0
0
(61)
th (x)
Finally, F is a smooth skew-symmetric matrix field such that
aFis
Vi n,, f:
(62)
Ozy
j=1
19

---

Published as a conference paper at ICLR 2026
D ADDITIONAL QUANTITATIVE RESULTS FOR GRAYSCALE IMAGES
Table 3: The PSNR (dB) results of self- self-supervised methods grayscale datasets corrupted by synthetic white Gaussian noise. Training was conducted with unknown non-constant o E [0, 50] (a single model handle all noise levels). The best and second best results in each category highlighted in red and blue colors, respectively. Non-constraint- based methods are denoted with
Dataset
Sct12
BSD68
Noise level a
15
25
50
15 / 25 / 50
supervised
DRUNet light
33.13/30.84/7..3
31.84 / 29.41 26.49
Noise2Score*
32.70 30.27 26.37
31.43 / 28 28.94 94 25.43
Paayaaa fian---jaa fjas
Neighbor2Neighborf
32.29 / 30.38 27.44
31.11 / 29.07 / 26.30
UNSURE (r = 10-3)
26.32 / 20.81 14.99
26.25 / 20.91 / 15.14
UNSURE
10
23.10 18.63 13.11
23.08 / 18.70 / 13.21
Noise2Self
31,05 29.47 26.94
29.24 / 27.79 25.69
CENSURE (ours)
31.62 / 29.80 27.05
29.73 / 28.13 25.81
Table 4: The PSNR (dB) results of deep learning-basee methods applied grayscale datasets corrupted by synthetic white Gaussian noise. Training was conducted with constant noise level (one model per noise level). The best and second best results in each category highlighted red and blue colors, respectively. Non-constraint-based methods are denoted with Models converted into noise-level-aware variants, using Propositions and are indicated by the symbol (no additional training).
Dataset
Set12
BSD68
Noise level a
15
25
50
15 / 25 / 50
supervised
DRUNet light
33.24 30.92 27.84
31.91 / 29.46 26.55
Noise2Score*
32.93 30.58 27.40
31.61/20 26.1/2 26.29
Neighbor2Neighhor
32.84 30.62 27.59
31.63 29.26 26.40
Unknown constant
UNSURE (r = 10-3)
31.88 29.84 27.15
30.90 / 28.72 26.08
Noise2Self
31.15 29.55 27.02
29.29 / 27.83 / 25.73
selfisupervised
CENSURE (ours)
31.65 29.81 27.05
29.87 / 28.14 25.78
UNSURE (r oracle)
22.16/30.23 27.33
31.14/28.99/ 26.19
Noise2Score
33.04
30.65 27.41
31.80 / 29.33 26.34
MC-SURE 10 3)
32.13 / 29.97 27.27
31.20 / 28.86 26.22
UNSUREt (T = 10 10
31.80/ 29.49 27.14
30.58 28.39 / 26.01
Known constant
Noise2Selft
32.07
33.05 27.25
30.80 / 28.66 / 26.09
CENSUREt (ours)
32.46
30.28 27.27
31.20 / 28.91 / 26.13
MC-SURE (7 oracle)
33.22
30.87 27.77
31.90 / 29.44 26.51
UNSURE (7 oracle)
33.05 30.79 27.58
31.47 / 29.22 / 26.34
20

---

Published as a conference paper at ICLR 2026
E TOY EXAMPLES
To gain some intuition about optimal divergence-constant estimators for denoising in Süe and their difference with their counterparts in Sis and SĒED extend the toy examples from Tachella al. 2025a the two-dimensional case In all experiments, the noise level is assumed constant.
E.
TWo DELTAS
Let = 201 + 2 denote the distribution of the clean signal and Py [1 be the distribution of the signal corrupted by AWGN (see equation 2 where is the all-ones vector in I2a and I denotes the 2 x identity matrix. Since the prior distribution is explicit, we can compute closed-form solutions for the following estimators:
NN(11, +
MMSS arg min (x x: f
JCD arg min Eyllf(y) -11
(63)
fES
pBS arg min ff -vll
SESSs
Standard calculus rules give:
+ 9/2
V y logpy(y) = 0²
tanh
y-y)
y
(64)
0²
MMSE (y) (y)= y 0²V, log pr)) = tanh
(=2
91 + y22
)
1,
(65)
o² tanh
CCD = cy + (1-c) 14 logmo( = 4 (1 -1-00 (1-c)n
y/1 + y22 o²
1
y
(66)
o²
and
(tanh(y2/o*)) (anh(yy//02)
(BS (1) cy + (1 c)
E(121011
cy + +(1 cc
(67)
with n = (E, | V log py (2)113)
For FDC = arg min Ey l1 (v) y there exists no closed-form solution, to the best of our knowledge. However, using Theorem we can train a neural network of the form AV where
0 1
A =
and v/a R + R. to solve
-1 0
0r arg min Ey, AVVl(v)))))) y113.
(68)
so that PDC well approximated by y - cy + (1 - eV In practice, we use a simple Multi-Layer Perceptron (MLP) parameterize the
The mean squared error (MSE) results, obtained by simulation, are reported in Table and Figure
E.2 GAUSSIAN AND SPIKE & SLAB
We do not detail the derivations of the closed-form solutions but the mean squared error (MSE) results are reported in T
In the one-dimensional case, we have Sbc Süs, which prevents from highlighting the differences between the estimators.
21

---

Published as a conference paper at ICLR 2026
dudlemin welt
Saat 1200
p-
Figure 3: Comparison between MMSE, CED, DC and BS estimators (for unknown a setting) for the toy signal distribution "Two Deltas". Each row correspond to a different denoiser (from top to bottom: id, MMSE. fMMSE, JCED, fCED fDC (ours). fRS fBS)). The denoisers MMSE, fCED and fBS are the closed-form solutions, while fMMSS JCED jDC and jBS refer to the approximated solutions learned from data leveraging a Multi-Layer Perceptron (MLP). Columns show, from left to right: (i) denoised samples (ty), with the mean squared eтror Ez. ( æ reported in the title and computed over test dataset; (ii) the vector field induced by the denoiser y + æ(y): (iii) the difference between this field and the identity function; and (iv) the divergence of the denoiser-
induced field. Note that PDC and fBS yield vector fields with identically zero divergence, visible in the last column. The unknown constant noise level a is set to 0.75.
22

---

Published as a conference paper at ICLR 2026
Two Deltas
Gaussian (O,
Spike & Slab
50
20
(. 1) Eōu NO (1 x²)
Py
W 1, 1)
OO (1 +
I 1)= (N(0,0²1)
MMSE
0.089
0.360
0.243
MSE CED
0.T06/0.083
1.000/0.60
0.427/0.243
MSE DC
0.218/0.15 157
1. 1.000/0.600
0.500/0.266 0.500/0.265
MSE BS
0.266/0.181
1. 1.000/0.360
Table 5: Toy signal distributions and their mean squared error (MSE) results for different estimators (left: c = 0, right: optimal c). The constant noise level a is set to 0.75.
F ADDITIONAL EXPERIMENTS
25.0
22.5
20.0
Figure 5: Ablation study on the number of terms K' in CENSURE. Average PSNR (in dB) results on Set12 (grayscale) with unknown constant noise level 25 is reported for each model. As ex- pected, performance improves as the number of terms increases.
Nerege PONR
17.5
Supervied
MC-SURE 10"1)
UNSURE (s 10-%)
12.5
UNSURE tom 1011, craciel
:0.0
300000
200000 300000
400000
SC0000
600000
Number Herrations
K
2
4
Figure 4: Stability issues during training for Monte Carlo approximation methods (see equation (6) for grayscale images with con- stant 25.
Set12
29.44
29.67
29.74
29.81
Ground truth
Noisy (a 15)
Noise2Self
CENSURE (ours)
Figure 6: Divergence-free estimators are more expressive than their blind-spot counterparts. Blind- spot networks, by construction, enforee their receptive field to exclude the center pixel y/1 when estimating 2;, since the component function f, cannot depend on 3/s by definition. In contrast, divergence-frr free networks pixel yı, which is typically highly informative about x; It is illustrated here as Noise2Self removes all isolated white data points, whereas divergence-free estimator preserve them more aceu- rately, leading to improved denoising performance.
exploit the full image context without masking, including the central
23

---

Published as a conference paper at ICLR 2026
FSNR 28 1368
FSNR 55
PSNR B3TDEB
PENKI 34N0 dB
Noisy
Supervised
Neighbor 2Neighbor
Noise2Score
FSNR: 0250
PSNR
BS
Noisy
Ground Truth
Noil
UNSURE
CENSURE (ours)
PSNR: T8.9
PSNR 26.66
PSNR: 20.33
PSNR: 20.03
300
Noisy
Supervised
Neighbor2Neighbor
2oise2Score
FSNR cB
FSNR 28.34
PENR: 1891 1891 air
PSNR: 28 33
Noisy
Ground Truth
2ell
UNSURE
CENSURE (ours)
FSNK 20
N. 25
PSNR 25
Noisy BSNR dB
Supervised
Neighbor2Neighhuu PSNR Lamn
Noise2Score
FSNR 25
PSNRR25136d1
Noisy
Ground Truth
Noise2Self
UNSURE
CENSURE (ours)
Figure 7: Qualitative blind image denoising results. All models were trained on dataset with
unknown random [0] (a single model handle all noise levels). Best viewed by zooming.
ESNR
PSNRA CB
FSNRS PSNRASISS CB
ESNRS SESS &B
Ground Truth
Noise2Self
UNSURE (T oracle)
CENSURE (ours)
PSNK 52
PSNK 30:30
PSNK: 52832
PSNK: 31
Noisy
SURE (7 oracle)
Noise2Selff
UNSURE (7 oracle)
CENSURE: (ours)
PSNR JBdB
VSNR:294161
PSNR: 24.8BB
PSNR: 24.30
25
Ground Truth
Noise2Self
UNSURE orucle)
CENSURE (ours)
PSNR: s0S
PSNK 900B
PSNK: 50.3008
PSNK 2B
Noisy
MC-S SURE (7 oracle)
Noise2Selft
UNSURE (7 oracle)
CENSURE: (ours)
FSNK
250
PSNK: 25
25
Ground Truth PSNK
Noisc2Self
UNSURE oracle)
CENSURE (ours)
25.500
PSNR
25.5
Noisy
MC-SURE (7 oracle)
Noise2Selft
UNSUREt (7 oracle) CENSURE; (ours)
Figure 8: Qualitative image denoising results. models were trained on dataset with constant noise level (one model per noise level). Models converted into noise-level-aware variants. ing Propositions and are indicated by the symbol f (no additional training). Best viewed by zooming.
24

---

Published as a conference paper at ICLR 2026
PSNR: 22.86 dB
PSNR: 32.13 1SNN1322.1.40 dB
PSNR: 31-67 dB
(a)
Noisy
Supervised
UNSURE
PSNR: OC dR
PSNR: 32:25 dB
PSNR: 32:40 dB
Noisy
Ground Truth
Neighbor
CENSURE (ours)
PSNR: 25.07 dB
PSNR: 35.01 dB
PSNR: 34.59
(
Noisy
Supervised
UNSURE
PSNR: dB
PSNR: 34.95 dB
PSNR: 34.88 dB
Noisy
Ground Truth
Neighbor2Neighboor
CENSURE (ours)
PSNR: 26.15 dB
PSNR: 40.00 dB
PSNR: dB
(c)
Noisy
Supervised
UNSURE
PSNR: (c dR
PSNR: 39.98 dB
PSNR: dB
Noisy
Ground Truth
Neighbor2Neighhor
CENSURE (ours)
PSNR 25.08 dB
PSNR: 39.02 dB
PSNR: 35.25
(d)
Noisy
Supervised
PSNR: dB
PSNR: 38.97 dB
PSNR: 38.66 dB
Noisy
Ground Truth
Neighbor2Neighhoor
CENSURE (ours)
Figure 9: Demonstration of the applicability self-supervised Gaussian denoisers trained on syn- thetic data (with constant noise level 25) to real-world fluorescence microscopy images, using Noise2VST framework (Herbreteau & Unser 2025). (a) Confocal fish from FMD dataset (Zhang et al. 2019); (b) two-photon mice from the FMD dataset; (c-d) fluorescence mieroscopy images from the W2S dataset (Zhou et al.) 2020). The denoisers are the same as those reported in 'able under the unknown-noise setting view by zooming.
25). Colors are for aesthetic purposes only. Best
25


Reviews:
[{"review_id": "zHqIyq9CKT", "reviewer": "Reviewer_7HQs", "summary": "This paper presents a **Divergence-Free Neural Network (DivFree)** designed for self-supervised Gaussian image denoising.  \nThe network is parameterized with skew-symmetric bases so that ∇·f(x)=0 holds exactly, thereby eliminating the need for Monte-Carlo divergence estimation used in SURE-based losses.  \nThe authors claim this leads to more stable training and improved performance compared to MC-SURE and UNSURE.  \nExperiments are limited to grayscale Gaussian noise (σ=15–50), with an additional “unknown σ” case that is not truly blind.", "questions": "1. How is the “unknown σ” setting defined? Was σ randomly sampled during training or fixed to one value?  \n2. Could the same architecture be evaluated on **Poisson denoising** using **PURE** to demonstrate generality beyond Gaussian SURE?  \n3. Why not apply divergence-free constraints to other tasks (e.g., optical flow, physics-based fields) where this property is physically meaningful?  \n4. Does enforcing div(f)=0 introduce measurable bias compared to the true SURE optimum?  \n5. Given the reported 9× inference cost, is the stability improvement practically worthwhile?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 1, "strengths": "- Mathematically consistent derivation of divergence-free vector fields using a skew-symmetric representation.  \n- Conceptually simple idea that connects physical conservation laws with SURE-based denoising.  \n- Reduces gradient variance by removing stochastic divergence estimation.", "weaknesses": "- **Misleading “unknown σ” claim.** The model is trained with a fixed σ and only tested on slightly different noise levels; there is no random-σ training or blind estimation, so it cannot handle unknown noise.  \n- **Extremely narrow scope.** All experiments use grayscale Gaussian noise. There are no color image... The method is not validated on **Poisson**, **Gamma**, or real-world datasets, although its principle directly relates to unbiased risk estimators like **PURE (Poisson Unbiased Risk Estimator)**.  \n- **High computational cost.** The paper itself states that inference is about **9× slower** than DRUNet due to repeated gradient evaluations for enforcing divergence-free constraints.  \n- **Limited novelty.** The method merely re-parameterizes the denoising backbone to set div(f)=0, optimizing the same SURE objective as MC-SURE. The observed differences stem from reduced variance but increased structural bias.  \n- **Lack of generality.** Divergence-free constraints could naturally apply to flow, optical-flow, or magnetic-field problems, yet the paper confines itself to toy Gaussian denoising.\n\n## Minor\n- No visualization verifying that learned vector fields are actually divergence-free.  \n- Bias–variance trade-off introduced by the constraint is never quantified.  \n- Comparison baselines (Noise2Score, Blind2Unblind, Neighbor2Neighbor, diffusion-based denoisers) are missing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Toercmx0QB", "reviewer": "Reviewer_G54J", "summary": "This paper proposes a new class of divergence-free denoisers by interpolating the constraints of two prior classes of divergence-free denoisers and also a method for constructing denoisers that belong to the new class directly, without the need to use additional loss functions during training like prior methods. The authors show improved performance with known noise levels, and claim this is due to closer adherence to the divergence-free constraint, as well as comparable performance in the unknown noise level setting.", "questions": "How does the following depend on K'?\n- denoising performance\n- runtime\n- divergence \n\nI would be willing to upgrade my score if the authors could provide a more thorough empirical validation of their method, in particular with respect to the major approximation that causes deviations from theory, the number of basis components of their divergence-free denoiser field.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "The idea of examining a class of objects (in this case, denoisers) in between two other existing classes in the literature is very well motivated. I could not find any issues with the theory, which is built on existing solid results in a reasonable way. The results also seem consistent with the understanding of the method presented in the paper, namely that in the known noise-level regime their more constrained method achieves higher performance than the less constrained baseline which does not enforce zero divergence strictly.", "weaknesses": "I'm not sure about the resource efficiency claims. It seems that mostly the case rests on using far fewer terms than necessary as a basis to represent the divergence-free denoiser. I think the paper would be stronger if the authors could show what the exact influence of the number of components, K', is on the denoising performance and how close their denoiser is to being truly divergence free. My understanding of the theory is that it only guarantees a divergence-free field if the number of basis terms is n(n+1)/2, where n is input size, but far fewer basis terms are used for the sake of computational tractability. Additionally, the authors should compare the runtime of the various methods, since it appears that their denoiser requires effectively K'+1 network evaluations, combining forward and backward passes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "udrJF8ObgG", "reviewer": "Reviewer_zsZU", "summary": "SURE needs divergence. To get rid of the divergence, they introduce a constant divergence field S_DC (Eqn 10). Section 4 is about how to generate such a divergence field. The core trick is by means of a universal approximation (Theorem 1). Realization is given in Section 4.2.", "questions": "No major concerns.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Solid contribution.\n\nThe coverage of the prior work is excellent.\n\nIntuition of the field construction can be done better. But basically I get the idea that they want to ensure symmetry so that the field is valid (Eqn. 13 and Eqn 14).\n\nEmpirical results are okay. This is a piece of theoretical contribution.", "weaknesses": "This is not a criticism but a general comment. Will not contribute to my judgment of the paper.\n\nSURE is a proxy for MSE, which assumes that the underlying noise is iid Gaussian. But if I know that the noise is iid Gaussian, I can literally just simulate training data and train my model in a supervised way. The cost of doing that is null. Even if we say that the noise level is unknown, worst case I just train a larger model with more data augmentation. So I have a hard time to convince myself that SURE is the right way to go. \n\nI can appreciate unsupervised learning, but it must be some bizarre unknown noise type that I cannot easily calibrate and know from my camera. But for this case, SURE doesn't seem to be the right approach to go.\n\nSo I am not sure about the utility down the road. Perhaps denoising is not a good showcase?? No need to address this in the rebuttal. Just some thoughts to share with the authors.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "za8JEdCEMR", "reviewer": "Reviewer_o8EU", "summary": "This paper introduces a theoretically grounded framework for constructing neural networks that are divergence-free for image denoising. The authors propose a resource-efficient parameterization that represents the network output as a structured combination of conservative fields. The proposed divergence-free property is used to simplify SURE loss, avoiding the instability of Monte Carlo approximations and the expressivity limitations of blind-spot methods. The experimental results show that the proposed method achieves better performance against other divergence-based approaches", "questions": "- The paper argues that the proposed method is more expressive than blind-spot methods because it doesn't ignore the central pixel. However, blind-spot methods are typically blind to the noise distribution, whereas DivFree's theoretical advantage requires knowing the noise level. Could you discuss this trade-off between expressivity and the need for prior information? How does your method perform if the provided noise level is not very precise", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ The main strength of this paper is its rigorous mathematical analysis.\n+ The method for constructing the sparse, parameterized skew-symmetric matrices and sharing the scalar potential network is reasonable. It successfully makes the theoretical representer theorem computationally tractable for high-dimensional data like images.", "weaknesses": "- The experimental evaluation is focused only on divergence-based self-supervised methods (MC-SURE, Noise2Self, UNSURE). The field of self-supervised (or unsupervised single-image) denoising is much broader and has developed significantly. The paper is missing crucial comparisons to other state-of-the-art methods that are not based on SURE, which makes it impossible to assess the true competitiveness of the proposed approach, e.g., [1-3]\n[1] High-quality self-supervised deep image denoising, NIPS 2019\n[2] Iterative denoiser and noise estimator for self-supervised image denoising, ICCV 2023\n[3] Positive2Negative: Breaking the Information-Lossy Barrier in Self-Supervised Single Image Denoising, CVPR 2025\n- The entire framework is demonstrated only for additive white Gaussian noise. While this is a standard setting for theoretical exploration, its practical use is unexplored. The paper itself acknowledges this limitation but does not provide any discussion or preliminary results on how the divergence-free constraint might perform with more complex, realistic noise models (e.g., Poisson-Gaussian, spatially correlated, or signal-dependent noise). The heavy reliance on SURE makes the direct extension non-trivial\n- The proposed parameterization is complex, though more efficient than the full model. This significant overhead could be a barrier to practical adoption, and this trade-off is not sufficiently emphasized in the results", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)