UNSURE: self-supervised learning with Unknown Noise level  and Stein's Unbiased Risk Estimate
Authors: Julián Tachella
Mike Davies
Laurent Jacques
Venue: ICLR 2025 Poster (2025)
Keywords: self-supervised learning
imaging inverse problems
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: We present an extension of SURE that doesn't require knowledge of the noise level, and a new mathematical framework to understand self-supervised learning losses.

Abstract: Recently, many self-supervised learning methods for image reconstruction have been proposed that can learn from noisy data alone, bypassing the need for ground-truth references.  Most existing methods cluster around two classes: i) Stein's Unbiased Risk Estimate (SURE) and similar approaches that assume full knowledge of the noise distribution, and ii) Noise2Self and similar cross-validation methods that require very mild knowledge about the noise distribution. The first class of methods tends to be impractical, as the noise level is often unknown in real-world applications, and the second class is often suboptimal compared to supervised learning.
In this paper, we provide a theoretical framework that characterizes this expressivity-robustness trade-off and propose a new approach based on SURE, but unlike the standard SURE, does not require knowledge about the noise level. Throughout a series of experiments, we show that the proposed estimator outperforms other existing self-supervised methods on various imaging inverse problems.

Published as a conference paper at ICLR 2025
UNSURE: SELF-SUPERVISED LEARNING WITH UN- KNOWN NOISE LEVEL AND STEIN'S UNBIASED RISK ESTIMATE
Julián Tachella'
Mike Davies
CNRS & ENS Lyon
University of Edinburgh
Lyon, France jul ian. tachettccellln waacchelll@ensseee.
Edinburgh, UK
m ke davies ac..uk
Laurent Jacques
UCLouvain Louvain-la-Neuve, Belgium
laurent. fateent.jacquusee jacqueséuclouvaing
ABSTRACT
Recently, many self-supervised learning methods for image reconstruction have been proposed that can learn from noisy data alone, bypassing the need for ground-truth references. Most existing methods cluster around two classes: i) Stein's Unbiased Risk Estimate (SURE) and similar approaches that assume full knowledge the noise distribution, and ii) Noise2Self and similar cross- validation methods that require very mild knowledge about the noise distribu- tion. The first class of methods tends to be impractical, as the noise level is of- ten unknown in real-world applications, and the second class is often suboptimal compared to supervised learning. this paper, we provide a theoretical frame- work that characterizes this expressivity-robusnness trade-off and propose new approach based on SURE, but unlike the standard SURE, does not require knowl- edge about the noise level. Throughout series of experiments, we show that the proposed estimator outperforms other existing self-supervised methods on various imaging inverse problems.
INTRODUCTION
Learning-based reconstruction methods have recently shown state-of-the-art results in wide variety of imaging inverse problems, from medical imaging to computational photography (Ongic et al., 2020). However, most methods rely on ground-truth reference data for training, which is often expensive or even impossible to obtain (eg. in medical and scientific imaging applications). This limitation can be overcome by employing self-supervised learning losses, which only require access to noisy (and possibly incomplete) measurement data (Tachella et al. 2023a).
Self-supervised denoising methods cluster around classes: i) SURE Metzler et al. 2020). Nois- ier2Noise (Moran et al. (020) and similar methods which assume full knowledge about the noise distribution, and ii) Noise2Self (Batson & Royer 2019) and other cross-validane validation methods which only require independence of the noise across pixels. While networks trained via SURE perform better than cross-validation methods, they are brittle to misspecification of the noise level, which is often not fully known in real-world imaging settings. This phenomenon can be seen robustness- expressivity trade-off, where leveraging more information about the noise distribution results in more optimal estimators, which are at the same time less robust to errors about the noise model. In this paper, we show that this trade-off can be understood through the set constraints imposed on the derivatives of the estimator: SURE-lik like estimators do not impose any constraints derivatives, whereas cross-vass validation methods add strong constraints on them. Our analysis paves the way for
"Code associated this paper is available at www.gi...oo/acccellaaaaa.

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new family of self-supervised estimators, which we name UNSURE, which lie between these two extremes, by only constraining the expected divergence of the estimator to be zero.
The contributions of the paper are the following:
1. We present theoretical framework for understanding the robustness-eepressivis trade-off of different self-supervised learning methods.
2. We propose a new self-super objective that extends the SURE loss for the case where the noise level is unknown and provide generalizations to spatially correlated Gaussian noise (with unknown correlation structure), Poisson-Gaussian noise, and certain noise dis- tributions in the exponential family.
3. Throughout a series of experiments imaging inverse problems, we demonstrate state-of- the-an art self- sulervupe learning performance in settings where the noise level (or its spatial correlation) is unknown.
1.1 RELATED WORK
SURE Stein's unbiased risk estimate is a popular strategy for learning from noisy measurement data alone, which requires knowledge of the noise distribution (Stein [981] Efron 2004 Aggarwal
et al. 2023). The SURE loss has been extended for large classes of noise distributions (Hudson
1978
Raphan & Simoncelli 2011: | Le Montagner et al. 2014), and has been used to train networks
in self-supervised way when the noise distribution is known (Metzler al. 2020 Chen et al.
2022). ENSURE (Aggarwal et al. 2023), despite having a similar name to this work, proposes weighting correction to SURE in the case where we obtain observations from multiple operators, and does not handle unknown noise levels. Zhussip et al. (2019) presents a variant of SURE that takes into account two noisy images instead of a single one, and assumes that the noise level is known. SURE belongs more broadly to the class of empirical Bayes methods (Raphan & Simoncelli] [ Efron 2011: Robbins 1964; Efron 2012), which build estimators from measurement data alone for a large class of noise distributions, however, to the best of our knowledge, the case of partially unknown noise distribution has not been considered.
Noise2Noise It is possible to learn estimator in self- way if two independent noisy realizations of the same image are available for training, without explicit knowledge the noise dis- tribution (Mallows) 1973). This idea was popularized in imaging by the Noise2Noise method (Lehti- nen et al. 2018). However, obtaining two independent noise realizations of the same signal is often impossible.
Noisier2Noise and Related Methods Noisier2Noise (Moran et al. 2020). the coupled bootstrap estimator (Oliveira et al.) 2022) and Recorupted2Rcorupeee (R2R) Pang et al. (2021), leverage the fact that two independent noisy images can be obtained from a single noisy image by adding additional noise. However, the added noise must have the same noise covariance the original noise in the image, and thus full information about the noise distribution is required. Noise2Score (Kim) & Ye. 2021), propose learn the score of the noisy data distribution, and then denoise the images via Tweedie's 's formula which requires knowledge about the noise level. (Kim et al.) (2022) proposes way to estimate the noise level using the learned score function. However, the method relies on Noisier2Noise approximation, whereas we provide closed-form formulas for the noise level estimation.
Noise2 and Validaaaliotio Methods When the noise is iid, a denoiser that does not take into account the input pixel to estimate the denoised version of the pixel cannot overfit the noise. This idea goes back to tross-validation-bases estimators (Efron) 2004). One line of work (Krull al. 2019) leverages this idea to build self-supervised losses that remove the center pixel from the input, whereas another line of work builds network architectures whose output does not depend on the center input pixel (Laine et al. 2019). These methods require mild knowledge about the noise distribution (Batson & Royer 2019) (in particular, that the distribution is independent across pixels), but often provide suboptimal performances due to the discarded information. Several extensions have been proposed: Neighbor2Neighhor (Huang al. 2021 is among the best-performing for denoising, and SSDU (Yaman et al. 2020) and Noise2Inverse (Hendriksen et al., 2020) generalize the idea for linear inverse problems.
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Learning From Incomplete Data In many inverse problems, such as pparse-angle tomography, image inpainting and magnetic resonance imaging, the forward operator is incomplete as there are fewer measurements than pixels to reconstruct. In this setting, most superviie denoising meth- ods fail to provide information in the nullspace of the forward operator. There are two ways to overcome this limitation (Tachella al. 2023ap: using measurements from different forward opera- tors (Bora et al., 2018; Tachella et 2022 Yaman et al., 2020 Daras et al. 22024), or leveraging the
invariance of typical signal distributions to rotations and/or translations (Chen al. 2021 2022).
Approximate Message Passing The approximate message-passing framework (Donoho et al., 2009) for compressed sensing inverse problems relies on an Onsager correction term, which results divergence- free denoisers the large system limit (Xue et al. 2016 Ma & Ping 2017 Skuratovs & Davies 2021). However, different from the current study, the purpose of the Onsager correction is to ensure that the output error each iteration appears uncorrelated subsequent iterations. this work, we show that optimal self-supervised denoisers that are blind to noise level are divergence-free expectation.
2 SURE AND CROSS- VALIDATION
We first consider the Gaussian denoising problems of the form
+
(I)
where y R" are the observed measurements, R" is the image that we want recover, (O I) is the Gaussian noise affecting the measurements. This problem can be solved by learning estimator from a dataset of supervised data pairs (æ,, y) and minimizing the following supervised loss
arg min Es (xwl(f()) - x=1
(2)
whose minimizer is the minimum mean squared error (MMSE) estimator f(y) = E( y). How- ever, many real-world applications, do not have access to ground-truth data æ for training, and instead only a dataset of noisy measurements y. SURE (Stein 1981) provides way to bypass the need for ground-truth references since we have tha
F F (1y) 2= = Fuy [1l.(()) - y/1² + 20 div)()))no no?].
where the divergence is defined as div f(y) = Eis ay. self-supervised loss
0s: (y). Thus, we can optimize the following
arg min Fy [1l/(y) y/ + 20 div div)
(SURE)
JEC
where L is the space of weakly differentiable functions, and whose solution is given by Tweedie's formula, i.c.,
B(xly} = y + o2 log
where Py is the distribution of the noise data y. While this approach removes the requirement of ground-truth data, it still requires knowledge about the noise level o², which is unknown in many applications.
Noise2Void (Krull et al., 2019). Noise2Self (Batson & Royer 2019) and similar cross- validation (CV) approaches do not require knowledge about the noise level by using estimators whose ith output (1)) does not depend on the ith input yi. which in turns implies that 0f (y) 0
Zy.
for all pixels i 1,..., n and all y R", These estimators have thus zero divergence div f for all y E R", and thus can be learned by simply minimizing data consistency term
arg min Ey (l) y
(5)
fEScv
'In practice, we replace the expectations over y by sum over a tinite dataset {yr obtaining empir- ical estimators. The theoretical analysis focuses the asymptotic case of N oo provide stronger characterization of the resulting estimators.
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SURE, R2R
UNSURE
8ft Eyy Egu
Noise2Selff Noise2lnverse, BlindSpol Nets
8fi By 8y:
(2)
Bay 8y:
5fi (g)
Af. B3 Byi
öf
(v) vy
Less expressive estimators
öy:
y = + de
y x
y
YPY
oo
y/2 p(ua)Tm)) Bylæ =
known
unknown scalar
unknown vector o
unknown vectors 0,7 unknown Pt
trade-off in self-supervised denoising. the noise dis- tribution is fully known, SURE provides the most expressive estimator, matching the performance of supervised learning. As the assumptions on the noise are relaxed, the learned estimator needs to be less expressive avoid over-fitting the noise. In this work, show that popular 'Noise2x' strategies impose too restrictive conditions on the learned denoiser, and propose an alternative that strikes better trade-off.
Figure The expressivity-robustii
where the function is restricted the space
0f:
Scv (f E E'
(y) for almost every y E R").
oy:
Using the SURE identity in 3 we have the equivalence with the following constrained supervised loss
Jcv arg min Ex, ar aarsll.. (1)()))))))))))))){{)) z=|²
fESev
whose solution is f.v(3)) Bl excluding the ith entry Vi- The constraint can be enforced in the architecture using convolutional networks whose receptive field does not look the center pixel (Laine al., 2019), via training. by choosing a random set of pixels at each training iteration, setting random values (or the value of of their neighbors) the input of the network and computing the loss only those pixels (Krull et al. 2019).
E(r;ly-i} for 1,..., where y-r denotes
The zero derivative constraint results in suboptimal estimators. since the ith measurement yt gener- ally carries significant information about the value of r₁. The expected mean squared eтror of the cross-validation estimator
En.yy - (1)) x x112 n2
1 n Ey V(alyy 72 Ey
with V the variance operator and the performance of this estimator can arbitrarily bad if there is little correlation between pixels, ie. V(zily-i) ~ Vxx,), in which case the mean squared error is simply the average variance the ground-truth data Ez,y 1 ee (y) - x1|² E. V(z)).
UNSURE
In this work, we propose to relax the zero-derivative constraint of cross-validation, by only requiring that the estimator has zero expected divergence (ZED), ie. Ey fly) O. We can then minimize the following problem:
7ED arg min Eyl1 yy1
f
where SZED (f E Lt: Ey, fzy) 0}. Since have that Sev C SZED, the estimator that is divergence-free expectation more expressive than the cross-validation counterpart. Again due to SURE, (7) is equivalent to the following constrained supervised loss
ZED arg min Ez,yy f x=12
min fESm
(8)
The constrained self-supervised loss in 7 can be formulated using Lagrange multiplier ry E IR
min max F,y [ll/(y)) - yll² + 2n div/dvvie /1))
(UNSURE)
/ 7
which has simple closed-form solution:
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Proposition 1. The solution of (UNSURE)
is given by
FZED
② y fjV (o)
where nj ( By V log (g) E R+ is the optimal multiplier.
natural question at this point is, how good can a ZED denoiser be? The theoretical performance of optimal ZED denoiser is provided by the following result:
Proposition 2. The mean squared error of the optimal divergence-free in expectation denoiser is given by
Ez.y -11 11 2ED (2) 11
x
1)
(10)
MMSE
where MMSE Er,y 7 [arly) x212 is the minimum mean squared error.
The proofs of both propositions are included in Appendix
We derive several important observations from these propositions:
1. As with Tweedie's formula, the optimal divergence- ffrec in expectation estimator in (9) can also interpreted as doing gradient descent step on log Py, but the formula agnostic to the noise level as only requires knowledge of V log Py (y) to compute the estimator.
2. The step size f) is a conservative estimate of the noise level, as we have that
0²
n
2 0²
(11)
1
MMSE
3. As we have that By log 1o (y)
(1)
8 log Pyly)
99
ressembles the natural
Oy;
gradient descent algorithm (Amari 2016).
4. Combining (4) and we can write f2ED as the convex combinatior of the MMSE esti- mator and the noisy input:
ZED (y) (Bly)) + w)y
(12)
where
[).
Ey (( M
5. We can expand the formula
(10 as a geometric series since
MMSE o²
< 1 to obtain
MMSE
Em.gy
7ED yy) - x|| 2 MMSE +
(13)
If the support of the signal distribution p(z) has dimension < n, we have that MMSE R σ 02k/n (Chandrasekaran & Jordan 2013), and thus i) the estimation of the noise level is accurate as 0² + (ok) ( and ii) the ZED estimator performs similarly to the minimum mean squared one,
Ex.y f (y) æ1|² x MMSE + (ak)² Perhaps surprisingly, in this case, the ZED estimator close to optimal without adding any explicit constraints about the low-dimensional nature of Pa-
The ZED estimator fails with high-entropy distributions, for example Pe N/ Io?). In this case, we have ń = x² + 02. as the noise level estimator cannot tell apart the variance coming from the ground truth distribution o from that coming from the noise a², and ZED estimator the trivial guess (2ED O. Fortunately, this worst-case setting is not encountered in practice as most natural signal distributions are diw-dimensional. To illustrate this, we evaluate the performance of a pretrained deep denoiser in comparison with its ZED version on the DIV2K dataset (see Figure 2 Remarkably, the ZED denoiser performs very similarly to the original denoiser (less than dB difference across most noise levels), and its error is very well approximated by Proposition 2
A presents the ZED estimators associated to popular separable signal distributions: spike- and-slab, Gaussian, and two deltas, illustrating the differences with MMSE and cross-vas cros-vvalidation estimators.
"The cross-validation estimator can be written convex combination of MMSE evaluations, as have that f.y (v) =S,, ( (dy, for 1,...,7.
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DRUNet
Test PSNR [dB]
ZED DRUNet
ZED theoretical MSE
0.1
0.3 0.4
Figure 2: Removing the expected divergence of pretrained MMSE denoisers. Considering the DRUNet as approximate MMSE denoiser, we use the formula in (12 to evaluate its ZED version and plot the expected theoretical error according to (TO). The ZED correction only results in a loss of less than dB for most noise levels. Moreover, the theoretical emror follows very closely the empirical one.
3.1 BEYOND ISOTROPIC GAUSSIAN NOISE
Correlated Gaussian noise many applications, the noise might be correlated across different pixels. For example, the most popular noise model is
y Nxx
where E is the covariance matrix capturing the correlations, which is often partially unknown. This setting is particularly challenging where most self-supervised methods fail. If the noise covariance known, the SURE formula in (3) is generalized
0f (v) 0y tr (Σ)
())
Exry lf(())
2|| = Ey [l.f(w) v/1 2t
Σ
the exact form of noise covariance is unknown, we can consider an s-dimensional set of plausible covariance matrices
R R= [E, E Ruxn, E, = Σ n E
for some basis matrices (vr [axm] Jj=1, with the hope that the true covariance belongs to this set, that E R. This is equivalent to restricting the learning estimators in the set
(S E L Ey V (f (v) 0, 8). Here, the dimension 2 offers a trade- 1
off between optimality of the resulting estimator and robustness to a misspecified covariance. We can thus generalize (UNSURE) as
Of Dy
min max E, (If(y) - y11² 2tr "
Σn
(v)
(14)
The following theorem provides the solution for this learning with Lagrange multipliers 1 E R problem:
Theorem 3. Let Pa Pa No, Σ) and assume that { E Raxnys are linearly independent.
The optimal solution of problem (14) is given by
f(y) y + En log
(15)
where à = =Q- = with Qú, = tr v Ey [V log (y) log ( J) and 8s tr (v.) for j ,.....8.
The proof is included in Appendix Proposition is special case with = 1 and L I.
For example, an interesting family of estimators that are more flexible than fCV and can handle different unknown noise level per pixel, is obtained by considering the diagonal parameterization
Σ, diag(1). In this case, we have 7 =EE,
8 llogss
2) for 1,..., and thus
Oy:
i log Pa
bog Py (y)
f(()) = + (,,
(y)
Oy:
Oy:

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for i 1,..., The estimator verifies the constraint By: By. is still more flexible than cross-validation, the gradients are only in expectation.
of (y) 0 for all i = .,...,
.,....... and thus
Another important family is related to spatially correlated noise, which is generally modeled as
y = x + a
where * denotes the convolution operator, E R a vector-valued and e ~ No I). If we don't know the exact noise correlation, we can consider the set of covariances with correlations up to ir taps/pixel we can minimize
min Eylls(y) - y/
(16)
SEBc
with Sc If L' :B. Ei 02(++))ond
04
(y) 0, jE {-r, ................................ r}} equivalently
t-r.....
Of
min max Ey (1f(y) y112 +
(y)
(C-UNSURE)
(y( mod
whose solution is f') y + f V log Py(()) (3) where the optimal multipliers are given by v F (1/F)))) where the division is performed elementwise, h E g is the =r tap autocorrelation of the score and F ((2++1)x(2+11) is the discrete Fourier transform (see Appendix A for more details).
noise In many applications. the noise has a multiplicative nature due to the dis- crete nature of photon-counting detectors. The Doisson-Gaussian noise model is stated as (Le Mon- tagner et al. 2014)
x + y y= -
(17)
where denotes the Poisson distribution, and the variance of the noise dependent on the signal level, since (1ilzz)) = xey + o². In this case, we minimize
min E, (1f(y) yll SESM:
(18)
with SPG = If C Ey divf(y) = 0, By y ff(z) = 0}. or equivalently
Of Dy.
min max Ey
l1f(y) - y + Σ 2 + uur)
(V)
(PG-UNSURE)
f 7.7
1=1
whose solution is f(y) y + (1n + yn) V Pog (3) 1 where i) and & are included in Ap- pendix C
Exponential family noise distributions We can generalize UNSURE to other noise distributions
with unknown variance by considering generalization of Stein's lemma, introduced by Hudson
1978). In this case, we consider the constrained set {f E L Ey Σn a(y;) 8f some function a R + R which depends on the noise distribution. For example, the isotropic Gaussian noise case is recovered with a(y;) 1. See Appendix B for more details.
By, (y) = 0} for
General inverse problems We consider problems beyond denoising = Ax + e, where y, E E I.T x E R" and A E guxx where generally we have fewer measurements than pixels, ie. m n. We can adapt UNSURE to learn in the range space of A T via
min max Ey A|! (Af(y) y) 11 2n div A'A]] (1)
(General UNSURE)
where At E RR is the linear pseudoinverse of A Eldar 2009), or stable approximation thereof. As with SURE, the proposed loss only provides estimate of the error in the nullspace of A. To learn in the nullspace A, we use General UNSURE with the equivariant imaging (EI) loss (Chen) et al. 2021), which leverages the invariance of natural image distributions to geometrical transfor- mations (shifts, rotations, etc.), please see Appendix F for more details. A theoretical analysis about learning in incomplete problems can be found in Tachella al. (023a).
Here consider 1-dimensional signals for simplicity but the result extends trivially to the 2-dimensional
case.

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4 METHOD
We propose two alternatives for learning the optimal ZED estimators: the first solves the Lagrange problem in [4], whereas the second one approximates the score during training, and uses the for- mula in Theorem for inference.
UNSURE We can solve the optimization problem in (4) by parametrizing the estimator So us- ing a deep neural network with weights 0 E I", and approximating the expectation over y by a sum over a dataset of noisy images {y, j=1 During training, we search for saddle point of the loss on 0 and 17 by alternating between a gradient descent step on and a gradient ascent step on 77. The gradient with respect to n does not require additional backpropagation through fo and thus adds a minimal computational overhead. We approximate the divergence using a Monte Carlo approximation (Ramani et al.
2008
tr M a (g) dy
(M (Jaly + 7b) (oo
(19)
where M& E Raxn b (O, 7) and 0.01 is small constant, and for computing the divergence in UNSURE we choose M = I, and for computing the second term in (PG-UNSURE) we choose
M diag(7 7 + rg). A pseudocode for computing the loss is presented in Appendix
UNSURE via score Alternatively, we can approximate the score of the measurement data with a
deep network so(y) VlogPy(3) using the AR-DAE loss proposed in Lim et al. (2020); Kim &
Ye 2021).
arg min ((g,b.)) ||b + rso(y) + Tb)l
(20)
where b ~ (0, 1) and d²). Following Lim et al. 2020), we start with large standard deviation by setting δmax and anneal it to a smaller value dmin during training At inference time,
reconstruct the measurements y using Theorem 3 as it only requires access to the score:
so(y) = y + En s0(y)
where Σg is computed using Theorem 3 with H ~ Ey (y()))((()) The expectation can be computed over the whole dataset or a single noisy image if the noise varies across the dataset. As we will see in the following section, this approach requires a single model evaluation per gradient descent step, compared to two evaluations when learning fo for computing (19). but it requires more epochs to converge and obtained slightly worse reconstruction results in our experiments.
5 EXPERIMENTS
We show the performance of the proposed loss in various inverse problems and compare it with state-of-the-art self-supervised methods. All our experiments are performed using the deep inverse library (Tachella et al., 2023b). We use the AdamW optimizer for optimizing network weights with step size x 10 and default momentum parameters and set 0.01, μ 0.9 and 0.01 for computing the UNSURE loss in Algorithm [ Examples of reconstructed images are shown in Figure 4 Results on real Cryo-EM EM data (Bepler al. 2020) are included in Appendix E
Gaussian denoising on MNIST We evaluate the proposed loss for different noise lev- els {0.05, 0.1, 0.2, 0.3, 0.4, 0.5) of Gaussian noise the MNIST dataset. We com- pare with Noise2Score (Kim et al. 2022), Neighbor2Neighbor (Huang et al.) [021]. recor- rupted2recoruuped (Pang et al. 1 and SURE (Metzler al., 2020) and use the same U-Net architecture for all experiments (see Appendix F for more details). The results are shown in Fig- ure The Lagrange multiplier 7) converges in few epochs to a slightly larger value of for all evaluated noise levels. SURE and R2R are highly sensitive to the choice of a. providing large errors when the noise level is wrongly specified. The proposed UNSURE loss and UNSURE via score do not require knowledge about the noise level and perform only slightly worse than the supervised case. We also evaluate the UNSURE via score approach using image-wise noise level estimations, obtaining similar performance than averaging over the whole dataset. As the best results are ob- tained by the UNSURE loss, we use this variant in the rest of our experiments.
'In our Gaussian denoising experiments, we set 6max 0.1 and 6min. 0.001 in Kim
2022
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Estimation y during training Test PSNR during training
Estimated noise level
0.20
0.5
30
0.15
PSNR
PSNR
20
20
15
75
50
100
0.2
0.4
0.2
0.4
epoch
epoch
0
0
supervised unsure neighbor2neighbor R2R
unsure
score
noise2score
unsure imagewise
SURE a= 0.2
Figure 3: MNIST denoising experiments. From left to right: i) evolution of the Lagrange multiplier
7 for the case with 0.2, ii) test PSNR during training for UNSURE and UNSURE via score with 0.1, iii) estimated Lagrange multiplier as function of a, iv) average test PSNR for various methods and various noise levels a.
Colored Gaussian noise on DIV2K We evaluate the performance of the proposed method on correlated noise on the DIV2K dataset (Zhang et al., 2017]. by adding Gaussian noise convolved with box blur kernel of 3 x 3 pixels. To capture the spatial structure of noise, we parameterize E, (14) as circulant (convolution) matrix, as explained in Section 3.1 bound on the support of the blur is needed for the proposed method, while the specific variance in each direction is not necessary. We train all the models on 900 noisy patches of 128 x 128 pixels extracted from the training set and test on the full validation which contains images of more than 512 x 512 pixels.
worth noting that only
Table I shows the results for the different evaluated methods (Neighbor2Neighhor, Noise2 and SURE). The proposed method significantly outperforms other self-supervised baselines which fail capture correlation, performing and supervised learning. Table 2 shows the impact of the choice of the kernel size in the proposed method, where overspecifying the kernel size provides a significant improvement over underspeci- fication, and a mild performance reduction compared to choosing the exact kernel size.
dB below the SURE method with known noise covariance
UNSURE
SURE
Method
Noise2 Void
Neighbor2Neighhor
Supervised
(unknown E)
(known Σ)
PSNR [dB)
19.09 + 1.79
23.61 + 0.13
28.72 1.03
29.77 = 1.22 29.91 + 1.26
Table 1: Average test PSNR of learning methods for correlated Gaussian noise.
Kernel size
x
3 3
x 5
PSNR [dB]
23.62
28.72
1.03
27.38 + 0.88
Table 2: Average PSNR of the proposed method as function of the chosen blur kernel size, where the ground-truth kernel had size 3 x 3.
Computed tomography with Poisson-Gaussian noise on LIDC evaluate a tomography prob- lem where (resized) images of 128 x 128 pixels taken from the LIDC dataset, are measured using a parallel beam tomography operator with 128 equispaced angles (thus does not have signifi- cant nullspace). The measurements are corrupted by Poisson-Gaussian noise distribution in (17) of levels 7
0.005. We compare our method with Poisson-Gaussiann SURE loss proposed in (Le Montagner et al. 2014), and cross-validation approach similar to Noise2lnverse (Hendriksen et al. 2020). In this case, we use the loss in (PG-UNSURE) with 77 and y as Lagrange multipliers, together with the pseudoinverse correction in GGeneral UNSURE We evaluate all methods using unrolled network with U-Net denoiser backbone (see Appendix F for details). Table presentss the test PSNR of the compared methods. The proposed loss outperforms the cross-validation approach while being close to SURE with known noise levels.
Accelerated magnetic resonance imaging with FastMRI We evaluate a single-coil 2x- accelerated MRI problem using subset of 900 images of the FastMRI dataset for training and

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Published as a conference paper at ICLR 2025
PG-UNSURE (unknown 7) 33.31 - 0.57
PG-SURE (known a, 7)
Method
Noise2Inverse
Supervised
PSNR [dB]
32.54 + 0.71
33.76 0.61
34.67 + 0.68
Table 3: Average test PSNR of learning methods for computed tomography with Poisson-Gaussian Gaussian noise.
Ground-rruth
Meesurement
Noves2Void
NeighoorZNeghhorr
UNSURE
SURE known 23
Supervised
Ground-rruhh
Messuremmnt
Ioveo2inverro
SURE+EI
Supervised
Ground-truth
Measurement
Noise2mivrree
PG-UNBURE
PG-SURE
Supervised
Figure 4: Image reconstruction results for various imaging problems. Top: colored Gaussian noise DIV2K. Middle: Accelerated magnetic resonance imaging with FastMRI. Bottom: com- puted tomography with Poisson-Gaussian noise on LIDC
100 for testing (Chen al. 2021). and adding Gaussian noise with 0.03 to the k-space mea- surements. We evaluate all methods using an unrolled network with a U-Net denoiser backbone (see appendix for more details). We compare with cross-validation, simple measurement consis- tency (1) - ") and SURE with slightly wrong noise level o = 0.05. The cross-validation method is equivalent to the SSDU method (aaman et al. 2020). We add the El loss with rotations as proposed in (Chen et al. (02T) to all methods since this problem has non-trivial nullspace. Table 4 shows the test PSNR for the compared methods. The proposed method performs better than all other self-supervised methods, and around dB worse than supervised learning.
BI (assumes 0)
UNSURE EI (unknown 35.73 = 1.45
SURE EI (assumes 0.05)
Method
CV + EI
Supervised 36.63 1.38
PSNR [dB]
33.25 1.14
34.32 + 0.91
28.05 4.73
Table 4: Average test PSNR of learning methods for accelerated MRI with noise level 0.03.
6
LIMITATIONS
The analysis in this paper is restricted to the & loss, which is linked to minimum squared error dis- tortion. In certain applications, more perceptual reconstructions might be desired (Blau & Michaeli 2018), and we do not address this goal here. Both SURE and the proposed UNSURE method re- quire an additional evaluation of the estimator during training than supervised learning, and thus training is more computationally intensive than supervised learning.
7 CONCLUSION
We present new framework to understand the robustness-exppessivvty trade-off of self-supervised learning methods, which characterizes different families of estimators according to the constraints in their (expected) derivatives. This analysis results in family of estimators which call UNSURE, which do not require prior knowledge about the noise level, and are more expressive than cross-validation methods.
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ACKNOWLEDGMENTS
Julián Tachella is supported by the ANR grant UNLIP (ANR-23-CC222-00113) Part of this work is funded by the Belgian F.R.S.FERRS (PDR project QuadSense; T.0160.24). The authors would like acknowledge ENS Lyon for supporting the research visits of Mike Davies and Laurent Jacques.
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A PROOFS
We start by proving Theorem Propositions
[ad can be then derived as simple corollaries.
Proof. We want to find the solution of the max-min problem
min max Ey (I((v) - y112 + tr En 0f Dy (w)
"
Since the problem is convex with respect to f and has affine equality constraints, it verifies strong duality (Luenberger 1969 page 236, Problem 7, Chapter 8), and thus we can rewrite as max max-min min problem, that
as min Ey [I((v) y112 + 2 En
(v) By
7
Choosing f(v) f(v). rewrite the problem
max min Ey 1 + 2tr En of By (3)
+ 2tr (En))
(21)
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Using integration by parts and the facts that i) Py Pe *N Σ) is differentiable and supported on IR", and ii) dou By. /Py 0 log Ph, we have that
By.
0f((z) dyj
/ py(y) Oyj
0f: y dy
By
(22)
/
Op (y) dy
f(()))
(23)
Oy;
|
f.()))
Opu
Dyg
y
Py (3) dy
(24)
Py (y(
① log Pu (y).
EBy (1()
(25)
Oyj
and thus By By (v) V (ogp,(y) (y) Plugging this result into 21
we obtain
Ey [1li(a)] 2tr En, f (3y ) log logPy(()))
max min
+ 2trr
(26)
" fu...hh
max min Ey (1 - -2f(y) T Σn J.....h
+ 2 (En)
(27)
where the first term can be rewritten in a quadratic form as
max Ey (f(()) En V log (ug)(()))l tr - (E, + 2tr ttnr
(28)
with H = Ey V Py Vog (v) Since the density Py(y) > 0 for all y IR" (as it is the
convolution of pa with Gaussian density), the minimum of 28 with respect to is
f(y) E,, V log Py())
(29)
Replacing this solution in 28 we get
max - tr EE H + 2tr (E,)
which can be rewritten as
min tr (E, 2 (En))
(30)
7
In particular, if we have E, Ej=o flj L for some matrices { E [R ) then the problem simplifies to
min Σ
nmn tr (VHHV)) - 27, tr (V)
6,3
Setting the gradient with respect to r to zero results in the quadratic problem Qn = v where Qi, := tr ( (V. and v; : (.) for i,j s. The matrix Q can be seen as Gram matrix since its entries can be written as inner products (L, 3 where H is positive definite due to the assumption that Py * (0, E). Thus, since by assumption the set [ ,1}=1 is linearly independent, Q is invertible and the optimal v is given by
Q'' 'v.
(31)
We now analyze the mean squared error of ZED estimators: We can replace the solution fly)
+ Σ,, v (og in SURE's formula to obtain
0² log
loggy Py (8)) tr (E)
Exy)))(y)) 21 E,, E., V (2)l 2t E(1 + En
Oy²
which can be written as
8 log PygPyy. tr
Eg u (1( x|12 = tr (E,, E.y V Vogg (logp,)))) E(T + E,,
) (r
Oy²
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Using the fact that By
0² log ay
(y)
-By V VlogPy (y) log (y) (1)
-H, we have
Ez, f x=1² En H) 2 H2 (2)
Proof of Propositions and 2
Proof. In this case, we have En = 11 and thus 31 reduces to
(I) tr (H)
(32)
Ey V Py
and the mean squared error is given by
E x [[( =012² = ii o²
which can be rewritten as Noting that MMSE o² - o E V (
Ez,y f(y) x0|² o²
(33)
(1) || = 0² 7. we have that
0²
(34)
0² MMSE
Combining (33 and 112 we have
0
(l - -0012
MMSE o²
(35)
02
MMSE 1).
(36)
Anisotropic noise In this case, we have Σ, = diag(n) = 2==1n/e;e and thus we have Q diag H) and v n1 in 31 giving the solution
1
(37)
LE
log Pa (y)
8%
for I...., n.
Correlated noise If we consider the 1-dimensional signal setting, we have Σ, circ(n) = Eh ni TjT is circulant matrix, where T3 is the j-tap shift matrix, and thus er+1 and Q n circ(h) where p22+1 the autocorrelation of log (1) (considering up to ビý taps) that is
alog Py
i O Py
h
y
(y)
n
Oy:
0y(++j-)) mad
for j 1, 1,...,2r 2r + and e denotes the (r + 1)th canonical vector. Thus, we have n - or cquivalently
hhr e,41,
== (1/FF)
where the division is performed elementwise and F E 1(2++1)x(2++) is the discrete Fourier trans- form.
B EXPONENTIAL FAMILY
We now consider separable noise distributions p(y|z) II ((3](1)), where g(<l)) belongs to the exponential family
((yla)= = h(y) exp(r(x)b(y)) - ((x))
where v²xx) log J (v)) (x))) is the normalization function.
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Lemma 4. Let y p(y) = IT (ys(2z)) be random variable with mean E "y a, where
the distribution q belongs to the set whose definition is included in Appendix A
Then,
05e (y)
Byl - f(y) =
Σ By ((y))
(38)
Oy.
where a R > oc) is a non-negative scalar function and x² = a/ k yl - a0l2 is the noise variance.
Stein's lemma can be seen as a special case of Lemma 4 by setting a(v) = 1, which corresponds to the case of Gaussian noise. If the noise variance o² is unknown, we can minimize
min Ey (1) - y
(39)
FESu
(v) = 0}. Again, we consider the Lagrange formulation where Sn E Ei= By by. of the problem:
0f:
max min Ey lIf(y) y11² + 2n f
a(ys) yy:
(y)
(40)
where we only need to know a(-) up proportionality constant. The solution to this problem is given by
((y) = y + 7 (a(y) a V log + Va(y)).
where fj is also available in closed form (see Appendix A for details). Note that when a(y) 1, 40 boils down to the isotropic Gaussian noise case (UNSURE)
Lemma Hudson 1978) applies to a subset of the exponential family where r(x) = x, h(y)
By (y) exy(b J b du) and By (w) = 1/a(y), with f bu)ddu interpreted as an indefinite integral. In this case, we aim to minimize the following problem:
0f By,
max Eylly f(y))1 + 2n
Σ Ea(ve)
(z)
(41)
1=1
for some non-negative function a + oo). As with the Gaussian case, we can look for a solution ((y) = + f(y), that is
Of. max min E2 11(11) + 2n a(y.) + a(yx) Oy. (2))
(42)
"
and follow the same steps, using the following generalization of 12
0f(())
f py Pu(y)a(v))
3i. (v)dy dy
Ey a(v))
(43)
oy
Dy:
/ M())aaa
Opu
da
+
(y:) ) [y)] dy
(44)
dy.
Oy.
Opu
/
Oa
[(W)al(s)))
+
(w:)]p, (2) dy
(45)
Py
Dy.
Ölog Py
da
= Ey f((x)(a(y))
(y) + By:
(3:)
(46)
Oy
obtain the solution f(y) ny(y) with s(y)
a(y) V log Py (3) Voly). Replacing the
optimal f in (41). we have the following problem for r/:
as: Ey Cya(v)) (vi)
m || ( (v) 12 2n Ey a(x + 2n²
(z)
(47)
Oz.
1=1
which is a quadratic problem whose solution is
E y. a(v)
(48)
Σm E.
(1)2
2
(1:)
ds:
(1)
Øy:
Thus, the minimizer of 41) is
f(y) y (a(y) Vlogy( Valy))
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C POISSON-GAUSSIAN NOISE
The PG-SURE estimator for the Poisson-Gaussian noise model in (17 is given by (ee Montagner al. 2014)
0f: (y ex,) 2y:fily -er) YY6 o²
By + 20
(49)
oy:
where e, denotes the ith canonical vector. proximation
Following Le Montagner et al. 2014), we use the ap-
Of: ayj
fily (v7) R fi(y)
(y)
and
05, ly enn) ~ Oy,
05i (y) 7 Oy.
02 fi (v)
0y
to obtain
0s: (v) + 20 J2 Oy:
Ey
llf(v) y/1² + > (2o² + 2yrr)
Oy,
(b)
YV/
(50)
We obtain the PG-UNSURE estimator by replacing (7, 0²) for the Lagrange multipliers (7,0) to obtain
0f: By: yv)
max min Ey (I[(y) y/12 Σ (2
2
(51)
9:7
f
where we drop the second order derivative, since we observe that in practice we have Of: (3)) Note that this problem is equivalent to
8
(y)
By.
09:
arg min max Ey (1) y/l
(52)
SESe
9.7
where SpG {f Eydivf(y) = 0, E,y V fz)) 0} and thus we have that Sey SpG C S2ED-
Setting f(y) y + f(y), we can rewrite problem 51
0f. (y) + 2n + 2YM
max arg min 9.7
E. f1())2 + +2n) + 2nr7)
(53)
Oy:
Using integration by parts as in
22 we get
max arg min Σ By f(u)² -2(())))(()))) (in
2 alog Pa (y) +7)
y17)
+ +22+
+ 2Y84
9,7
By.
After completing squares, we obtain
max arg min
Σ Ey (f(w) (n + 907)
Olog Py (y) )
- (n (n
unr)
Olog Py (3) -7) +2m++ Oy.
n.7
Oy:
thus have
f(y) (1n + yr) o v log + 17
and
f(y) (1n + y
v VoggPy + 17
where (n. 7) are given by
log Pu (4) + yur Oy:
2 log Py (y) -r) + 2m + 24y
max
Ey
Oy:
18

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Published as a conference paper at ICLR 2025
Defining the expectations ha,b = Ei=1 Ey 0 log Pv (y) for a. E {0, have Ou:
ho, h2,2 - ny² 271h122 2 + 21 + 2nn + 2
max 9/7
We can take derivatives with respect to (17. to obtain the optimality conditions
{o
n/o,2 7/1,2 yho,1
(54)
?22,2 ny n/1,2 + nho,1 + 2 + h1
which are equivalent to
mho.2 711,22 hg.1)
(55)
h1o n(h1,2 - ho,1) + (12222 + n + h122 - 2h1,1)
The solution to this system of equations is
7/112 ho,1)
(56)
h
ha,2
where
h1,0 - n(h1,22 ho,1)
(57)
20,2 2h1,1) (h1,2 1
In particular, if we remove the y-constraint (7 0), we obtain the UNSURE formula in Proposi- ie.
tion
Ey V log
Ta.2
D EXAMPLES
To gain some intuition about optimal ZED estimators and their differences with cross-validation, consider various popular signal distributions that admit elementwise decomposition (xx) Ih ax (2;) where 4e a one-dimensional probability distribution. Since the noise distribution is also separable, we have that (y) = II (y:) where qg = q. (0, 1). All estimators are applied in
seement-wise fashion, since there is correlation between entries, ie. g(y)) f(()) for all i = 1.... n, where : + is one-dimensional function. The cross poass-validation-bee based estimator cannot exploit any neighbouring information, and will just return constant value for all inputs, ie. g(M) Fy, {ys} ). the observed mean. In contrast, both the MMSE and ZED estimators use the input y: improve the estimation. In particular, the optimal ZED estimator
log 9, (vr)
g(y)) = y4 + Ey.. V log qy (v:))²
(O, 1). Table 5 summarizes the distributions, and Figure 5 shows the estima- tors for each case. In all cases, we have that the cross- vooss-vali estimator is trivial (Cy = where the mean squared error simply the signal variance, whereas the ZED estimator depends on y, except for the (non-sparse) Gaussian signal distribution. Interestingly, while the MMSE estima- tor always has non-negative derivatives (Gribonval, 201T), the optimal ZED estimator can have a negative slope.
where 4y = --
Two Deltas
Gaussian (o,
Spike & Slab
1 5-_
18
7. 1) + 1.6
4x
0²) + (L o²)
(O0 NN 0²) + (N
4N
MMSE
0.017
0.059
0.043
MSE CV
0.250
0.500
MSE ZED
0.024
0.135
Table 5: Examples of popular signal distributions, and the mean squared errors (MSE) for different estimators.
19

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Published as a conference paper at ICLR 2025
Two Deltas
Gaussian
Spike & Slab
ply)
MMSE estimator
fly)
ZED estimator
-1
-1
-1
cross-validation
-2
2
-2
y
y
Figure 5: Comparison between MMSE, ZED, and cross-validation estimators for different toy signal
distributions.
E REAL CRYO-EM DATA EXPERIMENT
We evaluate the proposed UNSURE method on real Cryo-EM data (provided by the Topaz-EM open-source library (Bepler et al. 2020i) whose noise distribution is unknown. The dataset con- sists images of 7676 x 7420 pixels. We applied the Poisson-Gaussian noise variant of our method (PG-UNSURE) For comparison,
included results using the Noise2Noise approach in- troduced by Bepler et al. (020), which leverages videos static images to obtain pairs of noisy images of the same underlying clean image. Both methods evaluated using U-Net architecture of the deep inverse library (Tachella et al.) (023b) with scales and no biases. Training is done using random crops of 512 512 pixels the Adam optimizer with a learning rate of 5 x 10-4 and standard hyperparameters. Our method receives as input the whole video averaged over time, and thus benefits from a higher signal- gegnal-to-nois to-n ratio input image than the Noise2Noise variant, whose input only averages over half of the video frames to obtain the pairs. Figure O shows the raw noisy image and the denoised counterparts. The proposed method produces good visual results, with fewer artifacts than the Noise2Noise approach.
Measurement
PG-UNSURE
Noise2Noise
DRANT
Figure 6: Blind denoising results of Cryo-EM data using the PG-UNSURE method and Noise2Noise (Bepler et al.) 2020). Since the image is very large 7676 7420 pixels, is recom- mended zoom in observe the details.
20

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Published as a conference paper at ICLR 2025
F EXPERIMENTAL DETAILS
We use the U-Net architecture of the deep inverse library (Tachella et al., 2023b) with no biases and an overall skip-connection as a backbone network in all our experiments, only varying the number of scales of the network across experiments.
MNIST denoising
We use the U-Net architecture with 3 scales.
DIV2K denoising
We use the U-Net architecture with scales.
Computed Tomography on LIDC We use an unrolled proximal gradient algorithm with 4 iter- ations and weight-tying across iterations. The denoiser is set the U-Net architecture with scales.
Accelerated MRI on FastMRI We use unrolled half-quadratic splitting algorithm with it- erations and no weight-tying across iterations. The denoiser is set as the U-Net architecture with scales.
Algorithm UNSURE loss. The standard SURE loss is recovered by setting a = 0, Σ, =
Require: step size a, momentum Monte Carlo approximation constant residual []y(( y||²
~ (0. 1)
div - [,,b)_ (fo (3) + rb) fo(v))
loss residual div
Odiv < ng + (1 μ1) an
# average stochastic gradients w.r.t. ")
- + ag gradient ascent Lagrange multipliers
return loss(0)
21


Reviews:
[{"review_id": "CHbTXhyFsz", "reviewer": "Reviewer_4fc2", "summary": "The paper presents a theoretical framework aimed at addressing the limitations of existing self-supervised learning (SSL) methods for image reconstruction. However, several critical issues undermine its contributions and overall validity.\nThe proposed estimator builds on Stein’s Unbiased Risk Estimator (SURE) without offering significant advancements beyond existing methodologies. While the authors claim their approach allows for learning in scenarios with unknown noise structures, similar claims have been made in prior works without the need for extensive computational resources. This redundancy raises questions about the paper's originality and its contribution to the field.\nThe theoretical framework lacks clarity, making it difficult for readers to grasp the practical implications of the proposed methods. The authors introduce constrained self-supervised losses with Lagrange multipliers, but fail to provide sufficient detail on how these constraints improve learning outcomes compared to traditional methods. This omission hampers reproducibility and understanding, which are essential in scientific discourse.\nAlthough the authors present evaluations claiming near state-of-the-art results, they do not adequately benchmark against a comprehensive set of competitive methods. Their focus on a narrow range of tasks, particularly in MRI reconstruction, limits the generalizability of their findings. The evidence provided does not convincingly demonstrate that their method consistently outperforms other self-supervised techniques across diverse applications.\nThe paper acknowledges that the proposed method requires more computational resources than existing approaches. This raises practical concerns regarding its applicability in real-world scenarios, especially considering that many researchers and practitioners may not have access to such resources. The trade-off between computational efficiency and performance is inadequately addressed, leaving potential users with unresolved questions about feasibility.\nWhile the authors assert that their method generalizes well to complex tasks like MRI reconstruction, they provide insufficient evidence to substantiate these claims. Generalization is a critical aspect of any learning framework; without robust validation across varied datasets and tasks, such assertions remain speculative at best.\nIn summary, while the paper attempts to tackle relevant issues in self-supervised learning for image reconstruction, it ultimately falls short due to a lack of novelty, clarity, empirical validation, and robust generalization evidence. These shortcomings warrant rejection as they detract from the potential impact and applicability of the research within the broader field of machine learning and computer vision.", "questions": "1)\tIn case of correlated Gaussian noise, the construction in (14) assumes that the set of s-dimensional covariance matrices includes the true covariance, i.e., the plausible covariance matrices. This intrinsically assumes a low dimensional structure, but this was not mentioned in the text. Also in practice, this assumption might not hold, and as such it would only do a low-dimensional projection of the true covariance onto s-dimensional space? The consequences and possible solutions of this case is not discussed.\n2)\tTable 1 results claim that the proposed method perform close (≈ 1dB) to the SURE with known noise model. However, 1dB in PSNR is a distinct difference, and claiming that performing close to the SURE is misleading.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1)\tThe paper presents, a novel family of self-supervised estimators that effectively addresses the challenges of unknown noise levels, extending the capabilities of existing methods like SURE, and well-written.\n2)\tIt offers a clear theoretical framework that elucidates the robustness-expressivity trade-off in self-supervised learning, enhancing the understanding of estimator dynamics.\n3)\tThe paper includes thorough experiments on established datasets, providing solid empirical evidence for the effectiveness of their method compared to other self-supervised methods and showing its applicability across diverse imaging problems.", "weaknesses": "1) Computational complexity was already mentioned by the authors. Proposed method requires additional evaluations during training, leading to increased computational costs and potential implementation challenges.\n2) The proposed ZED estimator struggles with high-entropy noise distributions, and the method's effectiveness may vary depending on optimizer settings and hyperparameter tuning. It would be better to provide possible strategies to remedy this \n3) While the experiments demonstrate strong performance in specific tasks, there is limited exploration of diverse imaging applications and noise scenarios, which may affect generalizability. \n4) The related work section can be summarized to better discuss the actual contribution of the current work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KoWZB9MOFx", "reviewer": "Reviewer_W3qG", "summary": "This paper addresses recent advances in self-supervised learning methods for image reconstruction that learn directly from noisy data, eliminating the need for ground-truth data. The authors introduce a theoretical framework that examines the trade-off between expressivity and robustness in noise-handling techniques.", "questions": "- On top of my previous comment, the authors did not discuss if the estimation of noise level has limitations in extreme cases? Could the authors discuss whether the noise level estimator is still accurate in low SNR cases which are also common in imaging applications?\n\n- For the case of General inverse problems, it is interesting the proposed method is able to adapt to the range of $A^{T}$. However, it posses the question, could the authors discuss the stability of the proposed method where the dimension of the range is small? What is the limit in terms of range dimension so the propose method is able to accurately estimate the noise level?\n\n- Based on my previous comments some discussion is needed on the performance of the proposed method when the estimation of noise level is performed in extreme cases, and low dimension range of $A^{T}$.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "-  The authors introduce a theoretical framework that examines the trade-off between expressivity and robustness in noise-handling techniques.\n\n- The method does not require knowledge about the noise level to denoise images.\n\n- Numerical validation is presented to verify the ability of the proposed methodology to estimate the noise level.", "weaknesses": "- A primary limitation of this paper is that the denoising experiments are carried out on synthetic scenarios i.e. a clean image is corrupted by noise.  Could the authors consider testing their denoising method and noise level estimation in a more realistic scenario? For instance Noise2Noise (method the authors claim to outperform) has been tested in experimental cryo-em images (see link below). Testing the proposed method in such a scenario would provide valuable insights into its practical applicability, particularly in terms of noise level estimation and the quality of the denoised output.\n\t\t\nhttps://github.com/tbepler/topaz/blob/master/tutorial/04_denoising.ipynb\n\n- The image dataset studied in this paper are relatively of low level of details (mostly low frequency images), raising an important question about robustness. How well would the algorithm perform when trained on images with higher levels of detail? Using the cryo-em images would be interesting scenario since resolution it is the most important factor to determine the performance of any denoising method.\n\n**Minor Comment**\n\n- I suggest the authors to change the notation of matrices in capital boldface to be consistent with the current notation of vectors.", "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": "5jn9iNp8Sz", "reviewer": "Reviewer_bm6s", "summary": "This paper introduces a novel framework that bridges this gap by proposing a SURE-inspired method that can effectively learn from noisy data without explicit noise-level information. The authors provide theoretical analysis of their proposed method and through experiments, the authors demonstrate good performance compared to existing self-supervised methods on various imaging inverse problems.", "questions": "(n/a)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This manuscript presents a well-structured and comprehensive review of existing denoising techniques. The clear and logical organization of the content enhances readability.", "weaknesses": "While the contribution of this manuscript may not be groundbreaking, it offers a valuable insight into denoising techniques. Regarding the title, a more descriptive title such as \"Unknown Noise Level Estimation via Self-supervised Stein's Unbiased Risk Estimator\" would provide a clearer indication of the paper's focus. Additionally, minor punctuation errors are present throughout the manuscript, which could be addressed in a future revision.", "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": "FMGEYXUXqC", "reviewer": "Reviewer_K2y6", "summary": "The paper deals with the problem of designing an image/signal denoising algorithm given a sample of noisy images/signals $y_1,\\ldots,y_N$ from $\\mathbb R^n$. It is assumed that each $y_i$ satisfies the relation $y_i = x_i + \\sigma\\epsilon_i$, $i=1,\\ldots,N+1$, where $\\epsilon_i$ is independent of $x_i$ and is drawn from the Gaussian distribution $\\mathcal N(0,\\mathbf I)$. The proposed method is solve with respect to $f$ and $\\eta\\in\\mathbb R$ \nthe optimization problem\n$$\n\\min_f \\max_{\\eta} \\frac1N \\sum_{i=1}^N \\Big(|f(y_i) - y_i|^2 + 2\\eta \\ \\textrm{div} f(y_i)\\Big),\n$$\nwhere $f$ is chosen from a parametric class of functions, such as neural networks with a given architecture and varying weights. \n\nExtensions to the setting of a noise covariance matrix of a more general form are also considered. In particular, if $y_i = x_i + \\Sigma^{1/2}\\epsilon_i$ with $\\Sigma = \\sum_{j=1}^s \\eta_j \\Psi_j$, where $\\Psi_1,\\ldots,\\Psi_s$ are some known matrices and $\\eta\\in\\mathbb R^s$ is an unknown vector, the proposal is to solve the saddle point problem\n$$\n\\min_f \\max_{\\eta\\in\\mathbb R^s} \\frac1N \\sum_{i=1}^N \\Big(|f(y_i) - y_i|^2 + 2\\sum_{j=1}^s \\eta_j \\ \\textrm{Tr} [\\Psi_jf(y_i)]\\Big).\n$$\n(There is a mistake in this formula as it is written in the paper, the correct form should be\n$$\n\\min_f \\max_{\\eta\\in\\mathbb R^s} \\frac1N \\sum_{i=1}^N \\Big(|f(y_i) - y_i|^2 + 2\\sum_{j=1}^s \\eta_j \\ \\textrm{Tr} \\big[\\Psi_j \\ Df(y_i)\\big]\\Big)\n$$\nwhere $Df$ is the Jacobian matrix of $f$. \n\nThe main mathematical result of the paper, stated in Theorem 3, provides the explicit form of the minimizer of the problem above.", "questions": "An important argument in the proof of Theorem 3 is the use of strong duality. The authors refer to (Luenberger, 1969, Chapter 8). I believe they are referencing Theorem 1 on page 224. (If this is indeed the case, it would be helpful to highlight it in the paper.) I have a question related to the application of this theorem. Assuming that the constraint is of the form $G(x) \\leqslant \\theta$, the theorem requires the existence of $x_1$ such that the strict inequality $G(x_1) < \\theta$ holds. My impression is that this condition cannot be satisfied in the setting under consideration. However, I may have overlooked something. Could the authors please clarify this?", "limitations": "", "rating": 3, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper is pleasant to read. The proposed method is based on a simple idea and the experiments show that it improves on some methods that have been recently proposed in the literature (such as Noise2Void, Neighbor2Neighbor).", "weaknesses": "1. Mathematical analysis of the proposed methodology is very light. In particular, neither the properties of the proposed estimator nor the convergence of the optimization algorithm are studied.  \n\n2. The framework for the learning problem is not clearly defined. The abstract and introduction reference supervised and self-supervised settings multiple times, but how these settings are interpreted in image reconstruction remains unclear. This only becomes explicit at the bottom of page 7, where it is clarified that the available data consist of $y_1, \\ldots, y_N$. I recommend moving this clarification to the beginning of Section 2.\n\n3. In Sections 2 and 3, the term estimator is repeatedly used to describe objects that cannot be computed directly from the data, as they rely on knowledge of the distribution of $y$. To avoid this misuse of terminology, it would be preferable to use a more accurate term.\n\n4. Minor comments and typos:\n- In the second paragraph of Section 1 on page 1, the two items starting with \"i) SURE\" and \"ii) Noise2Self\" are presented in a different order than in the abstract. For the sake of consistency throughout the paper, it would be beneficial to align their order.\n- Page 3, the two lines just above Eq. (5): I guess $\\mathbb R^m$ should be replaced with $\\mathbb R^n$.\n- Page 3, Eq. (4): This formula is easy to obtain, but it might be useful to provide a reference for non-experts. \n- Page 4, Eq. (UNSURE): I suggest putting parentheses around the expression within the expectation, to underline that the divergence is also under the expectation. \n- Something wrong with formula (14). The gradient of $f$ should appear in the second term.\n- Page 6: \"are more flexible\" -> \"are more flexible than\"\n- Page 6: \"we can considering\" -> \"we can consider\"\n\n5. The setting under consideration closely resembles what is commonly known as the empirical Bayes framework. A discussion of the relevant literature on empirical Bayes methods would be a valuable addition.\n\n6. The title of the paper appears somewhat misleading. It suggests that the paper will propose an unbiased estimator of the risk of an estimator $f(y)$ in the case where $\\sigma$ is unknown, implying that the unbiased risk estimator would not depend on $\\sigma$. However, the paper instead restricts attention to estimators $f$ for which the Stein unbiased risk estimate (SURE) happens to be independent of $\\sigma$.", "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)