Equivariant Splitting: Self-supervised learning from incomplete data
Authors: Victor Sechaud
Jérémy Scanvic
Quentin Barthélemy
Patrice Abry
Julián Tachella
Venue: ICLR 2026 Poster (2026)
Keywords: inverse problems
self-supervised imaging
equivariant neural networks
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Self-supervised learning for inverse problems allows to train a reconstruction network from noise and/or incomplete data alone. These methods have the potential of enabling learning-based solutions when obtaining ground-truth references for training is expensive or even impossible. In this paper, we propose a new self-supervised learning strategy devised for the challenging setting where measurements are observed via a single incomplete observation model. We introduce a new definition of equivariance in the context of reconstruction networks, and show that the combination of self-supervised splitting losses and equivariant reconstruction networks results in unbiased estimates of the supervised loss. Through a series of experiments on image inpainting, accelerated magnetic resonance imaging, sparse-view computed tomography, and compressive sensing, we demonstrate that the proposed loss achieves state-of-the-art performance in settings with highly rank-deficient forward models.

Published as a conference paper at ICLR 2026
EQUIVARIANT SPLITTING: SELF-SUPERVISED LEARN- ING FROM INCOMPLETE DATA
Victor Sechaud*1 Jérémy Scanvic 1,2, Quentin Barthélemy". Patrice Abry', and Julián Tachella*
LPENSI, CNRS, ENS de Lyon, France Pryymm, Lyon, France
ABSTRACT
Self-supervised learning for inverse problems allows train a reconstruction net- work from noise and/or incomplete data alone. These methods have the poten- tial of enabling learning-based solutions when obtaining ground-truth references for training is expensive or even impossible. In this paper, we propose new self-supervised learning strategy devised for the challenging setting where mea- surements are observed via single incomplete observation model. We intro- duce a new definition of equivariance in the context reconstruction networks, and show that the combination of self-supervised splitting losses and equivari- ant reconstruction networks results in unbiased estimates of the supervised loss. Through series of experiments image inpainting, accelerated magnetic nance imaging, sparse-view computed tomography, and compressive sensing, we demonstrate that the proposed loss achieves state-of-the-art performance in set- tings with highly rank-deficient forward models
INTRODUCTION
Inverse problems are ubiquitous in many sensing and imaging applications. They are written as
y Ax +
(1)
where A E IR is the known forward matrix, æ E IR" is the ground truth image to be estimated, y Rn is the observed measurement vector, and IRT is the unknown noise, generally assumed to follow a Gaussian distribution. This model is suitable for many imaging modalities, including magnetic resonance imaging (MRI) for medical imaging (Zbontar al., 2019), computed tomogra- phy (CT) (Withers et al. (021), microscopy (Ragone et al. 2023). remote sensing (Fassnacht et al. 2024), and astronomical imaging (Vojtekova et al.] 2021).
The number of linearly independent measurements is often smaller than the number of pixels in the target images due to physical and practical constraints. deficient, the forward process discards some of the information present in the target image. Further information about the target images is needed to solve the problem and different methods make different assumptions about the signal distribution.
this case, the forward matrix rank-
Modern Saaring--bsed solvers generally obtain state-of-the-art performance by training on super- vised pairs of ground-truth images and measurements. For certain applications, it is expensive or even impossible to obtain enough ground-truth data for supervised training (Belthangady & Royer 2019). This is notably the case in astronomical imaging, microscopy and medical imaging.
Recent sulf-supervised methods overcome this limitation by learning to reconstruct without ground- truth data, relying only dataset of measurements, and they generally differ in the assumption they make on the forward model. For denoising problems where the forward matrix is the identity mapping, certain methods rely on knowing the exact noise distribution (Eldar 2009 Pang et al. 2021 Monroy et al. 2025), others only assume it is entry-wise independent (Krull et al. 2019) while some make intermediate assumptions (Tachella et 2025a).
In settings where measurements are observed via multiple incomplete forward operators, such as accelerated MRI with masks varying across acquisitions or inpainting problems with missing pixels
"The first two authors have contributed equally to this paper.
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varying across images, the main approaches are splitting YYaman et al. 2020; Millard & Chiew 2023) and consistency across operators (Tachella et al. (022). Splitting losses divide the measure- ments into input and target components, and are unbiased estimators of the supervised loss if there is enough diversity of operators in the dataset (Daras et al. 2023 Millard & Chiew 2023).
In the more challenging case of measurement data obtained via a single incomplete forward opera- tor, the main self-supervised approach is equivariant imaging (Chen et al. [002]] (022) which makes the assumption that the target distribution is invariant a certain group of transformations including geometric transformations (Wang & Davies, 2024, Scanvic et al. 2025) and range transformations such intensity scalings (Sechaud et al., 2024). Experimental results show that it can obtain com- petitive performances to supervised methods even though it does not require ground-truth references for training (Chen al. 2022). However, training with is typically slower than supervised learn- ing, as it requires two to three evaluations of the model for every iteration, and can obtain subpar performances if the operator is highly incomplete.
In this work, propose equivariant splitting (ES), new self-supervised method for learning from measurements obtained via a single forward operator that combines the invariance to transforma- tions assumption of equívariant imaging and the simplicity and computational efficiency of splitring methods. The key idea behind our method is the use recent developments in equivariant architec- tures (Chaman& & Dokmanić 2021a Puny al. 2022) design a training loss that performs implicit ground truth data augmentation without any transformation overhead. Our theoretical results show that method yields (in expectation) the minimum mean squared error (MMSE) estimator as long as the model is expressive enough, which is not guaranteed for equivariant imaging and pre- vious splitting methods. Additionally, our experiments demonstrate state-of-the-ar performance on a wide range of self-supervised imaging problems including compressed sensing, image inpainting, accelerated MRI and sparse-vies view CT. This work is additional evidence that architectural constraints built upon equivariance are a powerful tool to solve ill- ill-posed imaging inverse problems.
Our contributions are the following:
1. We propose a new definition for equivariance in inverse problems, and propose architec- tures that satisfy this property, including unrolled architectures.
2. We propose new self-supervised loss that leverages equivariant networks (according to our definition above) whose global minimizer is the gold standard MMSE estimator under the assumption of an invariant signal distribution.
3. We demonstrate the performance of our method on a wide range of inverse problems in- cluding compressed sensing, inpainting, accelerated MRI and sparse-view computed to- mography.
RELATED WORK
Measurement splitting Various self-supervised losses consist of dividing measurement vectors into two components, one used input and the other as target. It has been used to solve oversam- pled single-operator inverse problems including full-view CT (Hendriksen et al., 2020). and also undersampled multi-operator inverse problems with theoretical guarantees of producing estimates equivalent to supervised estimates in expectation, notably accelerated MRI and image inpainting with varying masks (Daras et al.) 2023 Millard & Chiew 2023). To the best of our knowledge, this work is the first to extend these methods to the more challenging single-operator undersampled setting, which notably includes sparse- view CT and accelerated MRI (with fixed mask).
Equivariant imaging It is possible to learn from incomplete data associated with a single degra- dation operator, as long as the underlying signal distribution is invariant to group of transforma- tions (Tachella et al. 2023). Equivariant imaging (Chen et al., 2021: 2022) assumes that the dis- tribution of clean images remains unchanged under certain transformations, including translations, rotations and flips, and introduces a training loss that enforces the equivariance to these transfor- mations of the entire meessurement-reoonstuuuciie models. It has been used effectively to solve various inverse problems using adequate groups of transformations (Wang & Davies 2024; (Sechaud et al., 2024)), but it is computationally expensive due to the two three network evaluations. Moreover, the El loss is not necessarily unbiased
process, thereby constraining the set of learnable
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estimator of the supervised loss, and it is unclear whether this approach recovers the optimal MMSE estimator in expectation.
Equivariant neural networks The design of equivariant networks is active research topic and many different approaches exist. Some rely on data augmentation to make neural networks more equivariant to rotations and flips, while other rely averaging the group of transformations
test time (Rivera et al,, 2021 Puny et al.] 2022 Kaba et al.] 2023] Sannai et al., (024). A third line of work focuses on architectures that are equivariant by design; Cohen & Welling (20 6 2017) pro-
pose convolutional layers that are equivariant to rotations and flips, and other works also rely on the
design of more equivariant layers to improve translation -auittionn-ivviiinnneee [hhang (2019); Chaman & Dokmanić (2021b) propose equivariant pooling and downsaapplig/uupsappiin layers, and Karras et al. (2021) Michaeli et al. (2023) equívariant non-linearities and activation layers. Closer to our work, in the specific setting of inverse problems, Celledoni et al. (2021) propose the use of equiv- ariant denoising blocks within unrolled architectures, and Terris et al. (2024) similarly propose to render aadg-and denoisers more equivariant by averaging over transformations at test time. We take these analyses further by providing a clear definition of equivariance inverse problems and showing which popular posterior estimators and related architectures verify these properties.
BACKGROUND
Solving the inverse problem in eq. ① amounts to designing a reconstruction function f(y A) RS æ estimating ground truth signal TR" from its measurement vector y E Ih" and forward matrix A E Rmxx In practice, it is often implemented as a neural network parametrized by a set of weights. Supervised methods assume the existence of a finite dataset containing pairs of ground truth signals and measurements (( yee that are used to learn the reconstruction function fly, A). The main approach is to minimize a training loss equal to the mean squared error (Ongie
2019)
min
Σ FE
CSUP(E; (I,, Vi, A.1). LSUP(x, y, y. = (lf A) - x012.
(2)
While this approach obtains ofate-of-thh-art performance, it cannot be used in the absence of ground- truth data. Self-supervised methods overcome this limitation using a finite dataset containing only measurements (v: lizz, and training loss L(y. A, f)) that need no ground truth data to be evaluated, and which is designed to approximate well the supervised objective in eq. (2).
In denoising problems (A I) with Gaussian noise of known variance, RecoruptedZReoounnte (R2R) (Pang et al. 2021) and SURE (Metzler et al., 2020) provide unbiased estimators of the su- pervised loss. The R2R loss is computed by adding synthetic Gaussian noise ~ ~N o² to the measurements, creating input-target pairs as (y + aw, y #) for some a E (0, +oo). However, measurements are observed by an incomplete operator A with non-trivial nullspace, these losses fail to learn in the nullspace, approximating f fy. A) = A A Egjy.. {x) } + r(y A), with v being any function taking values in the nullspace of A (Chen et al. 2021).
In the rest of this section, we present the two main self-supervised losses that can learn beyond the nullspace, measurement splitting in Section 3.] and equivariant imaging in Section 3.2 By combining their main ideas, we obtain our new self-supervised loss introduced in Section
3.1 MEASUREMENT SPLITTING
One strategy to address the limitations imposed by the nullspace of a single operator is to employ multiple operators (Millard & Chiew 2023). The key idea is that different operators generally do not share the same nullspace; thus, observing measurements through the image spaces of multiple oper- ators allows access to the whole space R². Formally, they assume that measurements are obtained according y ~ p(yl Az) where the measurement operator A is itself drawn from a distribution P(A), and differ for each acquisition.
Splitting losses divide the measurements into two components y Ty yz T with correspond- ing operators A = [A] A, where A₁ E T x and A₂ E p==2X" with m, + m2 = m. The network is then trained to predict the entire measurements vector y from only of its two
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components y1, using the training loss
(S A, f) Ail.A. {IlAs(m, Ai) -1)- -
where 11 A1 y: 4) random splitting distribution, which chosen on per-problem ba- sis. The loss encourages the model learn in the nullspace of each operator by predicting the unobserved part. In practice, the expectation in eq. (3) is estimated using a single split yı for each training batch.
3.2 EQUIVARIANT IMAGING
Equivariant imaging (EI) relies on the assumption that the distribution of images is invariant under a group of transformations T |= 9 E G. to learn beyond the nullspace measurements ob- tained via a single operator (Chen et al. 2021 2022). In this setting, the reconstruction model is expected to be able to estimate ground truth images x as well as their transformations Tyæ in a co- herent manner, i.e., such that the entire with respect to the transformations f fAT,, A) = (Az, A). In order to achieve this, they pro- pose a self-supervised loss which consists in a traditional measurement consistency term (replaced by SURE in the presence of noise), along with an equivariance-promotiin term
pipeline f (Ax) is equivariant
Lm(y f) = |As( A) yll² + XB, ( A) f [ A). )))
(4)
where À > 0 is trade-off coefficient. Even though it has been shown to be particularly effective a wide variety of problems (Wang & Davies 2024 Sechaud et al., 2024), it typically requires from two three evaluations of the neural network which makes it very computationally expensive (Xu ett al. (2025). Moreover, the equivariant loss is only effective at enforcing equivariance when the learned estimator achieves almost perfect reconstructions, fly, A) is not the case for very ill-conditioned problems and which leads to the method having multiple
(Chen al. (021), which
possible solutions in general.
METHOD
In this section, present method that combines measurement splitting and El introduced in Section In order to learn from incomplete measurements obtained via a single operator, we rely on the same assumption of El:
Assumption 1. The distribution of ground truth images p(æ) is invariant to the transformations (Tg)eeg
(T Vg E G, Vo R..
(5)
The set of transformations is design choice of the method be chosen per-problem basis. Corollary [ helps to choose them for specific problems, and we use it the experiments. This assumption applies in many different settings, natural image distributions are generally invariant to rigid transformations (translations, rotations, flips, scalings), especially microscopic, acrial or re- mote sensing images which have no privileged orientation. Moreover, our experiments show that our method performs well even in the presence of approximate invariance, e.g., for medical im- ages. Refer to Appendix A.2 for more details about how to choose the transformations for a given application.
In Section 4.1 we present our proposed loss and state optimality under mild assumptions in Theo- rem and Proposition [ In Section 4.2 present new definition of equivariance for reconstruc- tion functions and state sufficient conditions for common architectures to equivariant in The- orem 2 In Section 4 we finally present computational synergy between our loss and these equivariant architectures stated in Theorem 3 See Appendix A.3 for a description of the end-to-end algorithm, and Appendix for the detailed proofs.
4.1 PROPOSED LOSS
Under Assumption I the measurements can be understood in a different way than they are tradi- tionally. Indeed, measurements y are generally thought of as being associated to the ground truth
We the notation Emp()((a)) for g(a)p(alb)da.
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image and the forward matrix A, but they can equally be understood being associated to the virtual ground truth image yy = T x and the virtual forward matrix Ag = ATg. g. with
Ax + AT,T + = Ag te.
(6)
The implicit multi-operator structure (with operators J of the problem hints that we should be able to leverage the splitting approaches presented in Section Moreover, since we use the same invariance assumption as EI, we can combine the two approaches obtain equivariant splitting (ES), new self- sulf-vuue loss LEs that has the advantages of both methods.
Noiseless measurements
The ES self-supervised loss is expressed as Lys f) - E, ( (CsPLr(y AT, g: f]}
= Eg Ea A AT, {Il ,T, (11, (y,1,44 A1) -113}}
(8)
where A p(AI|AT,) A ((A is a random splitting of AT Theorem 1. In the case of noiseless measurements with p(x:) G-invariant (Assumption D. if the matrix QA, A EglA, {(AT) AT,), E mRnx has full rank for some split At, then the splitting method yields the same MMSS-optimat reconstructions as the supervised method, i.e.
(y A1) A }
While sufficient for the matrix QA, to be invertible for the reconstructions to be almost or exactly optimal, in practice the spectrum (number of its non-negligible eigenvalues) of the matrix determines how close to optimal they are. In the experiments, we use a single (random) split per batch element and we average the reconstructions corresponding
splits inference.
Proposition 1. If the matrix QA By (Es A, f)). Then the reconstruction function
BaJaa (QA,) E R is invertible and f minimizes
Fly, A) A E. A.
Q QA, A1)
(10)
satisfies
fly, A) E.l QA, {x}} where eq. 11 is convex combination of MMSE estimators for different splittings.
(11)
In practice often neither QA nor QA, can computed closed-form, and we use a non-weighted average over random splittings
A). -) Σ (1)) with (2l A A1y)
(12)
where g chosen randomly over the group of transformations for each split.
As with El, the forward operator should not be equivariant with respect to the choice of transforma- tions, order learn beyond the nullspace of the operator:
Corollary 1. In order for the matrices QA, or QA to have full rank, it is necessary that A is not equivariant:
3g AT, TTA.A
(13)
Noisy measurements
The ES loss can be split into separate terms, one enforcing measure- ment consistency, and the other prediction accuracy:
Ces(y, A) = ( (Ey..Ailv.Az { [1l, A.) y 12 || A1) - - y
where A₁ and A₂ are a splitting of ATG If the measurements are noisy, the first term can be re- placed by self-supervised denoising loss. In particular, if measurements are corrupted by Gaussian noise of standard deviation a, we replace the first term by the R2R loss, yielding:
Ais( + aw, A1) (y) w
Lars(y, A.
Ar Al
+ |12 + tow, A1) -22l2²
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with NN o2) and hyper-parameter estimate of the clean measurement consistency term (Png et al. 2021). we can apply Theorem ( show that minimizing this loss (in expectation) also results in MMSE estimators (if the conditions on QA, or Q are verified). In the case of non-Gaussian noise, the R2R loss can be replaced by its non-Gaussian extension (Monroy al. (025). As with random splits, the expectation over is computed using a random realization per batch. At test time, we modify eq. (12) to average over both splits and synthetic noise additions.
(0, (0)+ Since R2R provides unbiased
4.2 EQUIVARIANT RECONSTRUCTORS
The ES loss requires a model evaluation for every mask and transformation. We show that, instead of sampling random transformation each evaluation, imposing architectural equivariance constraints removes the need to explicitly compute the transforms.
Image-to-ime to-image functions Φ( (x) are equivariant if they satisfy (Cohen Welling 2016)
oT, x) = Tg Φ(œ), Vo E R", Vg E G.
(14)
In this work, we introduce an extension of this definition reconstruction functions ((y.A). To the best of our knowledge, this is the first work that introduces this definition.
Definition 1. We say that the reconstruction function fly, A) an equivariant reconstructor
fly. AT,) ff. A), Vy R Vg E G. R
(15)
This property very general and the class of classical reconstruction functions that satisfy it is large. Theorem 2. The reconstruction, functions defined in points below are all equivariant as in eq.
(15)
1. Artifact removal network. For denoiser o(x) equivariant in the sense of eq. (14
f(y, A) (A y) fly. A) 6 ()
(16)
2. Unrolled network. For ((x) equivariant, any R and data fidelity di Ax, y), with
4n 0, 0b+1 = <b a V dl Axx, ())
(17)
for = L. and f A) LL-
3. Reynolds averaging. For a possibly non-equivariant reconstructor rly, A), with
f(y. A)
T Tyr(y. AT3).
(18)
9Eg
4. Maximum posteriori (MAP). For distribution p(a) invariant in eq. 5 with
fly. A) argmax {p(a y, A)
(19)
=ERe
5. Minimum mean squared error (MMSE). For distribution p(z) invariant as in eq.
5).
fly, A) Ex/yg, fxx)
(20)
For additional motivation and details about these reconstructor architectures, see Appendix A
4.3 EFFICIENT LOSS EVALUATION WITH EQUIVARIANT RECONSTRUCTORS
For equivariant reconstructors, the ES loss in eq. reduces to the splitting loss in eq. (3) Theorem 3. If (A, æ)) is an equivariant reconstructon then ES is equivalent to the splitting loss LES A) LSPLIT(y, A,
(21)
We emphasize that the condition for a reconstructor to be equivariant is different from the condi- tion enforced by the El loss, i.e., f(AT,a, A) f,,(Aa A). They only equivalent if the reconstruction function is a perfect one-to-ons mapping over all possible images.
In experiments, build equivariant reconstructor, using artifact removal networks with translation equivariant UNet denoiser Chaman & Dokmanić (2021b) and ii) unrolled networks with denoiser architecture equivariant rotations and flips via averaging (Sannai al. 2021).

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5 EXPERIMENTS
We assess the effectiveness of the proposed self-supervised loss using experiments conducted on different inverse problems. For each experiment, train model corresponding to method as well as baseline methods and compare their performance. The inverse problems we consider are 1) inpainting, 2) compressive sensing, 3) accelerated MRI and 4) sparse-view CT. We also validate our theoretical predictions by testing the effect of using an equivariant architecture. In Section 5.1 we detail our experiments on compressive sensing, in Section 5.2 on image inpainting, in Section 5.3 on accelerated MRI and in Section 5.44 on sparse-view CT, and in Section 5.5 we present an abla- tion study on the effect of equivariant architectures. For additional details about the experiments, see Appendix B
In each experiment, we compare against multiple baselines: a supervised baseline using the super- vised loss described in eq. (2) the EI (Chen et al. (20TT) baseline described in eq. (4), measurement consistency baseline (MC and SURE) Eldar (209) and a learning-fres baseline which are either measurements directly,
their image under the adjoint or the pseudo-inverse of the forward oper- ator. We use the same architecture and the same training procedure for every method to ensure a fair comparison. We report the peak signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) (Wang et al., 2004) of the final reconstructions for the different methods. They are distortion metries indicating how close the reconstructions are to the ground truth images. We do not include perception metrics which are known to be at odds with them (Blau & Michaeli] (2018). In each case, we also compute equivariance metries (EQUIV) for translations or rotations and flips.
We design a model architecture from the principles introduced in Section 4 with a variant equivari- shifts, one equivariant to rotations and flips, and without equivariance. It uses an existing unrolled architecture (Aggarwal et 2019) with a prior step implemented as standard UNet (Ron- neberger al. 2015). or that of Chamann Dokmanić (2021b) to enforce the equivariance shifts. It also optionally uses Reynolds' averaging to enforce the equivariance to rotations and flips. We use single equivariant variant dictated by Corollary [ for each problem, the shift-equivariant one for compressive sensing and inpainting, and that equivariant to rotations and flips for MRI and CT.
5.1 COMPRESSIVE SENSING
The 28 x 28 ground truth images are obtained from the MNIST dataset and are measured without additional noise through compressive matrices A m with < varying from training to training to assess the impact of the compression rate. These matrices are obtained by sampling 28 28. Figure shows
Aij (O, 1/m) for i 1, ...... m and j I,.... m, where the performance of the different methods for the different compression rates. Our method performs almost well as the supervised baseline, while the equivariant imaging baseline performs close to the supervised baseline only for higher compression rates.
40
PSNR (dB)
Supervised
30
ES (Ours)
EI
20
50
60
80
90
Compression level
Figure 1: Compressive sensing results. ES (ours) performs similarly as the supervised baseline,
unlike El (baseline) whose performance gap widens with higher compression levels.
5.2 IMAGE INPAINTING
The dataset consists of 128 x128 images from DIV2K (Agustsson & Timofte 2017 measured without additional noise through single subsampling matrix A E pmaxa selecting about 30% of the pixels. This matrix is obtained by sampling ai ai...., ., an B(0.3) where = 128 128 and letting Ar, = وي.رة for = 1...., m and j = 1..... n, where m s 0.3n the number of nonzero values in a, and je is the i-th index in corresponding nonzero value. Among the 900
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images in the dataset, 800 are used for training while the remaining 100 are used for testing. For the supervised method, we use different crops at each evaluation. Table and Figure show that ES performs almost well the supervised bascline, and better than El.
Ground Truth
Supervised
ES (ours)
Measurement
Figure 2:
Sample reconstructions for image inpainting, ES (ours) produces images perceptually closer to the supervised baseline than El (baseline) which appears blurry.
Table 1: Inpainting results. ES (ours) performs better than EI (baseline), both in terms of recon- struction quality (PSNR, SSIM) and measured equivariance (EQUIV), while performing competi- tively against the supervised baseline. In bold, the best self-supervised metrics (avg + st.d.).
Method
PSNR
SSIM f
EQUIV
Supervised
28.46 + 2.97
0.8982 + 0.0411
28.46 + 2.97
ES (Ours)
27.45 + 2.86
0.8737 + 0.0461
27.46 + 2.85
HI
25.89 + 2.65
0.8332 + 0.0521
25.89 + 2.65
MC
8.22 2.47
0.0983 + 0.0551
8.22 2.47
Incomplete image
8.22 2.47
0.0973 + 0.0542
N/A
5.3 MAGNETIC RESONANCE IMAGING
The dataset contains 320 x320 images from FastMRI (Zbontar et al. 2019) subsampled in the Fourier domain by a single binary mask corresponding to an acceleration of 8, as well as Gaus- sian noise with standard deviation of 0.005 corresponding
signal-to- noise ratio (SNR) of 40 dB. Mathematically, the forward operator E mmmx is expressed as A MF where F Raxx denotes the discrete Fourier transform matrix and where M E mR is the subsampling mask defined as Mr, = 55... for i = 1,... m and j = 1,..., n. where j; denotes the i-th component in pixel in one of the subsampled vertical lines of the 973 images in the full dataset, 900 are selected for training and the remaining 73 are used for testing. We also test our method on noise dominated setting using a different mask corresponding to an acceleration of 6, with higher noise level of 0. corresponding to an SNR of only 10 dB, see these results in Appendix [B.4] The variant of our proposed loss we use this experiment is the R2R one introduced in Section 4 with a 0.5. Table shows the performance of the different methods on the test set. Table 3 shows the synergy of our method with the equivariant architecture. Figure 3 shows sample reconstructions from the trained models.
1n corresponding
random Gaussian mask. Out
We also evaluate our method on real MRI measurements from FastMRI instead of synthetic mea- surements. We normalize the k-space data that are sampled different grids for different scans by resampling them using aliasing- free sinc interpolation on the 320 320 sampling grid of the ground truth scans. The resulting k-space data subsampled using the same x8 acceleration mask used in the setting with synthetic measurements. Table shows that our method performs competitively.
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Ground Truth
Supervised
ES (Ours)
SURE IDFT
Sample reconstructions for MRI (x8 Accel., 40 dB SNR). Unlike El (baseline) which suffers from dot-shaped artifacts, ES (ours) is perceptually closer to the supervised baseline. line with the theoretical predictions, SURE and IDFT (baselines) fail to recover information beyond the observed frequencies.
Figure 3:
5.4 COMPUTED TOMOGRAPHY
The dataset consists in pairs of ground truth CT scans and corresponding sinograms. In order to obtain the CT scans, we resize to 256 x256 pixels and clip between -1,000 and 1,000 HUs the ground truth scans from the LIDC-IDRI dataset (Armato III et (011). For each scan, we compute the corresponding sinogram using discrete Radon transform with 50 views and additive white Gaussian noise with a standard deviation of 0.001 corresponding to a SNR of about 50 dB. The resulting 1,010 pairs are further split into 900 training pairs and 110 test pairs. Table shows that ES performs almost as well as the supervised baseline, and better than EL.
Table 2: Medical imaging results. ES (ours) performs better than EI, SURE and MC (baselines), while performing almost as well as the supervised baseline in reconstruction quality (PSNR, SSIM) and measured equivariance (EQUIV), In bold, the best self-supervised metries (avg + st.d.).
MRI (x8 Accel,, 40 dB SNR)
Method
PSNR
SSIM
EQUIV
Supervised
28.74 + 2.81
0.6445 0.90
31.71 # 2.83
ES (Ours)
28.54 + 2.75
0.6195 0.1188
31.53 + 2.74
EI
27.88 + 2.64
0.5731 # 0.1299
30.79 + 2.64
SURE
24.45 + 1.86
0,5479 = 0.0740
27.35 + 1.90
IDFT
23.62 + 1.90
0.5052 + 0.0900
25.99 + 1.94
Real MRI measurements (x8 Accel.)
Method
PSNR 1
SSIM t
EQUIV
Supervised
28.81 + 2.85
0.6480 = 0.1
31.81 + 2.84
ES (Ours)
28.30 2.64
0.6151 0.1179
31.29 # 2.62
EI
27.88 + 2.61
0.5740 = 0. 1290
30.80 # 2.61
MC
23.63 + 1.90
0.5061 0.0904
26.00 + 1.94
IDFT
23.63 + 1.90
0.5060 0.0904
26.00 + 1.94
CT (50 views, dB SNR)
Method
PSNR t
SSIM f
EQUIV
Supervised
33.99 + 2.48
0.8819 = 0.0585
34.00 + 2.49
ES (Ours)
32.62 + 2.16
0.8570 # 0.0596
32.60 + 2.17
28.61 + 1.28
0.7400 = 0.0466
28.61 + 1.29
FBP
25.59 + 0.69
0,4805 # 0.0363
25.59 + 0.70

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5.5 ABLATION STUDY
Table shows that architectures designed to be equivariant are measurably more equivariant across imaging modalities and training losses. Moreover, it shows that networks trained using the splitting loss perform better for equivariant architectures than for non-equivariant architectures, with an even greater gap than for supervised inpainting baselines, which confirms the theoretical analysis made in Section Finally, we observe that non-equivariant architectures still lead to fairly measurably equivariant models. While surprising, this phenomenon has already been witnessed and usu- ally referred to as learned equivariance, whereby the training data and inductive biases lead to fairly equivariant learned models (Gruver et al., al.] (024)). We believe that this learned equivariance is respon- sible for the high performance of splitting methods even when using non-equivariant architectures. For additional results on the impact of equivariant architectures, see Appendix B.4
Table 3: Impact of using equivariant architectures. In accordance with the theoretical results described in Section 4 there is a synergy between the splitting loss and equivariant architectures resulting in higher performance. Non-equivariant models have surprisingly high equivariance mea- sures (EQUIV) which might explain their high performance when using the splitting loss. Eq. arch. denotes whether the architecture is equivariant. In bold, the best self-supervised metries (avg d: st.d.).
Image inpainting
Training loss
Eq. arch.
PSNR t
SSIM f
EQUIV 1
Supervised
V
28.46 2.97
0.8982 + 0.0411
28.46 2.97
28.62 = 3.03
0.9002 = 0.0414
27.85 77.8522777 2.71
Splitting (Ours)
V
27.45 2.86
0.8737 0.0461
27.46 2.85
X
27.20 + 2.83
0.8652 + 0.0461
26.52 + 2.60
MRI (x8 Accel., 40 dB SNR)
Training loss Supervised
Eq. arch.
PSNR t
SSIM f
EQUIV
28.74 2.81
0.6445 0.1094
31.71 2.83
x
28.48 + 2.68
0.6381 # 1082
28.78 + 1.95
Splitting (Ours) V 28.54 + 2.75
0.6195 # 0.1188
31.53 + 2.74
X 28.18 2.58
0.6104 = 0.1176
27.28 + 2.10
CONCLUSION
In this work, we propose new self-supervised loss for solving inverse problems which bridges the gap between existing equivariance and splitting-based self-supervised losses. We motivate the design of our loss by showing that minimizing the expected loss results in MMSE estimators. We further validate our method using numerical simulations on different image distributions and imag- ing modalities, including inpainting of natural images, MRI and CT. These results suggest that the proposed method compares favorably to the equivariant imaging baseline and is close supervised methods. To the best of our knowledge, this work is the first leverage equivariant networks to learn from incomplete data alone, going beyond the usual goal of improving the generalization of the networks to unseen transformations at test time. Our method provides a new way to evaluate the benefits of using different equivariant architectures, and benefit from future advances made in this field. More broadly, our work is further evidence that invariance is a promising prior for learning from incomplete data.
ACKNOWLEDGEMENTE
Victor Sechaud and Julian Tachella are supported by the ANR grant UNLIP This project was provided with computing HPC and storage resources by GENCI at IDRIS thanks to the grant 202--DD011016422 on the supercomputer Jean Zay's H100 partition. Part of the computations were performed on the machine cluster at the Pascal Blaise Center at the ENS de Lyon QQemmnner & Corvelled 2013).
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REFERENCES
Hemant Kumar Aggarwal, Merry P. Mani, and Mathews Jacob. MoDL: Model Based Deep Learning Architecture for Inverse Problems. IEEE Trans. Med. Imaging, 38(2):394-405, February 2019.
Eirikur Agustsson and Radu Timofte. NTIRE 2017 Challenge on Single Image Super-Resolution Dataset and Study. In CVPRW, pp. 1122-1131, July 2017.
Samuel G. Armato III, Geoffrey McLennan, Luc Bidaut, Michael F. McNitt-Gray Charles R. Meyer, Anthony P. Reeves, Binsheng Zhao, Denise R. Aberle, Claudia Henschke, Eric A. Hoffman, Ella A. Kazerooni, Heber MacMahon, Edwin J. R. van Beek, David Yankelevitz, Alberto M. Biancardi, Peyton H. Bland, Matthew S. Brown, Roger M. Engelmann. Gary E. Laderach, Daniel Max. Richard C. Pais, David P.-Y. Qing, Rachacl Y. Roberts, Amanda R. Smith, Adam Starkey, Poonam Batra, Philip Caligiuri, Ali Farooqi, Gregory W. Gladish, C. Matilda Jude, Reginald F. Munden, Iva Petkovska, Leslie E. Quint, Lawrence H. Schwartz, Baskaran Sundaram, Lori E. Dodd, Charles Fenimore, David Gur, Nicholas Petrick, John Freymann, Justin Kirby, Brian Hughes, Alessi Vande Casteele, Sangeeta Gupte, Maha Sallam, Michael D. Heath, Michael H. Kuhn, Ekta Dharaiya, Richard Burns, David S. Fryd, Marcos Salganicoff, Vikram Anand, Uri Shreter, Stephen Vastagh, Barbara Croft, and Laurence P. Clarke. The Lung Image Database Consortium (LIDC) and Image Database Resource Initiative (IDRI); A Completed Reference Database of Lung Nodules on CT Scans. Medical Physics, 38(2):115-9311 2011. ISSN 2473- 4209. doi: 10.1118/1.3528200
Chinmay Belthangady and Loic A Royer. Applications, promises, and pitfalls of deep learning for fluorescence image reconstruction. Nature methods, 16(12):115--1225, 2019.
Yochai Blau and Tomer Michaeli. The Perception-Distortion Tradeoff. In CVPR, pp. 6228-6237, 2018.
Elena Celledoni, Matthias J. Ehrhardt, Christian Etmann, Brynjulf Owren, Carola-Bibiane Schönlieb, and Ferdia Sherry. Equivariant neural networks for inverse problems. Inverse Prob- lems, 37(8):085006, July 2021.
Anadi Chaman and Ivan Dokmanić. Truly shift-invariant convolutional neural networks. In CVPR, pp. 3773-3783, 2021a.
Anadi Chaman and Ivan Dokmanić. Truly shift-equivariant convolutional neural networks with adaptive polyphase upsampling. Axilomar Conf Signals Sysı Comput, pp. 1113-1120, 120, 2021b.
Dongdong Chen, Julián Tachella, and Mike E. Davies. Equivariant Imaging: Learning Beyond the
Range Space. In ICCV, pp. 4379-4388, 2021.
Dongdong Chen, Julián Tachella, and Mike E. Davies. Robust Equivariant Imaging: A fully unsu- pervised framework for learning to image from noisy and partial measurements. In CVPR, pp. 5647-5656, 2022.
Taco S. Cohen and Max Welling. Group Equivariant Convolutional Networks. In ICML, pp. 2990- 2016.
Taco S. Cohen and Max Welling, Steerable CNNs. In ICLR, 2017.
Giannis Daras, Kulin Shah, Yuval Dagan, Aravind Gollakota, Alexandros G. Dimakis, and Adam Klivans. Ambient Diffusion: Learning Clean Distributions from Corrupted Data. NeurlPS, pp. 288-38 188-313,20 2023.
Leo Davy, Luis M. Briceno-Arias, and N. Pustelnik. Restarted contractive operators to learn at equilibrium, June 2025.
Yonina C. Eldar. Generalized SURE for Exponential Families: Applications to Regularization. IEEE Trans Signal Process, 57(2)4471441) 2009.
Fabian Ewald Fassnacht, Joanne C White. Michael A Wulder, and Erik Nasset. Remote sensing in forestry: current challenges, considerations and directions. Forestry: An International Journal of Forest Research, 97(1):11-31 2024.
11

---

Published as a conference paper at ICLR 2026
Nate Gruver, Mare Finzi, Micah Goldblum, and Andrew Gordon Wilson. The Lie Derivative for Measuring Learned Equivariance, 2024.
Allard A. Hendriksen, Daniel M. Pelt, and K. Joost Batenburg. Noise2Inverse: supervuue deep convolutional denoising for tomography. IEEE Trans. Comput. Imaging, 121320-1335, 2020.
Kyong Hwan Jin, Michael T McCann, Emmanuel Froustey, and Michael Unser. Deep convolutional neural network for inverse problems in imaging. IEEE Trans Image Process, 26(9):4509-455 4522, 2017.
Sékou-Oumar Kaba, Arnab Kumar Mondal, Yan Zhang, Yoshua Bengio, and Siamak Ravanbakhsh. Equivariance with Learned Canonicalization Functions. In ICML, pp. 15346-15566, 2023.
Tero Karras, Miika Aittala, Samuli Laine, Erik Härkönen, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Alias-free generative adversarial networks. In NeurIPS, volume 34, pp. 852-863, 2021.
Achim Klenke. Probability theory: comprehensive course. Springer, 2008.
Alexander Krull, Tim-Oliver Buchholz, and Florian Jug. Noise2Void Learning Denoising From Single Noisy Images. In CVPR, pp. 2129-2137, 2019.
Ilya Loshchilov and Frank Hutter. Decoupled Weight Decay Regularization, 2019.
Christopher A. Metzler, Ali Mousavi, Reinhard Heckel, and Richard G. Baraniuk. Unsupervised Learning with Stein's Unbiased Risk Estimator, July 2020.
Hagay Michaeli, Tomer Michaeli, and Daniel Soudry. Alias-Free Convnets: Fractional Shift Invari- ance via Polynomial Activations. In CVPR, pp. 16333-16342, 2023.
Charles Millard and Mark Chiew. A theoretical framework for self-supervised MR image recon- struction using sub-sampling via variable density Noisier2Noise. IEEE Trans Comput Imaging, 9:707-720, 2023.
Brayan Monroy, Jorge Bacca, and Julián Tachella. Generalized Supervised Learning Beyond Gaussian Noise. In CVPR, pp. 28155-28164, 2025.
Recorrupted: Self-
Greg Ongie. Rebecca Willett, Daniel Soudry, and Nathan Srebro. A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case, October 2019.
Tongyao Pang, Huan Zheng, Yuhui Quan, and Hui Ji. tecoruptee-to-RRcoruptdddd Unsupervised Deep Learning for Image Denoising. In CVPR, pp. 2043-2052, June 2021.
Javier Portilla and Eero Simoncelli. A parametric texture model based on joint statistics of com- plex wavelet coefficients. International journal of computer vision, 40(1):49-70, 2000.
Omri Puny, Matan Atzmon, Heli Ben-Hamu, Ishan Misra, Aditya Grover, Edward Smith, and Yaron Lipman. Frame Averaging for Invariant and Equivariant Network Design, March 2022.
Emmanuel Quemener and Marianne Corvellee. Sidus-the the solution for extreme deduplication of an operating system. Linux Journal, 2013(235):3, 2013.
Marco Ragone, Reza Shahabazian- Yassar, Farzad Mashayek, and Vitaliy Yurkiv. Deep learning modeling in microscopy imaging: review of materials science applications. Progress in Mate- rials Science, 138:101165. 2023.
Sebastián Rivera, Iván Ortíz, Tatiana Gelvez, Fernando Rojas, and Henry Arguello. Rotation invari- ant deep learning approach for image inpainting. In STS/VA, pp. 1-5, 2021.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional Networks for Biomed- ical Image Segmentation. In MICCAI, pp. 234-241, 2015.
Leonid Rudin, Stanley Osher, and Emad Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, (0(1))259-268, November 1992.
12

---

Published as a conference paper at ICLR 2026
Akiyoshi Sannai, Makoto Kawano, and Wataru Kumagai. Equivariant and Invariant Reynolds Net- works, October 2021.
Akiyoshi Sannai, Makoto Kawano, and Wataru Kumagai. Invariant and equivariant reynolds net- works. Journal of Machine Learning Research, 25(42):1-36, 2024.
Jérémy Scanvic, Mike Davies, Patrice Abry, and Julián Tachella. Scale-Fquivariant Imaging: Self- Supervised Learning for Image Super-Resolution and Deblurring, March 2025.
Victor Sechaud, Laurent Jacques, Patrice Abry, and Julián Tachella. Equivariance--asee self- supervised learning for audio signal recovery from clipped measurements. In EUSIPCO, August 2024.
Julián Tachella, Dongdong Chen, and Mike Davies. Unsupervised learning from incomplete mea- surements for inverse problems. NeurIPS, 35:4983-4995, 2022.
Julián Tachella, Dongdong Chen, and Mike Davies. Sensing Theorems for Unsupervised Learning
in Linear Inverse Problems. Journal of Machine Learning Research, 24(39):1-45 2023. ISSN
1533-7928.
Julián Tachella, Mike Davies, and Laurent Jacques. UNSURE: Sulf-super learning with Un- known Noise level and Stein's 's Unbiased Risk Estimate. In 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 Sar- ron, Victor Sechaud, Georg Schramm, Chao Tang, Romain Vo, and Pierre Weiss. DeepInverse: A Python package for solving imaging inverse problems with deep learning, May 2025b.
Matthieu Terris, Thomas Moreau, Nelly Pustelnik, and Julian Tachella. Equivariant plug-and-play image reconstruction. In CVPR, pp.
2024.
Singanallur V Venkatakrishhan, Charles A Bouman, and Brendt Wohlberg. aong-and-play priors for model based reconstruction. In JEEE Glob Conf Signal Inf Process, pp. 945-948, 2013.
Antonia Vojtekova, Maggie Lieu, Ivan Valtchanov, Bruno Altieri, Lyndsay Old, Qifeng Chen, and Filip Hroch. Learning to denoise astronomical images with u-nets. Monthly Notices of the Royal Astronomical Society, 503(3):3204-33255 2021.
Andrew Wang and Mike Davies.
for unsupervised imaging with camera geometry. In Alessio Del Bue, Cristian Canton, Jordi Pont-Tuset, and Tatiana Tommasi (eds.), ECCV Workshops, pp. 116-134, 2024.
Andrew Wang, Steven McDonagh, and Mike Davies. Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction, May 2025.
Zhou Wang, A.C. Bovik, H.R. Sheikh, and E.P. Simoncelli. Image quality assessment: From error visibility to structural similarity. IEEE Trans Image Process, 13(4)6600-61,, April 2004.
Philip J Withers, Charles Bouman, Simone Carmignato, Veerle Cnudde, David Grimaldi, Char- lotte K Hagen, Eric Maire, Marena Manley, Anton Du Plessis, and Stuart R Stock. X-ray com- puted tomography. Nature Reviews Methods Primers, 1(1):18, 2021.
Guixian Xu, Jinglai Li, and Junqi Tang. Fast Equivariant Imaging: Acceleration for Unsupervised Learning via Augmented Lagrangian and Auxiliary PnP Denoisers, July 2025.
Xuemiao Xu, Huaidong Zhang, Guoqiang Han, Kin Chung Kwan, Wai-Man Pang. Jiaming Fang, and Gansen Zhao. A two-phase space resection model for accurate topographic reconstruction from lunar imagery with pushbroomscanners. Sensors, 16(4):507, 2016.
Burhaneddin Yaman, Seyed Amir Hossein Hosseini, Steen Moeller, Jutta Ellermann, Kåmil Ugurbil, and Mehmet Akçakaya. Self-Supervised Learning of Physics- Guided Reconstruction Neural Net- works without Fully-Samples Reference Data. Magnetic Resonance in Med, 1466):31223111,,
2020.
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Published as a conference paper at ICLR 2026
Jure Zbontar, Florian Knoll, Anuroop Sriram, Tullie Murrell, Zhengnan Huang, Matthew J. Muck- ley, Aaron Defazio, Ruben Stem, Patricia Johnson, Mary Bruno, Mare Parente, Krzysztof Geras, Joe Katsnelson, Hersh Chandarana, Zizhao Zhang, Michal Drozdzal, Adriana Romero, Michael Rabbat, Pascal Vincent, Nafissa Yakubova, James Pinkerton, Duo Wang, Erich Owens, C. Lawrence Zitnick, Michael Recht, Daniel K. Sodickson, and Yvonne W. Lui. fastMRI: An Open Dataset and Benchmarks for Accelerated MRI, December 2019.
Richard Zhang. Making Convolutional Networks Shift-Invariant Again, ICML, pp. 7324-7334,
2019.
A DETAILS ABOUT THE METHOD
A.1
EQUIVARIANT RECONSTRUCTOR
Supervised
PSNR (dB)
30
ES (Ours)
EI
20
Pseudo-inv.
10
60
70
90
Compression level (%)
Figure 4: Compressive sensing results. Adds the pseudo-inverse reconstruction (pseudo-inv.)
baseline to Figure [
Equivariant reconstructor
1
Σ T
AT, AT,)
fly A)
9
gEG
A
DF
DF
DF
r(y, A)
1
UNet
(xx)
z - (A
A
A
vI)
A uz) x
Denoiser
Data-fidelity (DF)
MoDI.
Figure 5: Reconstructor equivariant to rotations and flips. It is the Reynolds averaging of 90º rotations and horizontal and vertical flips as in eq. (18) of non-equivariant reconstructor of the MAP type, as in eq. (19), implemented using MoDL unrolled algorithm (Aggarwal et al., 2019) with 3 iterations, shared weights, and using a non-equivariant residual UNet (Ronneberger et al. 2015 as the denoiser architecture.
In Theorem 2
we consider reconstruction functions of different structures. In this section, we give additional details about them.
The artifact removal reconstructor architecture (Jin et al., 2017) in eq. (16) consists in a projection step that maps the measurements back into the image space using the adjoint or the pseudo-inverse
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of the forward matrix, which is immediately followed by very general trainable network, often a UNet or another encoder-decoder type of network, or sometimes simply a fully convolutional network. Theorem 2 states that this reconstructor architecture produces equivariant reconstructors as long as the denoiser architecture is, itself, equivariant in the sense of eq. (14)
Reynolds averaging for possibly non-equivariant reconstruction functions ryy A) in eq. (88
1
f(y. A)
ITr(( AT₂). G
18
DEG
has, as far as we know, not been defined in previous works. is natural extension of Reynolds averaging for possibly non-equivariant image-to-image functions or denoisers (x()) (Sannai et al., 2024
Terris et al. 2024).
(TT (xx) = G
T 9Eg
(22)
which makes them equivariant in the sense of eq. (14), to reconstruction functions in order to make them equivariant in the sense of eq. (15). is fairly simple way to make a reconstructor equivariant and it is relatively inexpensive for small groups of transformations such as the group of 90º rotations and horizontal and vertical flips (Cohen Welling 2016). is however too expensive to be used in practice when the group is relatively large, like the group of shifts, since it would require to evaluate the neural network for as many times as there are pixels in the input image, for each image.
The maximum a posteriori (MAP) reconstructor architecture defined in eq. 19
f(y, A) = argmax (p(a) yy A)} #ER*
19
is classical reconstructor architecture that has been used in different ways, including iterative algorithms with a hand-crafted prior (Rudin et al. 1992, Davy et al. 2025), tectures using iterative approach but with the hand-crafted prior replaced with pre-trained de- noiser Venkatakrishhan et al., 2213), and unrolled architectures where the optimization problem is done with fixed number of steps and where parts of the algorithm are replaced with trainable mod- ules (Aggarwal et al.] 2019). It is often interpreted as a Bayesian maximum a posteriori estimator, but also commonly outside of a Bayesian framework as a variational approach with a data-fidelity term and a regularization term. It is one of the reconstructor architectures that we use for our ex- periments in Section 5 Theorem 2 states that these reconstructors are equivariant as long as the prior distribution is invariant in the sense of eq. (5). or equivalently, if the associated regularization function or negative log-prior is invariant.
archi-
The minimum mean squared error (MMSE) estimator in eq. 20
fly A) Emy.. {x)
20
is generally the target theoretical reconstructor as it achieves the highest theoretical PSNR TTchella
al. 2023). It is generally estimated by reconstructors trained with a mean squared error loss (Chn
et al. 202 l) which is notably how we train the supervised baselines in the experiments in Section 5
Theorem states that as long as the prior distribution is invariant in the sense of eq. (5), the MMSE
reconstructor is equivariant in the sense of eq. 15
A.2 How TO CHOOSE TRANSFORMATIONS FOR A GIVEN INVERSE PROBLEM
There are two major criteria for choosing the transformations for given application. First, the im- age distribution of interest should be invariant to the chosen transformations. Aerial, remote sensing and microscopic images are invariant translations and rotations as the scenes and subjects they measure exhibit no privileged position and orientation with respect to the image plane. Natural im- age distributions and texture distributions (Portilla & Simoncelli 2000) also generally invariant to translations but they are less invariant to rotations as natural images are typically oriented upward and texture distributions might be anisotropic.
Second, the transformations should also be chosen in accordance with the measurement operator. Corollary [ shows that transformations for which the operator is equivariant do not improve the
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reconstruction process. It is well-known criterion introduced in the original work equivari- ant imaging (Chen et al. (021) which remains correct in our setting where measurement splitting is added to the theoretical analysis. Table 4 lists correct choices of transformations for common measurement operators.
Table 4: Decision table for the transformations. Corollary shows that not all transformations are well-suited for all problems. Namely, transformations for which the operator equivariant introduce no additional information and should not be used. This table specifies which transformations are well-suited for which operator.
Operator
Translation
Rotation
Permutation
Amplitude
Isotropic blur
Image inpainting
Sparse-view CT
X
Accelerated MRI Compressive sensing
X
A.3 END-TO-END ALGORITHM
In this section, present the end-to-end ES algorithm. It consists in training step where equiv- ariant reconstructor is trained using backpropagation against a training dataset measurements only, after which it applied to obtain the reconstructions associated the test measurements. The ES loss used in the training step is computed using the expression in Equation (3) The expectations
estimated using Monte Carlo sampling where single sample is used training time and T 10 samples are used at inference. In the experiments, we use the splitting ratio = 0.8 corresponding mi
0.8m and m2 0.2m. Algorithms T and [
show the detailed algorithms pseudo-code.
Algorithm 1: Equivariant Splitting (Training procedure)
Input: Dataset D {y her forward operator E Rm split ratio s, learning rate 77.
number of epochs E, equivariant reconstructor fe
Output: Trained model fo for epoch = 1 to F. do
Shuffle dataset D:
foreach mini-batch C D do
init: L=
foreach in B do
init: M E [Rmixx xm a random split s.f m X m.
Split measurement and operator: y1 My, Ai MA;
Compute predictions: æ = fe y1, A1);
Compute sample loss f = AA y1l:
Aggregate loss: t L ++;;
end
Mean:
L
B3 L:
Backpropagation: compute gradients VoL: Update parameters: t- ADAM(O, VoL. 17) end end
B DETAILS ABOUT THE EXPERIMENTS
We use the optimizer Adam W (Loshchilov & Hutter 2019) for every training with different learning rates for the different inverse problems, weight decay of 10-8 beta coefficients equal 0.9 and
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Algorithm 2: Equivariant Splitting (Inference)
Input: Measurement E mR forward operator E Rmxx, split ratio 8, number of samples 7', trained reconstructor fe
Output: Reconstructed images è = -o A)
init: á =0
for i to (to/ T do
init: M E -R"1 a random split s.t my $ X m.
Split measurement and operator: y/ My, A1 MA; Compute one predictions: á; = fo A1);
Aggregate: t #;;;
end
Mean: T
0.999 and without the AMSGrad option. For longer trainings, use step schedulers that divide the learning rate by a factor ranging from to 10 at specific epochs, up to 3 or times.
In our experiments, make extensive use of the DeepInverse library (Tchella al. 2025b) that provides an implementation the various forward operators and training losses that we use. Every model is trained for up 50 hours single GPU, either NVIDIA H100, GH200 or RTX 4090. See Appendix B.4 for more details about the training durations.
Accel., 40 dB SNR
x6 Accel., 10 dB SNR
Figure 6: Examples of k-space measurements in MRL. (left) a less noisy problem with less mea- surements, (right) noisier problem with more measurements.
B. DATASETS
Image inpainting The ground truth images are images obtained from the dataset DIV2K (Agustsson & Timofte (017)) which contains pictures natural scenes (landscapes, animals) by first resizing them resolution of 256 x 256 pixels before extracting a central 128 128 pixels crop. We synthesize the measurements by corrupting the ground truth images using a single binary mask sampled from pixel-wise Bernoulli distribution with a 30% chance of keeping each pixel value. In this setting, we do not corrupt the measurements further with additional noise.
MRI The dataset consists in 973 pairs of ground truth images from the FastMRI dataset (Zoontar et al., (019) and associated k-space measurements synthesized using a single coil sensitivity map. The ground truth images have a size of 320 x 320 pixels and each corresponds the middle slice of a different 3d knee acquisition. The k-spaces are synthesized as discrete Fourier transforms sub- sampled on a single non-regular grid and further corrupted with additive white Gaussian noise with standard deviation of 0.005 corresponding to an average SNR of about 40 dB. The subsampling grid models the coil sensitivity map, is sampled from Gaussian distribution and corresponds to an acceleration of 8. The entire dataset is finally split into train/val split containing 900 images and test split containing the remaining 73 images. Our implementation uses the work from Wang et al. 2025)
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We also consider an additional dataset obtained in the same way except using different mask corresponding to an acceleration of and with a higher noise level corresponding to standard deviation of 0.1 or average SNR of 10 dB in the k-space domain.
B.2 NETWORK ARCHITECTURES
The unrolled architecture uses 3 iterations and the weights are shared across different iterations. Every UNet residual, has scales and has no normalization layer we find them to detrimental to the performance.
The transforms we use the model architecture and the metrics are grid- pree-pree:ving:: grid-aligned shifts, 90" rotations, and vertical and horizontal flips. The transforms we use in the El are grid- aligned shifts and 1 rotations following the prior art Chen al. (202I). Reynolds' averaging implemented using unbiased Monte-Carlo estimator whereby a single random transform is sam- pled at every evaluation to save on computational cost.
B.3 METRICS
In the experiments, metrics (PSNR, SSIM) and a new equivariance metric for reconstructors (EQUIV). The peak signal- to-noise ratio defined, for images with dynamic range normalized to [0, 1], the mean squared егror expressed in decibels (dB)
three different performance metrics including two standard distortion
PSNR Ezz { 1000o (I)f - ()) -=
(23)
The structural similarity index measure (Wang et al., 2004) is more perceptual metric which is combination of the empirical means, standard deviations and correlation coefficient associated to the reference and compared images. The complete definition of the metric is too long be included in this work and we refer the reader to the original publication for more details. In addition to these two standard distortion metrics, & Dokmanić (2021b) adapted for our proposed definition of equivariant reconstructors. It is the average squared error associated with eq. (15) and expressed in dB for readability
cquivariance metric similar to that used by Chaman
EQUIV - 100
(Fy.9 {11 AT,)) AT2) - {))))
f(y.
(24)
can roughly understood as a PSNR for equivariance. In every experiment, we use the same group of transformations for EQUIV for the equivariant reconstructor architectures.
B.4 RESULTS
Table 5: MRI results in another setting. Supplementary results for different MRI problem (x6
Accel,, 10 dB SNR) than in Table 2 In bold, the best self-supervised metries. Values: avg + st.d.
MRI (x6 Accel,, 10 dB SNR)
Method
PSNR 1
SSIM 1
EQUIV t
Supervised
27.39 # 2.44
0.5243 # 0.1373
30.38 2.43
ES (Ours)
27.33 # 2.45
0.5126 # 0.1444
30.32 2.44
EI
27.23 # 2,41
0.5110 # 0.1421
30.21 # 2.40
SURE
27.08 # 2.29
0.5097 # 0.1372
30.06 2.28
IDFT
23.85 1.05
0.3878 + 0.0272
25.14 0.79
In this section, we provide more details about the main experiments and present additional experi- ments.
For each training done in the main experiments, we report the average epoch duration and the number of epochs until the model has finished training. For the sake of comparability, conduct the different trainings for the same imaging modality on the same GPU. Namely, we used single NVIDIA RTX 3090 Ti GPU for every inpainting experiment and single NVIDIA H100
made sure to
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GPU for every MRI experiment. Table 10 shows that self-supervised methods including ours have generally longer epochs and require more epochs than the fully supervised gold standard. We believe that this is due to fundamental trade-off whereby the use of ground truth data accelerates the learning procedure while also enabling the use of simpler training algorithms. Moreover, unlike standard implementations of El and SURE, ES requires only single network pass per iteration resulting in significantly faster epochs. Overall, ES is computationally more efficient than EI, in addition to being more performant in terms of reconstruction quality.
Figure 4 shows additional compressive sensing results, Table 3 and Figure show results on a MRI experiment with settings different from the main one. Table shows extended results for the ablation study on equivariant architectures, and showw sample k-spaces from the MRI experiments. Figure shows the equivariant reconstructor architecture that we adopt in the MRI experiments.
Table 6: Extended results on the impact of equivariant architectures. Adds to Table 3 results for EI with a con-viuiva architecture for the inpainting task, results for the noise- doisn-domi MRI task. In bold, the best sulf-super metrics. Values: avg st.d.
Image inpainting
Eq. arch Training loss Supervised
PSNR 1
SSIM f
EQUIV 1
28.46 2.97
0.8982 # 0.0411
28.46 + 2.97
X
28.62 3.03
0.9001 = 0.0415
27.85 + 2.71
Splitting (Ours)
V
27.45 2.86
0.8737 # 0.0461
27.46 2.85
27.20 2.83
0.8651 0.0463
26.52 2.60
El loss
V
25.89 + 2.65
0,8332 # 0.0521
25.89
26.33 = 2.81
0.8451 # 0.0536
25.58
MC loss
8.22 + 2.47
0.098 = 0.055
8.22 2.44
8.24 2.48
0.100 = 0.056
8.24 + 2.48
MRI (x8 Accel., dB SNR)
Training loss
Eq. arch.
PSNR 1
SSIM f
EQUIV 1
Supervised
V
28.74 + 2.81
0.6445 + 0.1094
31.71 + 2.83
28.48 2.68
0.6381 # 0.1082
28.78 1.95
Splitting (Ours)
28.54 2.75
0.6195 0.1188
31.53 2.74
28.18 2.58
0.6104 = 0176
27.28 2.10
MRI (x6 Accel., 10 dB SNR)
Training loss Supervised
Eq. arch.
PSNR t
SSIM f
EQUIV 1
27.39 2.44
0.5243 + 0.1373
30.38 2.43
X
27.33 + 2.42
0.5174 # 0.1410
29.73 2.20
Splitting (Ours)
V 27.33 2.45
0,5126 # 0.1444
30.32 2.44
X 27.20 2.38
0.5095 + 0.1430
28.66 + 1.76
We present an additional inpainting experiment, designed to simulate more realistic acquisition scenario and study the consistency our method. Specifically, we consider noisy inpainting set- ting in which entire image columns are randomly removed. Such sampling patterns naturally arise in satellite imaging systems based on push-broom scanners (PBS) (Xu et al. 2016). As reported in Ta- ble 7 the proposed method achieves performance comparable to that of the El baseline, confirming its consistency under this more practical acquisition model.
To clarify the behavior of ES when the forward operator A is equivariant nearly equivariant, conducted the following experiment: we performed an PBS inpainting operation in which every other column of the image is removed. In this setting, the operator A becomes "almost" equivariant in the sense that for any even horizontal shift g, we have AT, = TT A, and the same property holds for all vertical shifts. Table shows the reconstruction performance deteriorates significantly.
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Ground Truth
Supervised
ES (Ours)
B1
SURE
IDFT
Figure 7: Sample reconstructions for noise dominated MRI (x6 Accel., 10 dB SNR). In the noise dominated setting, the different models perform more similarly than in the less noisy setting shown in Figure
Table 7: Inpainting with push-broom masks. We consider unevenly-spaced mask sampled randomly to test our method in a realistic inpainting setting, and an evenly-spaced mask to verify the claim in Corollary empirically in case of almost-equivariance.
Image inpainting (Randomly-spaced push-broom mask)
Method
PSNR 1
SSIM f
Supervised
23.72 = 2.10
0.743 + 0.051
ES (Ours)
23.04 + 1.81
0.734 + 0.050
EI
23.05 + 2.13
0.707 + 0.066
Incomplete image
9.44 + 2.31
0.141 = 0.048
Image inpainting (Evenly-spaced push-broom mask)
Method
PSNR 1
SSIM t
Supervised
28.37 = 2.15
0.873 + 0.035
ES (Ours)
21.94 + 2.12
0.617 = 0.091
Incomplete image
9.61 2.42
0.152 + 0.061
28
27
PSNR (dB)
26
Splitting loss (Eq. arch.) Splitting loss
El loss (Eq. arch.)
El loss
24
200
400
600
800
1,000
1,200
1,400
1,600
Epoch
Figure 8:
Performance evolution during training for inpainting. Splitting methods perform better than El independent of the network architecture.
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B.4.1 TESTING THE EFFECT UNKNOWN NOISE DISTRIBUTIONS
In addition to the main experiments where we use the knowledge of the noise distribution in the reconstruction algorithm, we test a more realistic scenario where the noise distribution is unknown and is estimated to zero for lack of better estimate. To do so, we train models using variants of ES and EI where the measurements are assumed to be noiseless even though the training and testing data are nonetheless corrupted with noise. Table Salle&shhn shows that ES also performs better than El when the noise distribution is unknown. Moreover, it shows that the performance of ES tends to be lower when the noise distribution is unknown, but it does not drop exceedingly which demonstrates the stability of ES to unexpected noise.
Table 8: Performance when the noise distribution is unknown. In the unknown noise scenarios, we use the variants of ES and El corresponding to assuming that the measurements are noiseless.
Image inpainting
Method
Known
PSNR f
SSIM 1
Supervised
23.72 2.10
0.743 + 0.051
ES (Ours)
23.04 + 1.81
0.734 + 0.050
ES
22.05 1.36
0.59 = 0.061
EI
23.05 2.13
0.707 = 0.066
EI
22.01 1.71
0.59 + 0.051
Incomplete image
9.44 = 2.31
0.141 0.048
MRI (x8 Accel 40 dB SNR)
Method
Known
PSNR f
SSIM 1
Supervised
28.74 + 2.81
0.6445 + 0. 1094
ES (Ours)
28.54 2.75
0.6195 0.1188
ES
X
28.52 2.75
0.6194 0.1191
EI
27.88 2.64
0.5731 0. 0.1999
EI
X
27.89 + 2.59
0.5755 + 0. 1285
IDFT
23.62 + 1.90
0,5052 + 0.0900
MRI (x6 Accel., 10 dB SNR)
Method
Known
PSNR f
SSIM 1
Supervised
27.39 2.44
0.5243 0.1373
ES (Ours)
27.33 2.45
0.5126 0.1444
ES
25.73 + 1.49
0,4566 = 0.0622
27.23 2.41
0.5110 0 0.12
EI
X
26.02 + 1.65
0.4706 # 0.0873
IDFT
23.85 1.05
0.3878 + 0.0272
B.4.2 EMPIRICAL VERIFICATION OF THE EQUIVARIANCE OF MAP RECONSTRUCTORS
We verify empirically that MAP reconstructors equivariant long as the prior is itself equiv- ariant, i.c., the claim made in Theorem 2 Since they cannot be computed exactly in general, we consider a specific scenario where they the two-dimensional decimation operator with decimation rate 2, 3) y
NN - II,), 2) A E mmx is Ax ~ No o² Imm. and 4) that T, denotes the rotation by angle g E (09, 90", 180", 270") ). Under these assumptions, the prior distribution is equivariant and the MAP estimator in eq. (9) can be expressed in closed-form as
We assume that 1)
f AT,) = 72 72
A AAy y
(25)
with fly. A) being the special where T, In. In the experiment, grayscale image with 128 rows and 128 columns and we compute the equivariance metric in eq. (24) (EQUIV) for the MAP reconstructor using 256 i.i.d. samples from the joint distribution. Table
128 x 128 for
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shows the results for every angle and for the average over all angles. As predicted theoretically, perfect equivariance is achieved.
Table 9: Empirical validation of the equivariance of MAP estimators.
0"
90"
180
270
Average
EQUIV
Training durations. For each training, we report the average epoch duration and the number of epochs until the model is trained. Inpainting trainings are conducted on a single NVIDIA RTX 3090 Ti GPU and MRI trainings on NVIDIA H100 GPU.
Table 10:
Image inpainting
Method
Epoch duration (s)
Epochs
Supervised ES (Ours)
12
200
12
1000
EI
14
1000
MRI (x8 Accel., 40 dB SNR)
Method
Epoch duration (s)
Epochs
Supervised ES (Ours)
29
200
24
13800
EI
53
7800
SURE
36
5700
MRI (x6 Accel., 10 dB SNR)
Method
Epoch duration (s)
Epochs
Supervised
19
70
ES (Ours)
19
3100
EI SURE
75
2400
35
1200
C
PROOFS
For the sake of clarity, we state the propositions and theorems a second time before their proofs. We also state and prove the additional Lemma
[ which helps prove Theorem
Lemma 1. The minimization problem
min Ex.y {1) Axtl²)
(26)
f
admits solutions the functions of the form
{x) + (I A' AAAo)) f(y) A AE
(27)
where u(y) is any function.
Proof. Let's start by stating that f(y) Eujy {Ø) the only solution of (Klenke min
2008
En.ay (1l(()) ar 1=) /
(28)
For f any solution of eq. 26 applying it to f(y) = Af and T Ax gives
Af = Al {x}
(29)
and applying f(y) A Af + (I At fz) with v()) f(v) yields eq. 27 Conversely,
the objective in eq. (26) has the same value no matter the f satisfying eq. (27) and since at least one of them is solution of eq. (26). they all are.
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Theorem 1. In the case of noiseless measurements with p(z) G-invariant (Assumption 1. if the matrix QA, - Ai {(AT,)) ATg TRux has fidl rank for some split A,, then the splitting method yields the same MMSE-optimal reconstructions as the supervised method, i.e, (y, A1) Relui {m))
Proof.
Ey CEs (y, f)}
=By (Eg (Ey1 ALu.
A1) y)}
EEyo yi A1 R. yT,, (1) A1) y
-Ey. A a ||T ff ; A1) - y1
Ex.yL.AL. g || (1/1 A1) ATz=11?
Ey,, A - E Al AT, (J)1, A1) - x) [2=]
Ey..A (1(1) A1) - - T (L))))))) x)};
where the third line use that p(y1, A y, AT2) P1y,, y.g) A is fixed. The fifth line use the noiseless measurements assumption and the invariance of the distribution p(z). The last line
uses definition of QA By applying Lemma] [ the global minimizer of the expected loss is given by:
("),, A1) Q'a, QA (x) -1-
Q'2, QA.)((1))
(30)
where ;: 12 + R" is any function. Moreover, since Q has full-rank, Qa, Q and then ("( AI) ExlEi {x).
(31)
Proposition 1. If the matrix QA EAJAA (QAi) E Ruxn invertible and f minimizes
Ey Ces((y A, f)). Then the reconstruction function funtiittiiooooiiiiinnnnnn
fly, A) A E A.lu.A QA Ai),
(10)
satisfies
Fly A) = Ey Ayl.A QQ E {x}} where eq. 11 is convex combination of MMSE estimators for different splittings.
(11)
Proof. By applying eq.
30 to f in the definition of f we obtain:
fly A) Ey lvv A Q QA. (Q' QA EllinA A, (0) + (I (I- (I-Q'A,QA) )
-E (Q QA, Amjen,A, {xt} + E. 1or QA. (I- ((11))
= -Ey,,Aily.A @Q QA, Emjyn,A {x}}.
Corollary 1. In order for the matrices QA, or QA to have full rank, it is necessary that A is not
equivariant:
3g G, AT, + T,A.A
(13)
Proof. Let's assume by contradiction that equivariant with respect T,
AT = T A,
(32)
and æ E ().
QA x EglA, {(AT,) AT, } x
EgJA, {(AT) AT,x)
EEIA, {(AT,) (TyT,
EgglA {(AT,) TT,0)
O
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Therefore,
ker( (AAA) (A)2 ker (0).
(33)
The matrix QA has non-trivial nullspace and thus cannot have full rank. Moreover, since this non-trivial nullspace is the same for all virtual operators ATg. then QA shares the same non-trivial nullspace.
Theorem 2. The reconstruction functions defined in points below are all equivariant as in eq. 15
1. Artifact removal network. For a denoiser ((x) equivariant in the sense of eq. 14
f(y. A) (A y) or fly, A) o (A
(16)
2. Unrolled network. For ((x) equivariant, any E and data fidelity d( Ax, y), with
30 0, 3h+1 o( Va, dl ((xxx,)))
(17)
for I. and (y A) Lt.
3. Reynolds averaging. For a possibly non-equivarianl reconstructons rly A), with
f(y, A) = |91
Σ Tor(y, ATg).
(18)
DEG
4. Maximum a posteriori (MAP). For a distribution p(x) invariant as in eq. 5 with
f(y. A) argmax xER"
(p(x 1p(=11,44 y, A)
(19)
5. Minimum mean squared error (MMSE). For distribution p(z:) invariant as in eq.
fly, A) Emly.A (w)
(20)
Proof. We prove each case separately.
1. Denoting A := A or Ax = At, eq. 16) gives, (AT,)* = -T A*
f AT,) (Tg A Axy)
(34)
and since ((x) is equivariant, i.e., eq. 14 holds, it simplifies to eq.
2. We start by making the notation show the explicit dependency on y and A:
zo ( A) O, 2x+1 A) = 6 (g. A) 7V Vv, (v. mu ((E A), A))
(35)
and proceed to show that for 1 O..... T. it holds that
It Ty AT,) T (x
(36)
For i holds
To AT,) T, Zuly. A).
(37)
Let's assume that eq.
(36) holds for k < L. Applying it and the chain rule in eq. (35 yields
#x+1 (g, AT,) ó T [xx ( A) - VV (e A), ()))
(38)
Finally, applying eq. [4) in this equation gives
xk++ (y.AT) AT,)
T o (y. A);
(39)
and by induction, fly, A) ( A), eq. 15) holds.
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3. From eq. 188 it holds that
f(y, ATg) 9
Σ AT
(40)
hEg
which, as the group action property holds To gh, rewrites as
[y AT,)
Thr(y, gTgh):
(41)
heg
Applying the change of variable h' = gh in this equation gives
1
S(y AT₂)
ryy ATr);
(42)
kEg
which finally, using group action property again T TTyTT gives eq. 15
TTg
4. Taking the negative natural logarithm in eq. I9 and using Bayes' theorem gives
fly, A) = argmin (d(Ax, y) + p(x)
(43)
=ER"
where d(Ax,y) y) log ply Ax) and p(xx = - p(z) Applying x' Tg x,
AT,) = Tyyl argmin
{
d(AAx, y) + + (Tg x) }.
(44)
#E
and eq. 5 makes p(æ)) invariant as well (T p(a:). Therefore, eq. (15) holds.
5. Let's assume that p(x) and (A) are invariant in the sense of eq. (5) We first prove that
ply, AT,) ply, A).
(45)
Using eq.
5 the invariance of p(A) and the independence of x and A, we compute
p(y. AT,)
Ex (p(y, AT, x)} = Ex (p(y. AT,, x)p( x)}
Ex {p(y) ATg. p))p(((.))))) = E. {p(y AT, (A)}
Ex (p(y) Ax)) Eg {p(y A, x) A x)}
Ex (p(y, A x)} ply
Next we start from
fly AT,) Eu AT, {Ø)
(46)
and applying the integral formula for expectations gives
/ |x p(ær y, AT,) dr.
AT,)
(47)
Using Bayes' formula and p(y A, = p(y Aw)) it becomes
p(y AT,x)
fly, AT,) x
pl p(AT,, , y) P(AT, g. x) dr.
(48)
By using eq. 45). the invariance of p( ()) and the independence of with æ, we obtain
fly AT,) = x p(y AT,x) p(A)p(z) de.
(49)
p(A,y)
With the change of variable x' -Tyx, and since T, is unitary, we arrive
fyy AT,)) = / p(y Ax)
p p(A)p(TT 1z)) dx.
(50)
p( p(A, y)
Finally, applying eq. 5 in this equation yields eq.
15
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Theorem 3. fA, x)) is equivariant reconstructon, then ES is equivalent the splitting loss
A) = f..
(21)
Proof. We start from eq. 7
Lys f) & Eg (y AT,:
(51)
By A Al AT, fI -
(52)
As A is splitting of ATg. g. we can write A MAT, g for M a splitting matrix. We obtain
LEs(y, A, By {EMIUS { g M MAT)) -y1²}}
(53)
Applying eq. 15) and cancelling out T3 with T yields
CES y A. f) = E, EEyys (IAf((M,, MA) - -yl
(54)
By dropping the expectation in and rewriting (My, MM) as (y1,A1), this yields in eq. (1p.
REPRODUCIBILITY STATEMENT
We share the implementation of method and experiments (code in Section T to make our work easier to reproduce.
26


Reviews:
[{"review_id": "S1gUClARqi", "reviewer": "Reviewer_46Uu", "summary": "This paper addresses self-supervised learning for inverse problems in settings where practitioners lack access to ground truth data and must train from incomplete measurements. The proposed  approach relies on the assumption that the ground truth distribution is invariant to a group of transformations, as formalized in Assumption 1. Building on this assumption, the authors propose a novel self-supervised loss function, Equivariant Splitting (ES), which trains the reconstruction network by optimizing consistency between full and partial measurements, averaged over symmetry transformations of the forward operator. The authors provide theoretical justification by showing that their expected loss converges to the MMSE estimator under certain conditions. Furthermore, they demonstrate empirically that their proposed method achieves strong performance on natural images and MRI reconstruction across a diverse range of inverse problems.", "questions": "**Questions:**\n\n1. The equivariance framework is quite general and could potentially apply to many problem domains beyond those explored in the paper. Could the authors provide additional guidance on selecting appropriate transformation groups for different applications? For instance, a discussion of which symmetries are valid for medical imaging, computational photography, or scientific imaging would be valuable for practitioners. Understanding the broader applicability and potential impact of this framework across different domains would strengthen the paper's contribution.\n\n2. The theoretical analysis provides helpful motivation for the proposed method. Could the authors clarify which aspects of the proof techniques represent novel contributions versus adaptations of existing results?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "**Strengths:**\n\n- The paper presents a novel method that addresses a more realistic setting than traditional generative prior approaches, which typically assume access to ground truth images or paired ground truth-measurement data. The combination of equivariant priors with measurement splitting is original and well-motivated.\n\n- The authors provide theoretical analysis demonstrating that the proposed loss converges in expectation to the MMSE estimator under certain conditions, offering solid justification for the approach.\n\n- The empirical results are strong, achieving performance competitive with supervised baselines that have access to ground truth data, which validates the practical effectiveness of the self-supervised approach.", "weaknesses": "**Weaknesses:**\n\n- Assumption 1 would benefit from additional discussion regarding the practical scope of applicable transformations for different problem domains. While the general framework is elegant, natural images often exhibit partial rather than full symmetries (e.g., small rotations of 15° are reasonable, while 180° rotations may be implausible). Providing examples for specific domains would strengthen the paper and provide valuable guidance for practitioners applying this method to new problems.\n\n\n- The theoretical analysis provides useful motivation for the proposed method, though the mathematical results could be presented differently to better reflect their contribution. The proofs largely build on established techniques. This is a significant portion of the paper. \n\n- Mirror Weakness Theorem 2 makes interesting claims about Reynolds averaging and MAP estimates that would benefit from empirical validation. To my knowledge I did not see anything.", "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": "0upFTbAxUc", "reviewer": "Reviewer_zSdB", "summary": "This paper discusses a study of self-supervised method to solve inverse problems, with the approach as designing loss functions with the \"SPLIT\" technique.", "questions": "- What does $\\mathbb{E}_{x | y, A} \\{ x \\}$ mean, exactly? Do you mean the reconstructed $\\widehat{x}$ given measurement matrix and measurement observation $y$?\n- What does $A_1 | y, AT_g$ mean, exactly?\n- In section 5.1, multiplying by \"measurement matrix with fewer columns than rows\" is not compressive, is it?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "Structure of the paper is clear.", "weaknesses": "- Theorem 1 is well expected. If $(AT_g)^\\top AT_g$ has full rank, then $AT_g$ itself does not dimensionally compress information, hence the ill-conditioning of the inverse problem is no-longer a concern.\n\n- Throughout this paper, the notation is largely unclear. \n\n    - It is hence by requested that authors specify dimension for all matrix quantities for clarity.\n    - With expect to all expectations, authors explicitly explain which quantities are being integrated with respect to its distribution and with respect to which quantities is the variable still stochastic.\n\n- In the experiment section, authors should very clearly define what the measurement matrix $A$ is for all scenarios. Explicitly explain its shape and why the measurement is a compressive measurement. \n    - Because of the proposed \"SPLIT\" technique, authors should also explain clearly what $A_2$ is in every testing scenario.\n\n- In the experiment section, authors should clearly list the citation of used baselines, and list the definition of all metrics, in the main text or in the appendix. Say, what is \"EI\"?\n\n-  In the methodology section, authors should state what the final proposed algorithm is, in a crystal clear way.", "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": "rPnNzpmSKF", "reviewer": "Reviewer_ZYHS", "summary": "This paper proposes Equivariant Splitting (ES), a self-supervised learning method for solving linear inverse problems when only incomplete measurements from a single forward operator are available. The method combines ideas from equivariant imaging (EI), which leverages invariance of the signal distribution under a group of transformations, and measurement splitting, which divides measurements into input/target pairs. The authors introduce a new definition of equivariance for reconstruction networks and show that, under mild conditions, minimizing their proposed loss yields the MMSE estimator. They validate ES on image inpainting, compressive sensing, and accelerated MRI, reporting performance close to supervised baselines and better than existing self-supervised methods like EI.", "questions": "Corollary 1 states that the forward operator A must not be equivariant w.r.t. $G$ for $Q_{A_1}$  to be full rank. In practice, how should one select $G$ given a fixed $A$ ? Are there automated or data-driven strategies to choose transformations that maximize the rank of $Q_{A_1}$?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The method targets a highly relevant and challenging problem: learning from incomplete data with a fixed forward operator (e.g., fixed MRI sampling mask, fixed inpainting mask). \n* The reported results show consistent improvements over EI (especially in highly ill-posed regimes like high CS compression) and a performance closely following supervised learning methods.\n* The ablation study in Table 3  demonstrates clearly the synergy between equivariant architectures and the ES loss, validating the theoretical claims.", "weaknesses": "* The proposed self-supervised learning strategy relies on strong assumptions on signal invariance and operator non-equivariance, while it does not explore what happens when these assumptions are violated.\n* Limited Robustnes to real-world distortions: All experiments assume perferct knowledge of the forward operator A (in the linear inverse imaging literature this assumption is known as an \"inverse crime”) and either idealized noise models (i.i.d Gaussian) or  the absence of noise. Morevoer, the studied problems are oversimplified and are not representative of problems met in real-world applications. For example, in image inpainting the authors consider a binary mask that keeps 30% of all the pixel and the absence of noise while more realistic scenarios would involve added noise and non-uniform masking.\n* Computational Overhead and Scalability Concerns\n  *  Although Theorem 3 shows that equivariant architectures reduce the ES loss to a standard splitting loss at training time, training still requires multiple random splits per sample, which increases memory and compute costs compared to simple supervised or consistency-based baselines.\n  * The use of Reynolds averaging in Eq. 18 for enforcing equivariance becomes prohibitively expensive for large groups (e.g., continuous shifts or rotations), as noted in Appendix A. The paper only evaluates small discrete groups (e.g., 90° rotations + flips), limiting generalizability.", "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": "SkrYdPygcF", "reviewer": "Reviewer_vQFi", "summary": "The paper proposes Equivariant Splitting (ES), a self-supervised training loss for inverse problems that combines measurement-splitting ideas with equivariant modelling. Under an invariance assumption on the image distribution (Assumption 1), ES averages splitting losses computed on virtual operators formed by composing the forward operator with group transforms and shows (Theorem 1 / Proposition 1) that, when certain matrix conditions hold, minimizers of the ES loss coincide (in expectation) with the MMSE estimator. The authors introduce a reconstructor notion of equivariance (Definition 1), prove that several common architecture families satisfy it (Theorem 2), and show ES reduces to standard splitting for equivariant reconstructors (Theorem 3). Empirical evaluation covers compressive sensing, image inpainting, and MRI, comparing ES to supervised, equivariant imaging (EI), SURE baselines, and reporting PSNR / SSIM / an equivariance metric.", "questions": "- In practice, how does one verify (or have you measured) whether $Q_{A_1}$ or $\\bar{Q}_A$ is full rank for realistic forward operators (e.g., the MRI coil-sensitivity map that you use)? \n- How sensitive is ES when the dataset only approximately satisfies Assumption 1? Can you provide quantitative experiments where invariance is gradually violated (e.g., introduce systematic asymmetries) and show performance degradation versus EI / supervised baselines? \n- What is the typical number of random transforms/splits used at training and at test time? What is the overhead relative to EI and supervised training? \n- Corollary 1 requires that the forward operator $A$ is not equivariant. Can you clarify and demonstrate what happens when $A$ is equivariant or nearly equivariant (e.g., radial sampling in MRI, which is equivariant with rotational transforms)? Does ES break down or merely degrade gracefully? \n- Reproducibility and released artifacts. The paper references DeepInverse and gives some training details, but can you (i) release code and trained models, (ii) provide exact seeds and the split/sampling scripts for splits/transforms, and (iii) include scripts to reproduce the key tables (Tables 1–4) and the equivariance metric computations (EQUIV)? Appendix B mentions some implementation choices, but the reproducibility checklist is incomplete.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper formulates a unifying view that connects splitting losses and equivariant imaging and states the assumptions under which ES is theoretically unbiased for the supervised objective. \n- Introducing a definition of equivariant reconstructors and enumerating common architectures that satisfy it is useful. \n- The proposed loss is practically implementable. The paper explains how to use Monte-Carlo sampling of transforms and splits, and how to replace the noiseless term with R2R in the noisy setting, making the method applicable to realistic measurement noise models. \n- Experiments span multiple inverse problems (compressive sensing, inpainting, MRI) and include an ablation on equivariant vs non-equivariant architectures, which demonstrates the claimed synergy between splitting losses and equivariant architectures in the tested settings.", "weaknesses": "- The main theoretical guarantee requires (i) the distributional invariance (Assumption 1) and (ii) invertibility conditions (full rank) on the matrix $Q_{A_1}$ or $\\bar{Q}_A$ for some split $A_1$. These conditions are sufficient but seem strong and may rarely hold in practice; the manuscript gives limited practical diagnostics or guidance on verifying these conditions for real forward operators. \n- The method depends critically on choosing an appropriate group (translations, rotations, flips, etc.). The paper does not sufficiently analyze robustness when the invariance assumption is only approximate or mis-specified (e.g., natural images that are not strictly invariant to some transforms). Practical recommendations for selecting $G$ per application are brief. \n- Although multiple tasks are included, some experimental choices raise concerns: (i) use of MNIST for compressive sensing (very simple data distribution), (ii) DIV2K for inpainting with synthetic masks, and (iii) synthetically generated k-space masks for MRI. The assertion that ES achieves “state-of-the-art” self-supervised performance is not fully established across diverse, realistic datasets or against the latest baselines in unsupervised/self-supervised imaging literature (such as DDRM). \n- Reynolds averaging over large groups is noted to be impractical, and the authors use Monte-Carlo sampling, but the runtime, per-iteration cost, and the number of samples are not clearly reported. Training budgets are mentioned (up to 50 hours on a single GPU), but fair comparisons to baselines in terms of wall-clock cost or memory are missing. \n- Ablations and failure modes are not discussed in detail. The ablation on equivariant architectures is informative, but there is little analysis of when ES fails (e.g., non-unitary transforms or operators that are nearly equivariant. Corollary 1 suggests A must not commute with transforms).", "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)