Equivariant Splitting: Self-supervised learning from incomplete data
_{Official PyTorch implementation, ICLR 2026}

ES reconstructions from incomplete measurements (Inpainting, MRI)

Equivariant Splitting: Self-supervised learning from incomplete data
Victor Sechaud, Jérémy Scanvic, Quentin Barthélemy, Patrice Abry, Julián Tachella

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 the same minimizer in expectation as the one of a supervised loss. Through a series of experiments on image inpainting, accelerated magnetic resonance imaging, and compressive sensing, we demonstrate that the proposed loss achieves state-of-the-art performance in settings with highly rank-deficient forward models.

News

April 25, 2026

• Equivariant splitting is now natively supported in DeepInverse!

Experiments

Equivariant splitting is evaluated on different imaging modalities: compressive sensing, image inpainting, accelerated MRI and sparse-view computed tomography. The code and instructions to reproduce the results for compressive sensing and inpainting are available here, for MRI here and for computed tomography here.

Accelerated MRI

Equivariant splitting learns to solve the challenging, ill-posed problem of accelerated MRI directly from noisy k-space measurements.

This section details how to run the experiments for accelerated MRI in the mri folder.

Setting up the environment

For better reproducibility, we recommend using conda to set up the environment using our provided environment.yml file.

cd mri
conda env create -f environment.yml

Note

The environment installs our fork of DeepInverse, not an official release.

Creating the dataset

To prepare the dataset used in train.py for the experiments, follow the instructions below.

1. Download FastMRI single-coil knee acquisitions knee_singlecoil_train (72.7 GB)
2. Place the downloaded data in FastMRI/knee/singlecoil_train
3. Generate the slice dataset using python create_dataset.py

If everything is set up correctly, it should create a directory FastMRI-Slices containing a file named fastmri_knee_singlecoil.pt.

Training a model

python train.py <config>

Configurations

The parameter <config> corresponds to one of the configuration names below.

Loss	Equivariant	Acceleration	Configuration name
Supervised	✅	x8	MRIx8_EQ_Supervised
Supervised	❌	x8	MRIx8_NEQ_Supervised
ES (Ours)	✅	x8	MRIx8_EQ_ES
ES	❌	x8	MRIx8_NEQ_ES
EI	✅	x8	MRIx8_EQ_EI
SURE	✅	x8	MRIx8_EQ_SURE
Supervised	✅	x6	MRIx6_EQ_Supervised
Supervised	❌	x6	MRIx6_NEQ_Supervised
ES (Ours)	✅	x6	MRIx6_EQ_ES
ES	❌	x6	MRIx6_NEQ_ES
EI	✅	x6	MRIx6_EQ_EI
SURE	✅	x6	MRIx6_EQ_SURE

Sparse-View Computed Tomography

Equivariant splitting learns to solve the challenging, ill-posed problem of sparse-view computed tomography directly from noisy sinograms.

This section details how to run the experiments for sparse-view computed tomography in the tomography folder.

Setting up the environment

For better reproducibility, we recommend using conda to set up the environment using our provided environment.yml file.

cd tomography
conda env create -f environment.yml

Note

The environment installs our fork of DeepInverse, not an official release.

Creating the dataset

To prepare the dataset used in train.py for the experiments, follow the instructions below.

1. Download the LIDC-IDRI CT scans using the NBIA Data Retriever
2. Place the downloaded data in LIDC_IDRI so that it contains the LIDC-IDRI directory and the metadata.csv file
3. Generate the sinogram dataset using python create_dataset.py

If everything is set up correctly, it should create a directory LIDC_IDRI-Tomography containing a file named dinv_dataset0.h5.

Training a model

python train.py <config>

Configurations

The parameter <config> corresponds to one of the configuration names below.

Loss	Equivariant	Configuration name
Supervised	✅	CT_EQ_Supervised
Supervised	❌	CT_NEQ_Supervised
ES (Ours)	✅	CT_EQ_ES
ES	❌	CT_NEQ_ES
EI	✅	CT_EQ_EI
EI	❌	CT_NEQ_EI

Compressive Sensing and Inpainting

This section details how to run the experiments for compressive sensing and inpainting in the cs+inp folder.

Usage

The run_train.py script calls the train function defined in train.py.

run_train.py expects a path to a YAML configuration file as an argument. Example configuration files can be found in the configs/ folder.

Example

cd cs+inp
python run_train.py --config_path configs/config_cs_split_equi.yaml

Acknowledgment

This work makes use of the efficient forward operators and training losses in DeepInverse and the convenient MRI training and comparison features of SSIBench.

Citation

Please consider citing this work if you find it useful in your research:

@inproceedings{sechaud2026equivariant,
    title={Equivariant Splitting: Self-supervised learning from incomplete data},
    author={Sechaud, Victor and Scanvic, J{\'e}r{\'e}my and Barth{\'e}lemy, Quentin and Abry, Patrice and Tachella, Juli{\'a}n},
    booktitle={The Fourteenth International Conference on Learning Representations},
    year={2026},
    url={https://openreview.net/forum?id=upMIVpe467}
}