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. The code is available at https://github.com/vsechaud/Equivariant-Splitting
@article{arxiv.2510.00929,
title = {Equivariant Splitting: Self-supervised learning from incomplete data},
author = {Victor Sechaud and Jérémy Scanvic and Quentin Barthélemy and Patrice Abry and Julián Tachella},
journal= {arXiv preprint arXiv:2510.00929},
year = {2026}
}