# Equivariant Splitting: Self-supervised learning from incomplete data

arXiv: https://arxiv.org/abs/2510.00929 · HTML: https://mlsanity.com/paper/2510.00929/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 58 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 30 of 35)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [Truly shift-invariant convolutional neural networks](https://mlsanity.com/paper/2011.14214/) (2020)
- [Unsupervised Learning with Stein's Unbiased Risk Estimator](https://mlsanity.com/paper/1805.10531/) (2018)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)
- [Equivariant neural networks for inverse problems](https://mlsanity.com/paper/2102.11504/) (2021)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [DeepInverse: A Python package for solving imaging inverse problems with deep learning](https://mlsanity.com/paper/2505.20160/) (2025)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [Unsupervised Learning From Incomplete Measurements for Inverse Problems](https://mlsanity.com/paper/2201.12151/) (2022)
- [UNSURE: self-supervised learning with Unknown Noise level and Stein's Unbiased Risk Estimate](https://mlsanity.com/paper/2409.01985/) (2024)
- [Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurring](https://mlsanity.com/paper/2312.11232/) (2023)
- [Generalized Recorrupted-to-Recorrupted: Self-Supervised Learning Beyond Gaussian Noise](https://mlsanity.com/paper/2412.04648/) (2024)
- [Alias-Free Convnets: Fractional Shift Invariance via Polynomial Activations](https://mlsanity.com/paper/2303.08085/) (2023)
- [Truly shift-equivariant convolutional neural networks with adaptive polyphase upsampling](https://mlsanity.com/paper/2105.04040/) (2021)

## Built on by (most cited first, 1 of 1)

- [UNet-AF: Alias-free UNet architectures](https://mlsanity.com/paper/2603.11323/) (2026)

## Abstract (arXiv)

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