# PE-CSNet: An equivariant network architecture with learnable patch-based sparse representation

arXiv: https://arxiv.org/abs/2608.14708 · HTML: https://mlsanity.com/paper/2608.14708/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 66 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [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)
- [A Deep Learning Approach to Structured Signal Recovery](https://mlsanity.com/paper/1508.04065/) (2015)
- [Group-based Sparse Representation for Image Restoration](https://mlsanity.com/paper/1405.3351/) (2014)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Data augmentation for deep learning based accelerated MRI reconstruction with limited data](https://mlsanity.com/paper/2106.14947/) (2021)
- [Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration](https://mlsanity.com/paper/2312.15701/) (2023)
- [Low Rank Regularization: A Review](https://mlsanity.com/paper/1808.04521/) (2018)
- [Fast Multi-class Dictionaries Learning with Geometrical Directions in MRI Reconstruction](https://mlsanity.com/paper/1503.02945/) (2015)
- [FRIST - Flipping and Rotation Invariant Sparsifying Transform Learning and Applications](https://mlsanity.com/paper/1511.06359/) (2015)
- [Learning Nonlocal Sparse and Low-Rank Models for Image Compressive Sensing](https://mlsanity.com/paper/2203.09656/) (2022)
- [A fast patch-dictionary method for whole image recovery](https://mlsanity.com/paper/1408.3740/) (2014)
- [Deep unfolding as iterative regularization for imaging inverse problems](https://mlsanity.com/paper/2211.13452/) (2022)
- [Practical Compact Deep Compressed Sensing](https://mlsanity.com/paper/2411.13081/) (2024)

## Abstract (arXiv)

Compressive sensing (CS) enables accurate signal reconstruction from sparse measurements and is widely applied in medical imaging, remote sensing, and image compression. However, designing an effective, task-specific sparse transform and the corresponding optimization procedure for high-quality CS remains challenging. This process typically requires expert domain knowledge and laborious parameter tuning. To address this issue, we present a Patch-based Equivariant deep unrolling architecture, termed PE-CSNet, for accurate CS recovery. While traditional CS methods generally use predefined patch-based transform sparsity, we generalize this idea by incorporating learnable transform sparsity that adapts to the specific CS task through an optimization-driven process. Specifically, we first establish a generalized patch-based CS model, which we solve via a block coordinate descent (BCD) algorithm. The BCD solver is then unrolled into a deep neural network, where all parameters of both the CS model and solver are learned through end-to-end training. To improve data efficiency, we introduce a stochastic equivariant training strategy that exploits the patch-wise structure of the network, enabling PE-CSNet to learn effectively even from limited data. We further provide a simpler, parameter-shared version of PE-CSNet and briefly discuss its convergence as an iterative solver. For practical applications, the network uses stage-specific (non-shared) parameters to enhance its expressive power and thereby improve its performance. On the tasks of CS magnetic resonance imaging (CS-MRI) and CS coded diffraction patterns (CS-CDP), PE-CSNet achieves state-of-the-art accuracy with fast computational speed, outperforming traditional methods and existing deep unrolling methods.