# Learning a Compressed Sensing Measurement Matrix via Gradient Unrolling

arXiv: https://arxiv.org/abs/1806.10175 · HTML: https://mlsanity.com/paper/1806.10175/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2018.

## Built on by (most cited first, 16 of 16)

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- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
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- [Adaptive Illumination based Depth Sensing using Deep Superpixel and Soft Sampling Approximation](https://mlsanity.com/paper/2103.12297/) (2021)
- [Regularized Optimal Transport Layers for Generalized Global Pooling Operations](https://mlsanity.com/paper/2212.06339/) (2022)
- [Mixture Model Auto-Encoders: Deep Clustering through Dictionary Learning](https://mlsanity.com/paper/2110.04683/) (2021)
- [Robust Deep Compressive Sensing with Recurrent-Residual Structural Constraints](https://mlsanity.com/paper/2207.07301/) (2022)
- [Learning to sample in Cartesian MRI](https://mlsanity.com/paper/2312.04327/) (2023)
- [Generalized Compressed Sensing for Image Reconstruction with Diffusion Probabilistic Models](https://mlsanity.com/paper/2405.17456/) (2024)
- [Learning Binary Sampling Patterns for Single-Pixel Imaging using Bilevel Optimisation](https://mlsanity.com/paper/2508.19068/) (2025)
- [Learning a Compressive Sensing Matrix with Structural Constraints via Maximum Mean Discrepancy Optimization](https://mlsanity.com/paper/2110.07221/) (2021)
- [Gradient-Based Learning of Discrete Structured Measurement Operators for Signal Recovery](https://mlsanity.com/paper/2202.03391/) (2022)
- [Learning Trees of ℓ0-Minimization Problems](https://mlsanity.com/paper/2302.02548/) (2023)
- [Outlier Detection Using Generative Models with Theoretical Performance Guarantees](https://mlsanity.com/paper/2310.09999/) (2023)
- [Learned Single-Pixel Fluorescence Microscopy](https://mlsanity.com/paper/2507.18740/) (2025)
- [Learning the Mathematical Property for Designing Low Mutual Coherence Binary Sensing Matrices](https://mlsanity.com/paper/2608.12982/) (2026)

## Abstract (arXiv)

Linear encoding of sparse vectors is widely popular, but is commonly data-independent -- missing any possible extra (but a priori unknown) structure beyond sparsity. In this paper we present a new method to learn linear encoders that adapt to data, while still performing well with the widely used ℓ1 decoder. The convex ℓ1 decoder prevents gradient propagation as needed in standard gradient-based training. Our method is based on the insight that unrolling the convex decoder into T projected subgradient steps can address this issue. Our method can be seen as a data-driven way to learn a compressed sensing measurement matrix. We compare the empirical performance of 10 algorithms over 6 sparse datasets (3 synthetic and 3 real). Our experiments show that there is indeed additional structure beyond sparsity in the real datasets; our method is able to discover it and exploit it to create excellent reconstructions with fewer measurements (by a factor of 1.1-3x) compared to the previous state-of-the-art methods. We illustrate an application of our method in learning label embeddings for extreme multi-label classification, and empirically show that our method is able to match or outperform the precision scores of SLEEC, which is one of the state-of-the-art embedding-based approaches.