# QCM-SGM+: Improved Quantized Compressed Sensing With Score-Based Generative Models

arXiv: https://arxiv.org/abs/2302.00919 · HTML: https://mlsanity.com/paper/2302.00919/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 23 of its 51 references that are in the map.
- First posted on arXiv 2 Feb 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 23 of 23)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Exploiting Deep Generative Prior for Versatile Image Restoration and Manipulation](https://mlsanity.com/paper/2003.13659/) (2020)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Invertible generative models for inverse problems: mitigating representation error and dataset bias](https://mlsanity.com/paper/1905.11672/) (2019)
- [Diffusion Model Based Posterior Sampling for Noisy Linear Inverse Problems](https://mlsanity.com/paper/2211.12343/) (2022)
- [Approximate Message Passing algorithms for rotationally invariant matrices](https://mlsanity.com/paper/2008.11892/) (2020)
- [Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors](https://mlsanity.com/paper/2002.01697/) (2020)
- [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021)
- [Quantized Compressed Sensing with Score-based Generative Models](https://mlsanity.com/paper/2211.13006/) (2022)
- [Global Guarantees for Blind Demodulation with Generative Priors](https://mlsanity.com/paper/1905.12576/) (2019)
- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Semi-analytic approximate stability selection for correlated data in generalized linear models](https://mlsanity.com/paper/2003.08670/) (2020)
- [A Unitary Transform Based Generalized Approximate Message Passing](https://mlsanity.com/paper/2210.08861/) (2022)

## Built on by (most cited first, 4 of 4)

- [Generative Spectrum Cartography: Unified Reconstruction and Active Sensing via Diffusion Models](https://mlsanity.com/paper/2512.20108/) (2025)
- [Score-Based Turbo Message Passing for Plug-and-Play Compressive Imaging](https://mlsanity.com/paper/2512.14435/) (2025)
- [Deep Probabilistic Unfolding for Quantized Compressive Sensing](https://mlsanity.com/paper/2605.11475/) (2026)
- [Posterior Information Dynamics of Diffusion Models for Linear Inverse Problems](https://mlsanity.com/paper/2608.21709/) (2026)

## Abstract (arXiv)

In practical compressed sensing (CS), the obtained measurements typically necessitate quantization to a limited number of bits prior to transmission or storage. This nonlinear quantization process poses significant recovery challenges, particularly with extreme coarse quantization such as 1-bit. Recently, an efficient algorithm called QCS-SGM was proposed for quantized CS (QCS) which utilizes score-based generative models (SGM) as an implicit prior. Due to the adeptness of SGM in capturing the intricate structures of natural signals, QCS-SGM substantially outperforms previous QCS methods. However, QCS-SGM is constrained to (approximately) row-orthogonal sensing matrices as the computation of the likelihood score becomes intractable otherwise. To address this limitation, we introduce an advanced variant of QCS-SGM, termed QCS-SGM+, capable of handling general matrices effectively. The key idea is a Bayesian inference perspective on the likelihood score computation, wherein expectation propagation is employed for its approximate computation. Extensive experiments are conducted, demonstrating the substantial superiority of QCS-SGM+ over QCS-SGM for general sensing matrices beyond mere row-orthogonality.