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

TL;DR

Quantized CS suffers from nonlinear quantization, especially in 1-bit settings, making faithful reconstruction challenging. The paper advances QCS-SGM by introducing QCS-SGM+, which uses expectation propagation to approximate the intractable likelihood for general sensing matrices and couples this with annealed Langevin dynamics guided by a score-based prior, enabling posterior sampling from $p(f x|f y)$ beyond row-orthogonal matrices. The approach yields explicit EP-derived gradients and a fast SVD-based implementation, delivering substantial PSNR/SSIM gains on MNIST, CIFAR-10, and CelebA across ill-conditioned, correlated, and multi-bit scenarios. This generalization broadens the applicability of diffusion-prior CS methods to practical sensing systems, with improved reconstruction under severe quantization and robust performance against noise.

Abstract

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.

Figures (21)

• Figure 1: A schematic of the basic idea of QCS-SGM+ in computing the intractable pseudo-likelihood $\tilde{p}({\bf{y}}| {\bf{z}}_t ={\bf{A}} {\bf{x}}_t)$ for general matrices. The subscript $t$ is dropped for simplicity. We resort to EP to obtain an effective factorized approximation of $f_b({\bf{\tilde{n}}})$ so that a closed-form solution of $\tilde{p}({\bf{y}}| {\bf{z}}_t ={\bf{A}} {\bf{x}}_t)$ can be achieved, which enables the computation of otherwise intractable pseudo-likelihood score.
• Figure 2: Qualitative comparisons of different methods under 1-bit CS on MNIST and CelebA for ill-conditioned $\bf{A}$ ($\kappa=10^3$ for MNIST and $\kappa=10^6$ for CelebA) when $M< N$. It can be seen that the proposed QCS-SGM+ achieves consistently better results than other methods.
• Figure 3: Reconstructed images on CIFAR-10 with QCS-SGM and QCS-SGM+, respectively, under 1-3 bit CS when the condition number of $\bf{A}$ is 1000, $M=2000, \sigma = 0.1$. It can be seen that the proposed QCS-SGM+ achieves consistently better results than QCS-SGM.
• Figure 4: Quantitative comparisons between QCS-SGM+ and existing methods under 1-bit CS for MNIST, CIFAR-10, and CelebA when the sensing matrices $\bf{A}$ are ill-conditioned and correlated, respectively. The number of measurements are $M=400,2000,4000$ for MNIST, CIFAR-10, and CelebA, respectively, all satisfying $M<N$. Standard error bars for the results are also shown.
• Figure 5: Quantitative comparisons between QCS-SGM+ and other methods under 1-bit CS with different levels of Gaussian noise for ill-conditioned $\bf{A}$ with condition number $\kappa=10^3$. The number of measurements are $M=400,2000,4000$ for MNIST, CIFAR-10, and CelebA, respectively, all satisfying $M< N$. Standard error bars for the results are also shown.