QUANTIZED COMPRESSED SENSING WITH SCORE-BASED GENERATIVE MODELS

ABSTRACT

We consider the general problem of recovering a high-dimensional signal from noisy quantized measurements. Quantization, especially coarse quantization such as 1-bit sign measurements, leads to severe information loss and thus a good prior knowledge of the unknown signal is helpful for accurate recovery. Motivated by the power of score-based generative models (SGM, also known as diffusion models) in capturing the rich structure of natural signals beyond simple sparsity, we propose an unsupervised data-driven approach called quantized compressed sensing with SGM (QCS-SGM), where the prior distribution is modeled by a pre-trained SGM. To perform posterior sampling, an annealed pseudo-likelihood score called noise perturbed pseudo-likelihood score is introduced and combined with the prior score of SGM. The proposed QCS-SGM applies to an arbitrary number of quantization bits. Experiments on a variety of baseline datasets demonstrate that the proposed QCS-SGM significantly outperforms existing state-of-the-art algorithms by a large margin for both in-distribution and out-of-distribution samples. Moreover, as a posterior sampling method, QCS-SGM can be easily used to obtain confidence intervals or uncertainty estimates of the reconstructed results. The code is available at https://github.com/mengxiangming/QCS-SGM. 

1. INTRODUCTION

Many problems in science and engineering such as signal processing, computer vision, machine learning, and statistics can be cast as linear inverse problems: y = Ax + n, where A ∈ R M ×N is a known linear mixing matrix, n ∼ N (n; 0, σ 2 I) is an i.i.d. additive Gaussian noise, and the goal is to recover the unknown signal x ∈ R N ×1 from the noisy linear measurements



Figure 1: Reconstructed images of our QCS-SGM for one FFHQ 256 × 256 high-resolution RGB test image (N = 256 × 256 × 3 = 196608 pixels) from noisy heavily quantized (1bit, 2-bit and 3-bit) CS 8× measurements y = Q(Ax + n), i.e., M = 24576 ≪ N . The measurement matrix A ∈ R M ×N is i.i.d. Gaussian, i.e., Aij ∼ N (0, 1 M ), and a Gaussian noise n is added with standard deviation σ = 10 -3 .

