Quantized Compressed Sensing with Score-Based Generative Models
Authors: Xiangming Meng
Yoshiyuki Kabashima
Venue: ICLR 2023 poster (2023)
Keywords: generative models
compressed sensing
linear inverse problems
quantization

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 ${\textit{noise perturbed pseudo-likelihood score}}$ 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. $\textit{The code  is available at}$ https://github.com/mengxiangming/QCS-SGM.

Published as a conference paper at ICLR 2023
QUANTIZED COMPRESSED SENSING WITH SCORE- BASED GENERATIVE MODELS
Xiangming Meng and Yoshiyuki Kabashima
Institute for Physics of Intelligence and Department of Physics
The University of Tokyo
7-3-1, Hongo, Tokyo 113-0033, Japan {meng, kaba @abaaagg... ecc. -- wtt.kc...ccooc...oooooo ac.
ABSTRACT
We consider the general problem of recovering a high-dimensional signal from noisy quantized measurements. Quantization, especially coarse quantization such 1-bit sign measurements, leads to severe information loss and thus 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 pre-trained SGM. To perform posterior sampling, an annealed pseudo- likuloooi score called noise perturbed sveudo-likelihone score is introduced and combined with the prior score SGM. The proposed QCS-SGM applies to an arbitrary number of quantization bits. Experiments variety of baseline datasets demonstrate that the proposed QCS-SGM significantly outperforms existing state-of-ther the-art algorithms by a large margin for both in distribbuiiinn and out-of-distribution samples. Moreover, as posterior sampling method, QCS-SGM can be easily used to obtain confidence intervals uncertainty estimates of the reconstructed results. The code is available at https:/ggghubb.mmmmeenngggnnnngo
1-bit
2-bit
3-bit
Ground Truth
3.3
PSNR: 11.64 CB, SSIM: 0.500
PSNR: 24, dB, SSIM: 0.695
PSNR: 25.71 dB, SSIM: 0.753
Figure 1: Reconstructed images of our QCS-SGM for one FFHQ 256 x 256 high-resolution RGB test image (N 256 x 256 3 = 196608 pixels) from noisy heavily quantized (Ibit, 2-bit and 3-bit) CS 8x measurements y Q(Ax n). ie.. M 24576 < N. The measurement matrix A M iid. Gaussian, i.e.. Ag ~ (0, ). and a Gaussian noise is added with standard deviation a 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 M N is known linear mixing matrix, n WNnn 0, I is an i... additive Gaussian noise, and the goal is recover the unknown signal Nx1 from the noisy linear measurements

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y E RMxx Among various applications, compressed sensing (CS) provides highly efficient paradigm that makes it possible to recover a high-dimensional signal from a far smaller number of M < NV measurements (Candès & Wakin 2008). The underlying wisdom of CS is to leverage the intrinsic structure of the unknown signal x to aid the recovery. One of the most widely used structures is sparsity, i.e., most elements of x are zero under certain transform domains, e.g., wavelet and Fourier transformation (Candès & Wakin 2008). In other words, the standard CS exploits the fact that many natural signals in real-world applications are (approximately) sparse. This direction has spurred hugely active field of research during the last two decades, including efficient algorithm
design (Tibshirani 1996, BBeck& & Teboulle| [2009] Tropp & Wright 2010, Kabashima 2003 Donoho
et al.
2009), theoretical analysis (Candès et al.) [0066] Donoho 2006, Kabashima et 20.09) Bach
et al.
2012). as well as all kinds of applications (Lustig et al. 2007] 2008). just to name a few.
Despite their remarkable success, traditional CS is still limited by the achievable rates since the sparsity assumptions, whether being naive sparsity or block sparsity (Duarte Eldar 201T), are still too simple to capture the complex and rich structure of natural signals. For example, most natural signals are not strictly sparse even in the specified transform domains, and thus simply relying on sparsity alone for reconstruction could lead to inaccurate results. Indeed, researchers have proposed to combine sparsity with additional structure assumptions, such as low-rank assumption (Fazel et al., 2008; Foygel & Mackey (2014), total variation (Candès et al., 2006); [Tang et al.) 2009). further improve the reconstruction performances. Nevertheless, these hand-crafted priors can only apply, if not exactly, to a very limited range of signals and are difficult to generalize to other cases. To address this problem, driven by the success of generative models (Goodfellow et al. [0114] Kingma & Welling 2013 Rezende & Mohamed, 2013), there has been a surge of interest in developing CS methods with data-driven priors (Bora et al. 2017 Hand & Joshi 2019 Asim et al., 2020 Pan et al. 2021). The basic idea is that, instead of relying on hand-crafted sparsity, the prior structure of the unknown signal is learned through a generative model, such as VAE (Kingma & Welling 2013) or GAN (Goodfellow et al., 2014). Notably, the past few years have witnessed a remarkable success of one new family of probabilistic generative models called diffusion models, in particular, score matching with Langevin dynamics (SMLD) (Song & Ermon] 2219 2020) and denoising diffusion probabilistic modeling (DDPM) (Ho et al. 2020 Nichol & Dhariwal, 2021), which have proven extremely effective and even outperform the state-of-the-art (SOTA) GAN (Goodfellow et al. 2014) and VAE (Kingma& & Welling 2013) in density estimation and generation of various natural sources. As both DDPM and SMLD estimate, implicitly or explicitly, the score (i.e., the gradient of the log probability density w.r.t. to data), they are also referred together as score-based generative models (SGM) (Song et al., 2020). Several CS methods with SGM have been proposed very recently (Jalal al. 2021a Kawar et al. [022] 2022) Chung et al. (022) and perform quite well in recovering x with only a few linear measurements in
Nevertheless, the linear model (T) ideally assumes that the measurements have infinite precision, which is not the case in realistic acquisition scenarios. In practice, the obtained measurements have to be quantized to a finite number of Q bits before transmission and/or storage (Zymnis et al.) 2009 Dai & Milenkovic (011). Quantization leads to information loss which makes the recovery particularly challenging. For moderate and high quantization resolutions, i.e., Q is large, the quantization impact is usually modeled as mere additive Gaussian noise whose variance is determined by quantization distortion (Dai & Milenkovic, 201T, Jacques et al., 2010). Subsequently, most CS algorithms originally designed for the linear model (1) can then be applied with some modifications. However, such an approach is apparently suboptimal since the information about the quantizer is not utilized to a full extent (Dai & Milenkovic (211I). This is true especially in the case of coarse quantization, i.e., Q is small. An extreme and important case of coarse quantization is 1-bit quantization, where Q 1 and only the sign of the measurements are observed (Boufounos & Baraniuk 2008). Apart from extremely low cost in storage, 1-bit quantization is particularly appealing in hardware implementations and has also proven robust to both nonlinear distortions and dynamic range issues (Boufounos & Baraniuk] 2008). Consequently, there have been extensive studies on quantized CS, particularly 1 1-bit CS, in the past decades and a variety of algorithms have been proposed, e.g., (Zymnis al. 2009)] Dai &
Milenkovic 2011 Plan & Vershynin 2012: 2013 Jacques al., 2013] Xu & Kabashima 2013 Xu et al. 20144] Awasthi et | Meng ct al., 2018 Jung et al. 2021 Liu et al. [220 Liu &
Liu 2022). However, most existing methods are based standard CS methods, which, therefore,
inevitably inherit their inability to capture rich structures of natural signals beyond sparsity. While
several recent works (Liu et al. 2020: Liu & Liu 2022) studied 1-bit CS using generative priors, their
main focuses are VAË and/or GAN, rather than SGM.
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1.1 CONTRIBUTIONS
We propose a novel framework, termed as Quantized Compressed Sensing with Score-based Generative Models (QCS- in short), to recover unknown signals from noisy quantized measurements. OCS-SGM applies to an arbitrary number of quantization bits, including the extremely coarse 1-bit sign measurements. To the best of our knowledge, this is the first time that SGM has been utilized for quantized CS (QCS).
We consider one popular SGM model called NCSNv2 (Song & Ermon 2020) and verify the effectiveness of the proposed QCS-SGM in QCS on various real-world datasets including MNIST, Cifar- 10, CelebA 64 x 64, and the high-resolution FFHQ 256 x 256. Using the pre- trained SGM as a generative prior, for both in-distribution and of-distribution samples, the proposed QCS- SGM significantly outperforms existing SOTA algorithms by a large margin in QCS. Perhaps surprisingly, even in the extreme case of 1-bit sign measurements, QCS-SGM can still faithfully recover the original images with much fewer measurements than the original signal dimension, as shown in Figure I Moreover, compared to existing methods, QCS-SGM tends to recover more natural images with good perceptual quality.
We propose a noise-perturbed pseudo- pieuudoo-iklliie likllihoood score as principled way to incorporate information from the measurements, which is one key to the success of QCS-SGM Inter- estingly, when degenerated to the linear case (T) if A is row-orthogonal, then QCS-SGM reduces to a form similar to Jalal et al. 2002 While the annealing term (denoted as ?? in (4) of Dalal et al.] (2021a) is added heuristically appears naturally within our framework and admits an analytical solution. More importantly, for general matrices A, our method generalizes and significantly outperforms Dalai et al.) (2021a). It is expected that the idea of noise- coiseeperturbed escudo-likelihon score can be generalized to other conditional generative models as a general rule.
one additional
As one sampling method, QCS-SGM can easily yield multiple samples with different random initializations, whereby one can obtain confidence intervals or uncertainty estimates.
1.2 RELATED WORKS
The proposed QCS-SGM is one kind of data-driven method for CS which generally include two lines of research: unsupervised and supervised. The supervised approach trains neural networks
from end to end using pairs of original signals and observations (Jin et al. 2017: 2017/Zhang & Ghanem
2018: Aggarwal et al. 2018 Wu et al. 2019; Yao et al.) 2019 Gilton et al. 2019: Yang et al.) 2020
Antun et al.) 2020). Despite good performance on specific problems, the supervised approach is severely limited and requires re- re-training even with slight change in the measurement model. The unsupervised approach, including the proposed QCS-SGM, only learns a prior and thus avoids such problem-specific training since the measurement model is only known and used for inference (Bora al. 2017 Hand & Joshi 2019; [Asim et al.] 2020 Pan et al.] [02T]. Among others, the CSGM framework Jii et al. 2017 is the most popular one which learns the prior with generative models.
Recent works Liu et al.] (020)]; Liu Liu (2022) extended CSGM to non-linear observations including 1-bit CS, which are the two most related studies to current work. However, the main focuses of et al. (2020); Liu & Liu (2022) are limited to VAE and GAN (in particular DCGAN (Radford et al. 2015) ). As is widely known, GAN suffers from unstable training and less diversity due to the adversarial training nature while VAE uses a surrogate loss. Another significant drawback of priors with VAE/GAN is that they can have large representation errors biases due architecture and training, which can be easily caused by inappropriate latent dimensionality and/or mode collapse (Asim et al., 2020). With the advent of SGM such as SMLD (Song & Ermon 2019 2020) and DDPM (Ho et al., 2020 Nichol & Dhhrrwwa wal 202T), there have been several recent studies (Jlll et al. 2021alb b and outperform conventional methods. Interestingly, Daras et al. (202)) performs posterior sampling in the "latent space" of pre-trained generative model. Nevertheless, all the existing works on SGM for CS focus on linear measurements (1) without quantization, which therefore lead to performance degradation when directly applied to the quantized case (2). Moreover, even in the linear case (1), as illustrated in Corollary I.2 and Remark 3 the likelihood score computed in Jalal et al. (202 b) is not accurate for general matrices A. By contrast, we provide a simple yet effective approximation of the likelihood score which significantly outperforms Dalal et al. 2021a) for general matrices even in the linear case. Another related work (Qiu et al., 2020) studied 1-bit CS where the sparsity is implicitly enforced via mapping low dimensional representation through a known ReL.U generative
Kawar al. 2021 2022/Chung et al. 2022] Daras et al. 2022) in CS using SGM as a prior
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network. Wei et al. (019) also considers non-linear recovery using generative model assuming that the original signal is generated from a now-dimensional latent vector.
2 BACKGROUND
2.1 INVERSE PROBLEMS FROM QUANTIZED MEASUREMENTS
The inverse problem from noisy quantized measurements is posed follows (Zymnis et al. 2009
y Q n),
where the goal to recover the unknown signal x E R&xx1 from quantized measurements y Mx1 where A E R g is known linear mixing matrix, noise with known variance, and Q(-) RM + Q A x1 is an element-wise quantizer function which
(n; o²1) is i.i.d. additive Gaussian
maps each element into finite (or countable) set of codewords Q, i.e., Vmn Q22m + amm) Q. or equivalently (2m + nm) E Q (ym), m = 1, M, where 2m. is the m-th element = Ax.
Usually, the quantization codewords Q correspond to intervals (ymmis et al. 2009), i.c., Q (pm) y Mym.) where ly... My.. denote prescribed lower and upper thresholds associated with the quan- tized measurement Vm. i.e., 1u... S (Zm Rhm) < by.... For example, for uniform quantizer with Q quantization bits (resolution), the quantization codewords Q = { ) consist of 22 elements, i.e.,
4r (2r 2 1) A/2, r 1, 1,.,,,,, 2?.
where A > 0 is the quantization interval, and the lower and upper thresholds associated are 1,..., 20:
1, - (r- 20-1 1) r 2, 2,.,, Ug. = { (r
-0, -00, 1;
+00, , [ 22.
22-29-11 (r-29-1) A,
(4)
In the extreme 1-bit case, i.e., Q = only the sign values are observed, i.e.,
y sign + n),
(5)
where the quantization codewords Q {-1, +1} and the associated thresholds are [-1 -0O, u = 0 and (11 = 0, u41 = +00, respectively.
2.2 SCORE-BASED GENERATIVE MODELS
For any continuously differentiable probability density function p(x), if we have access its score function, i.e., Vx log (x)., we can iteratively sample from it using Langevin dynamics (Turq et al. 1977: Bussi Parrinello, 2007. Welling & Teh 2011)
X, x 0re Vx logg(x(-1) 22,z,, 1 ISIST,
where 2, ((z,; 0, 04 > is the step size, and T' is the total number of iterations. It has been demonstrated that when 0g is sufficiently small and T is sufficiently large, the distribution of XT will converge p(x) (Roberts & Tweedie 1996 Welling & Teh 2011). In practice, the score function x log p x) is unknown and can be estimated using a score network se (x) via score matching (Hyvärinen 2006 Vincent 201T). However, the vanilla Langevin dynamics faces variety of challenges such as slow convergence. To address this challenge, inspired by simulated annealing (Kirkpatrick et al.] 1983] Neal] (001). [ong & Ermon (2019) proposed an annealed version of Langevin dynamics, which perturbs the data with Gaussian noise of different scales and jointly estimates the score functions of noise-perturbed data distributions. Accordingly, during the inference, an annealed Langevin dynamics (ALD) is performed to leverage the information from all noise scales.
Specifically, assume that p= x) = Mix;; x, I), and so we have p*( = f Paara (x)p=(> x x)dx, where (x) is the data distribution. Consider we have a sequence of noise scales [ satisfying Bmax B1 > Bg > ... 3T Amin > O. The 8min - 0 is small enough so that P3m. (x) Pdota (x), and 8max is large enough so that Pamas (X) RJ Nx;; 0, 4) The noise
While more sophisticated quantization schemes exist (Dirksen 2019). here we focus on memoryless scalar quantization due its popularity and simplicity where the quantizer clement-wise.
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conditional score network (NCSN) (x .3) proposed in Song & Ermon (019)) aims to estimate the score function of each Pa, (x) by optimizing the following weighted sum of score matching objective
0- = arg min
Σ pxpsaa (x) Pps, (8/x) [1180(2, 3c) Vx log (x x)][ 1=1
After training the NCSN, for each noise scale, we can run K steps of Langevin MCMC to obtain sample for each Pa, (x)
x x ars0 (x] 1-1
,8)) + V2izz k 1,
(8)
The sampling process is repeated for = 1.2..... T sequentially with xp (x; 0, 8 max I) and xf+1 = x when t < T. As shown in Song & Ermon (2019), when K + oo and a - for all t, the final sample x will become an exact sample from Pam. (x) = Pdua (x) under some regularity conditions. Later, by a theoretical analysis of the learning and sampling process of NCSN, an improved version of NCSN, termed NCSNv2, was proposed in [Son & Ermon (2020) which is more stable and can scale to various datasets with high resolutions.
QUANTIZED CS WITH SGM
In the case of quantized measurements with access y, the goal is sample from the posterior distribution p(x y) rather than p(x). Subsequently, the Langevin dynamics in 6 becomes
x = x a₁ V - log yy + /2azz, 1 ss
(9)
where the conditional (posterior) score Vx logp(x yy is required. One direct solution is specially train a score network work that matches the score Vx log p(x y). However, this method is rather inflexible and needs re-train the network when confronted with different measurements. Instead, inspired by conditional diffusion models (Jalal et al.) 2021a), we resort to computing the posterior score using the Bayesian rule from which Vx p(x y) is decomposed into two terms
Vx log p(x y) = V logp(x) + V logp(y x),
(10)
which include the unconditional score V logg(xx) p(x) (we call it prior score), and the conditional score Vx log x) (we call it likelihood score), respectively. The prior score Vx log p(x) can be casily obtained using a trained score network such as NCSN or NCSNv2, so the remaining goal is to
compute the likelihood score Vx logo p(y x) from the quantized measurements.
3.1 NOISE-PERTURBED PSEUDO-LIKELIHOG SCORE
In the case without noise perturbing in x, one can calculate the likelihood score Vx log p(y directly from the measurement process (2) and then substitute into (10) obtain the posterior score Vx log p(x y). However, this does not work well and might even diverge (Jalal et al. (202 The reason that NCSN/NCSNv2, as shown in (7. the estimated score not the original prior score Vx log Pdata (x) but rather noise-perturber prior score Vg log Pa, (x) for each noise scale 3. Therefore, the likelihood score directly computed from ( does not match the noise- perturred prior score. To address this problem, we propose to calculate the noise-perturbed pseudo- pxeudo-likellhood likelihwod score V log pa, (y x) by explicitly accounting for such the noise perturbation in NCSN/NCSNV2. Unfortunately, for NCSN/NCSN22, the V log Ps, (y x) is intractable in the general case. To this end, we propose a simple yet effective approximation by introducing two assumptions.
|x)
Assumption 1. The prior p(x) uninformative w.r.t p(X x) i.e. p(x x) x p(x x).
Remark 1. This assumption is usymptotically accurate when the perturbed noise in x becomes negligible, i.e., 3. + 0 when t is large. An illustration of this is shown in Appendix
Assumption 2. The matrix A is row-orthogonal, i.e., AAT is diagonal matrix.
Remark 2. The measurement matrices A are usually carefully designed in CS. Interestingly, most popular CS matrices A satisfy this assumption, e.g., discrete Fourier transform (DFT) in MRI, discrete cosine transform (DCT), Hadamard matrices, as well as general row-orthogonal matrices Vehkapera et al. (2016). For another class of widely used i.id. Gaussian matrices, this assumption is approximated satisfied, as verified in Appendix B Notably, it has been theoretically demonstrated
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that row-orthogonal matrices mostly preferred for CS due both superior performance and easy implementation (Vhkaaperä et al. 2016). Therefore, this assumption does not impose serious limitation in the case of CS, and leave the study of general A as a future work.
Subsequently, we obtain our main result as shown in Theorem !
Theorem 1. (noise-pen pertsseper pseudo pseudo-likelihon likelihood score) For each noise scale 3, 0, if ps. (x x) ( x, I) under Assumption and Assumption 2 the noise-perturbed pseudo-likelihond score Vx log Ps. (y for the quantized measurements y in (2) can be computed as
Vx Po ly x) = G...
(11)
where Y, [91, 92, ... garl" E RRxx1 with each element being
exp
(-
exp
9m 00 + 8f ||a
m7 1, 1,2,... M,
(12)
2
Jiym
exp
(- 2-2)ds
where a BIxN denotes the m-th row vector of A and
aT. x
T x 1ym
Uym
(13)
02
B7
a
V0² 3 ||at
Proof. The proof shown in Appendix C
In the extreme 1-bit case, more concise result can be obtained, as shown in Corollary
1.1
Corollary 1.1. (noise-perturbed pseudo-likelihood score in the 1-bit case) For each noise scale 3e O, ps, (x x) Ni;; x, 1)),, under Assumption I and Assumption
the noise-perturbed pssudo-likelihove score Vx log (y x) for the sign measurements y sign Ax + n) in 5 can be computed as
2
Vx log (y 8) G2
(14)
where G(B,,,, [91, 92, .... DMll TT E R MxI
with each element being
(-
1 + Vm
1 Vm
exp
9m
(mm)
2 21 $b (zmm))
m. 1,2, M,
(15)
2OT B8 (
1 1 e- - dt is the cumulative distribution function 2
where 7m Vo and (=)
(CDF) of the standard normal distribution.
Proof. It can be easily proved following the proof in Theorem Interestingly, if no quantization is used, or equivalently Q - oc, the model in (2) reduces to the standard linear model ( Accordingly, results in Theorem Caan be modified as in Corollary T.2
Corollary 1.2, moise--peruurble 3. 0, if pa, (x x) NN x, B),,, under Assumption! [ the noise-perturbed lekelo-ikeelhood score Vx log pa. (y x) for linear measurements y = Ax n in [ can be computed as
pseudo-likelinn likelihood score without quantization) For each noise scale
Vx logop, (y x = A AJ (o BAAT) Ax-
(16)
Moreover, under Assumption the V log P... (y x) can be further simplified as
V Po (y x) = A Glinear (Br.y X)x,
(17)
where Giinear (Br, A, x) = [g,, gml T E M with each element being
m = 1, M. Vma 2
Jm
(18)
o2+ + 82 1
Proof. The proof is shown in Appendix D
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Remark 3. In contrast the quantized case, here obtain closed-form result 16 for any kind of matrices A. Interestingly, if Assumption 2 is further imposed, it reduces to (18). which is similar to Jalal et al. (2021a) where the annealing term 33 | 12 in (18) corresponds to o in Jalal et al. (202la) However, oi in Jn.Jall al. (202Ta) is added heuristically as an additional hyper-paramett while we derive it in closed-form under the principled perturbed pseudo-likelihod score. More importantly, for general matrices A. our closed-form result (I6) significantly outperforms Jalal et al. (202la. as demonstrated in Appendix Therefore, we not only explain the necessity of an annealing term Jalal et al. (02la) but also extend and improve Jalal et al. (202la) in the general case.
3.2 POSTERIOR SAMPLING VIA ANNEALED LANGEVIN DYNAMICS
By combining the prior score from SGM and noise-perturbed pseudo- score in Theorem D we obtain the resultant Algorithm 1
namely, Quantized CS with SGM (QCS-SGM in short).
Algorithm 1: Quantized Compressed Sensing with SGM (QCS-SGM)
Input: { )T1 y, A, o², quantization codewords Q and thresholds (llg. u4)1 E Q} Initialization: xi ~ L (0, 1)
for f 1 to T do
ar < /3./33
for k 1 to do
Draw z I),
Compute G( y, A, x
1-1 as
12 (or 15 5 for 1-bit)
xf =x=x"==
k-1 + (x k-1
,,)
A ( A, xxx(11) + 22,z
x
x+1
Output:
X = XT
K
EXPERIMENTS
We empirically demonstrate the efficacy of the proposed QCS-SGM in various scenarios. The widely-used i..dd Gaussian matrix A, i.e., A.; ~ 10. are considered. The code is available al
Datasets: Three popular datasets are considered: MNIST (LeCun & Cortes 2010) Cifar- 10
(Krizhvskyy & Hinton 2009), and CelebA (Liu et al., 2015), , and the high-resolution Flickr Faces High Quality (FFHQ) (Karras et al., 2018). MNIST (LeCun & Cortes 2010) are grayscale images of size 28 x 28 pixels so that the input dimension for MNIST is N 28 x 28 784 per image. Cifar- 10 (Krizevskyy & Hinton, 2009) consists of natural RGB images of size 32 x 32 pixels, resulting in
N 32 x x 3072 inputs per image. For CelebA dataset (Liu et al. 2015), we cropped each face image a 64 x 64 RGB image, resulting in N 64 64 3 = 12288 inputs per image. The FFHQ are high-resolution RGB images of size 256 x 256, so that N = 256 x 256 x 196608 per image. Note that to evaluate the -iitt--dittiitttionn (OOD) performance, we also cropped FFHQ to size 64 x 64, as OOD samples for CclebA. All images are normalized to range [0, 1].
Regarding the SGM models, we adopt the popular NCSNv2 (Song & Ermon 2020 in all cases. Specifically, for MNIST, we train a NCSNv2 (Song & Ermon (0200) model on the MNIST training dataset with similar training set up Cifarlo& in (Song& & Ermon (2020), while directly use the for Cifar-10, and CelebA, and FFHQ, which are available in this Link Therefore, the prior score can be estimated using these pre-trained NCSNv2 models. After observing the quantized measurements as (2). we can infer the original x by posterior sampling via QCS-SGM in Algorithm I It important to note that, we select images x that are unseen by the pre-traineirec SGM models. For details of the experimental setting, please refer to Appendix F We have submitted the code in the supplementary material and will release upon acceptance.
QCS-SGM:
4.1 1-BIT QUANTIZATION
First, we perform experiments on the extreme case, i.e., 1-bit measurements. Specifically, we consider images of the MNIST (LeCun & Cortes 2010) and CelebA datasets (Liu et al.) 2015) in the same setting as Liu & Liu (2022). Apart from SCS. we also show results of LASSO (Tibshirani 1996) (for CelebA, Lasso with DCT basis (Ammdd et al. 1974 is used), CSGM (Bora et al. 2017
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(a) MNIST; .M 200, 0.05
(b) ColcbA, M 4000, 0.001
Typical reconstructed images from 1-bit measurements on MNIST and CelebA. QCS-SGM faithfully recovers original images from 1-bit measurements when M & N. Compared other methods, QCS-SGM recovers more natural images with good perceptual quality.
Figure 2:
WIITT, 1-8
Calebl
07
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300
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Figure 3: Quantitative comparisons based on different metrics for 1-bit MNIST and CelebA. QCS-SGM remarkably outperforms all other methods under all metrics.
BIPG (Liu et al. 2020p, and OneShot (Liu & Liu 2022). For Lasso (Lasso-DCT), CSGM, BIPG, and OneShot, we follow the default setting as the open-sourcee code of & Liu (022).
The typical reconstructed images from 1-bit measurements with fixed < are shown in Figure for both MNIST and CelebA. Perhaps surprisingly, QCS-SGM can faithfully recover the images from 1-bit measurements with < measurements while other methods fail or only recover quite vague images. To quantitatively evaluate the effect of M. compare different algorithms using three popular metrics, i.c., peak signal-to- -o-isse ratio (PSNR) and structural similarity index (SSIM) for different M, and the results shown in Figure It can be seen that, under both metrics, the proposed QCS-SGM significantly outperforms all the other algorithms by a large margin. An investigation of the effect of (pre-quantization) Gaussian noise shown in Figure 23 in Appendix Please refer to Appendix H for more results.
Multiple samples & Uncertainty Estimates: It worth pointing out that, as kind of sampling method, QCS-SGM can yield multiple samples with different random initialization so that we can easily obtain confidence intervals or uncertainty estimates of the reconstructed results. Please refer to Appendix G for more typical samples of QCS-SGM. In contrast, OneShot (Liu & Liu 2022) can easily get stuck in a local minimum with different random initializations (Liu & Liu 2022
4.2
MULTTI-ITT QUANTIZATION
Then, we evaluate the efficacy of QCS-SGM in the case of multi-bit quantization, e.g., 2-bit, 3-bit. The results on Cifar- 10 and CelebA are shown in Figure 4 The results of the linear case without quantization are also shown for comparison using (18) in Corollary 1.2 As expected, with the increase of quantization resolution, the reconstruction performances get better. One typical result for nigh-resolution FFHQ 256 x 256 images is shown in Figure Moreover, to evaluate the quantization effect under "fixed budget" setting, i.e., Q x M remain the same for different Q and M, we conduct experiments for CelebA in the fixed case of Q M = 12288. Interestingly, as shown in Table [ under 'fixed budget", while there is apparent increase in PSNR for the multi-bit case, the perception metric SSIM remains about the same as 1-bit case. For more results, please refer to Appendix H
4.3 OUT-OF-DISTE DISTRIBUTION PERFORMANCE
In practice, the signals of interest may d-dttsribbtttiinnn (OOD) because the available training dataset has bias and is unrepresentative of the true underlying distribution. To assess the performance
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CelebA (61 x 61)
I bit M
12288
2 bit, M 6144
3 bic M
4096
Method
PSNK 7
SSIM
PSNR
SSIM
PSNR 1
SSIM
QCS-SGM (ours)
20.1
0.81
24.3
0.78
25.8
0.80
Lasso-DCT
16.2
0.68
16.9
0.69
18.2
0.74
CSGM
17.6
0.68
18.6
0.70
18.9
0.72
BIPG
11.6
0.57
11.59
0.47
11.45
0.53
OneShot
16.1
0.62
13.3
0.55
14.75
0.54
Table Quantitative comparison of different methods for Q-bit CS with "fixed budget", ie., Q x M is fixed. Results are averaged over validation set of size 100. Reconstructed images are shown in Appendix H
(a) Cifar-10, M 2000, a 0.001
(b) CelebA, 4000 0.001
Figure Results QCS-SGM Cifar- CelebA images under different quantization bits.
CRSW, tot
BPQ 1M
OneBht 164
QC9-69K 1-81
QC3-89M, 2:12
QC9-69M
Figure 5: Results out-of-distribution (OOD) FFHQ dataset from 1-bit (2-bit and 3-bit also shown for QCS-SGM) measurements. 10000, a 0.001.
of QCS-SGM for OOD datasets, we choose the CelebA as the training set while evaluating results on OOD images randomly sampled from the FFHQ dataset, which have features of variation that are rare among CelebA images (c.g., skin tone, age, beards, and glasses) (Jalal et al.) (0221b). As shown Figure even on OOD images, the proposed QCS-SGM can still obtain significantly competitive results compared with other existing algorithms.
CONCLUSION
In this paper, novel framework called Quantized Compressed Sensing with Score-based Generative Models (QCS-SC SGM) is proposed to reconstruct nigh-dimensional signal from noisy quantized measurements. Thanks to the power of SGM in capturing the rich structure of natural signals beyond simple sparsity, the proposed QCS-SGM significantly outperforms existing SOTA algorithms by a large margin under coarsely quantized measurements on a variety of experiments, for both in- distribution and out-of-distribution datasets. There are still some limitations in QCS-SGM. For example, it is limited to the case where measurement matrix A is (approximately) row-orthogonal The generalization QCS-SGM to general matrices an important future work. In addition, at present the training of SGM models is still very expensive, which hinders the applicability of QCS-SGM the resource-limited scenario when no pre-trained models are available. Compared to traditional CS algorithms without data-driven prior, the possible lack of enough training data also imposes great challenge for OCS-SGM. therefore important to address these limitations in the future. Nevertheless, as it is research topie with rapid progress, believe that SGM could open up opportunities for quantized CS, and sincerely hope that our work as a first step could inspire more studies this fascinating topic.

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ACKNOWLEDGEMENTS
X. Meng would like to thank Jiulong Liu for his help in understanding their code of OneShot. This work was supported by JSPS KAKENHI Nos. 17H00764, and 19H01812, 22H05117, and JST CREST Grant Number JPMJCR1912, Japan.
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A VERIFICATION OF ASSUMPTION
Mathematically, we can write the exact p(y | x) as
ply x) = / ply x)p(x d),x
(19)
where from the Bayes' rule,
p(x x)p(x)
pxx x)
(20)
J p x x)p(x)dx
For NCSN/NCSNY2, recall that the likelihood p(k x) follows a Gaussian distribution / (5;;; x, I))). where / is the variance of the perturbed Gaussian noise and, by definition, 3 - 0 in later steps of the reverse diffusion process in NCSN/NCSNv2. As a result, from (00). the likelihood p(x x) will dominate the posterior p(x x) B - O and therefore p(x x) p(x x), indicating that the prior p(x) becomes uninformative. Note that this is particularly the case when the entropy of p(x) is high, which is usually the case for generative models capable of generating diverse images. While this assumption is crude at the beginning of the reverse diffusion process when the noise variance / is large, is asymptotically accurate as the process goes forward. The effectiveness of this assumption is also empirically supported by various experiments in Section Verilicttion plalE) p(al))
Vorilcution
10
10
4
mazr
10"
10
10" 100 200 300 400
100
200
300
400
500
500
Figure 6: Comparison of the exact mean and variance of p(a: 3) with the pseudo mean and variance under the uninformative assumption, i... p(r ž) p(ū x)) in the toy scalar Gaussian example. In this plot, we set 8max = 90, Bmin = 0.01, T' = 500, which are the same as the setting of NCSNy2 for CelebA. The ê and the prior standard deviation ou is set to be d. 15. It can be seen that the approximated values approach the exact values very quickly, verifying the effectiveness of the Assumption for this toy example.
toy example: In the following, we illustrate the assumption in a toy example where x reduces to a scalar random variable x and the associated prior p(z) follows a Gaussian distribution, i.c., p(x) (I; 0, o where o² is the prior variance. The likelihood p(x x) in this case simply p( x) = Ni; x. 82). Then, from (0) after some algebra, it can be computed that the posterior
distribution pz 8)
p( ()) = N(;; Mexact: Vexact)
(21)
where
o
02
Mexact 0 + 32
3, Vexnct of + 32
(22)
Under the Assumption
| i.c., p( ()) x p( x), we obtain an approximation of p(z
[i) as follows
p(x ()) = 5(x T)) Nax; Mpseuddo Upeeudo);
(23)
where
Mepeeudo 8, Upseuuko 182
(24)
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By comparing the exact result (22 and approximation result 24
it can be easily seen that for
fixed o > O, as 3 - 0, we have Mpacudo - Mpost and Upscudo Uposs In Figure 6 similar to
NCSN/NCSNV2, we anneal 3 as 3t /uaxx (min I geometrically and compare Mpscudo: Upscudo Bmaxx
with Mexact, Vexact as increase from to can be seen in Figure 6 that the approximated values Mpseuda, "psendo especially the variance "pueudo- approach to the exact values Mexxct, Mexaat very quickly,
verifying the effectiveness of the Assumption under this toy example.
B VERIFICATION OF ASSUMPTION
Before verifying it in the i.i.d. Gaussian case, we first briefly add comment on the general case when AA" not an exact diagonal matrix. As demonstrated later Appendix C the Assumption 2 is introduced to ensure that the covariance matrix o²I + B3 A of Gaussian noise is diagonal so that we can obtain a closed-form solution for the likelihood score in the quantized case, which is otherwise intractable.
Suppose that the elements of A follow i.i.d. Gaussian, i.e., Aij ~ ~N(0, Next, we investigate the elements of the matrix C AA" {Cig), =...M.
Regarding the diagonal elements Ca, by definition, it reads as
Cs Σ A i...M.
(25)
As Ca is the sum of square of N i.i.d. Gaussian random variables Agj, j j= I...,, Cj follows Gamma distribution, i.e., [ 2x²). The mean and variance of Ci can be computed as No and 2.
Regarding the off-diagonal elements Cıjs i + j. by definition, reads as
C₁. = Σ Amn i,j L...M,i j.
(26)
As Ain and Ajn are independent Gaussian for i - j. it can be computed that the mean and variance of Cij are 0 and No respectively. When x² = 1/M and M a.,, where > 0 is the constant measurement ratio, the variance of Cij is aN 0 as N - oc. As result, all the off-diagonal elements of AA will tend to zero as N > oc. From another perspective, in Figure 7 we compute the ratio between the average magnitude of off-diagonal elements Ci₃ and the diagonal elements C;: when M = aN with a 0.5. can be seen that as increases, the magnitude of the off-diagonal elements becomes negligible. As result, when A is i.i.d. Gaussian, the matrix AA? can be well approximated as a diagonal matrix in the high-dimensional case.
C PROOF OF THEOREM
Proof. Let us denote z = Ax. For each noise scale Br >> 0, under the Assumption I
we obtain
Ps. (x
x)
x Pa, ( x)
~ N (x; X, II
(27)
then we can equivalently write
X R 3ew,
(28)
where w ~ I). As a result, 2 obtain
Ax = A(* + B.w) = A + AAw. Then, from
we
= Q (AX+ n);
(29)
where ñ n + 3,A.. Since n~ ~ NN o') and NN, 1) and are independent to each other, it can be concluded that ñ is also Gaussian with mean zero and covariance I AAA.". i.e. (in 83 AAT).
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Aseeage Gu//CC
1071
10
12
107
107
10"
30*
10"
N
Figure 7: The ratio between the average magnitude of the diagonal elements Cj and off-diagonal elements City when M a.N with a = 0.5.
Subsequently, under the Assumption i.c., AA is a diagonal matrix, each element 7. (ñm 0, 8 || of ñ will be independent to each other and thus from (38) can obtain a closed-form solution for the likelihood distribution (we will refer as pseudo-likelihood due to the assumptions used) p(y/ž- Ax) as follows
M II m=1
plyl2 AX)
p (ym 2m xxxx
(30)
where, from the definition of quantizer Q
p (ym
2m xmmxx) p (lym
S 2m + Am Uymm)
(31)
im + Uym
2m + 1m1
+
(32)
V02 + 333 ||a
02 + B3 a
+ (- Uy..) (-(ggm))
(33)
where (2) distribution and
1 -0 e
2
dt is the cumulative distribution function of the standard normal
v2n
a x
a x
am
(34)
pm
pm
V 02 + 37 aa
0² + 33 ||
As result, it can be calculated that the noise-perturbed pseudo-likelihood score Vx log (y for the quantized measurements: y in (2) can be computed as
())
Vx log (y x) A (
(35)
where G,
[9 12, ... garl" E Mx with each element being
22 exp (-
exp
m 1,2, .... M,
(36)
V 02 + 83 || 1 1|
frm Jiym
exp
(- 12 )
which completes the proof.
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Cud Sarta
080
(a) Cosine Similarity
(b) PSNR
Figure 8: Averaged Cosine Similarity and PSNR of reconstructed MNIST images of ALD in Jalal et al. 2021a) and ours (with formula (T8) and (16), respectively) when M = 200, 0.1 for the different condition number of the matrix A cond A) = 1000. can be seen that our method with (16) significantly outperforms ALD Dalal al. (2021a) high condition number while performing similarly at low condition number. Ours with diagonal approximation (18) is about the same as ALD
as expected.
D PROOF OF COROLLARY
Proof. Similarly in the proof of Theorem under the Assumption we can equivalently write
let denote Ax. For each noise scale 3g >
x + Bew,
(37)
where w ~ (O, I). As a result, z = Ax = A(X Bew) = AR + A.Aw. Then, in the case of the linear model, from ①[. we obtain
= Ax n,
(38)
where ñ A,Aw. Since n ~ NN.,0 and w ~ NO0, I) and are independent to each other, it can be concluded that ñ is also Gaussian with mean zero and covariance o + AA'" i.e., ñ (, 030 + A.AA).. Therefore, closed-form solution for the likelihood distribution p(y 2 = Ax) can be obtained as follows
exp (-b(y- AX) (0 (y- AA)
ply AX)
(39)
(2T)M det (o2I+ SAAT)
As result, we can readily obtain a closed-form solution for the noise-perturbed pseudo-likelihood score V log (y x) as follows
Vx log 1x) (o (100+$AAAA)) (y-AX)
(40)
Furthermore, if AAT T is diagonal matrix, the inverse of matrix (o²1 + AAAA) become trivial since it is diagonal matrix with the m-th diagonal element being 0² 33 || After some simple algebra, we can obtain the equivalent representation in (18) which completes the proof.
E COMPARISON WITH ALD IN JALAL ET AL.
A221 IN THE LINEAR CASE
As shown in Corollary 1.2 in the special case without quantization, our results Theorem I can be reduced form similar to the ALD in Jalal al. 20) However, there several different important differences. First, our results are derived in a principled way as noise perturbed pseudo- likelihood score and admit closed-form solutions, while the results in Jalal al. (2021a) obtained
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2021a
Figure 9: Tyical recovered MNIST images of ALD in Jalal al. (2021a) and ours (with formula
(18) and [6). respectively) when M = 200, a 0.05 and the condition number of matrix A is cond
1000. can be seen that our method with (16) significantly outperforms ALD in Jalal
al. 202 which performs about the same
with diagonal approximation (18)
heuristically by adding additional hyper-parameter (and thus needs fine-tuning). Second, the results in Jalal et al. (2021a) are similar to an approximate version (88 of ours, which holds only when A is a diagonal matrix. For general matrices A, we have a closed-form approximation (16. We compare our results with ALD (Jalal et al. 2021a) and the results are shown in Figure and Figure It can be seen that when at low condition number of when AA? T is approximately diagonal matrix, our with diagonal approximation (18) performs similarly as (16), both of which are
similar ALD (Jalal al. 2021a), as expected. However, when the condition number of A is large so that AA? T is far from a diagonal matrix, our method with the (16) significantly outperforms ALD in Jalal et al. (2021a).
F DETAILED EXPERIMENTAL SETTINGS
In training NCSNx2 for MNIST, we used similar training setup as Song & Ermon 2020 for Cifar10 as follows.
Training: batch-size: 128 n-epochs: 500000 n-iters: 300001 snapshot-freq: 50000 snapshot-sampling: true anneal- power::
false.
Please refer to Song & Ermon (2020) and associated open-sourced code for details of training. For Cifar-10, CelebA, and FFHQ, we directly use the pre-traines models available in this Link
When performing posterior sampling using the QCS-SGM in T for simplicity, set constant value < 0.0002 for all quantized measurements (e.g., 1-bit, bit, 3-bit) for MNIST, Cifar10 10 and CelebA. For the high-resolution FFHQ 256 x 256, set 0.00005 for 1-bit and 0.00002 for 2-bit and 3-bit case, respectively. For all linear measurements for MNIST, Cifar 0, and CelebA, we set 0.00002. is believed that some improvement can be achieved with further fine-tuning of for different scenarios. For MNIST and C0, we set 31 = 50, Sr 0.01, T 232: for CelebA, we set 31 90, 37 0.01, T 500; for FFHQ, set 31 348, 3T 0.01, 2311 which are the same as Song & Ermon 2220)). The number of steps K in QCS-SGM for each noise scale is set to be K all experiments. For more details, please refer the submitted code.
G MULTIPLE SAMPLES AND UNCERTAINTY ESTIMATES
As one kind of posterior sampling method, QCS-SGM can yield multiple samples with different random initialization that we can easily obtain confidence intervals uncertainty estimates of the reconstructed results. For example, typical samples as well as mean and std are shown in Figure I0 for MNIST and CelebA in the 1-bit case.
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H ADDITIONAL RESULTS
Some additional results are shown this section.
Figure and Figure 12
show results with relatively large value of M in the same setting Figure respectively.
and Figure
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Figure 12: Results of QCS-SGM for C1 (M = 4000) and CelebA (M = 16000) images under different quantization bits.
Figure 13 shows the quantitative results of QCS-SGM for different quantization bits.
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Figure 14 and Figure 15 show the reconstructed images of QCS-SGR for Cifar and CelebA in the fixed budget case of M 3072 and Q M 12288, respectively.
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(a) Ground Truth
(b) 1-bit, M 3072
(c) b-b M 1536
(d) 3-bit, M 1024
Figure 14: Reconstructed images of QCS-QCCS for Cifar in the fixed budget case (Q x M =3072).
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(a) Ground Truth
1-bit, M 12288
(c) b-b M 6144
(d) 3-bit, M 4096
Figure 15: Reconstructed images of QCS- CCM- for CelebA in the fixed budget case (Q x M 12288) in the same setting as Table
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Cifar10 CS2
Cifar10 CS8
CelebA CS2
CelebA CS8
Method
SSIM t PSNR
S PSNR
SSIM
PSNR f SSIM f PSNR
QCS-SGM (ours)
0.9712
34.33
0.8634
26.20
0.9694
36.80
0.9156
31.13
Neumann networks (Gilton et al. 2019
33.83
25.15
35.12
28.38
Table 2:
Quantitative comparison (PSNR (dB) and SSIM) of QCS-SGM with the Neumann networks in Gilton et al. (219)) for CS in the linear case, i.e., without quantization. Both Cifar10 32 32 and CelebA 64 x 64 are considered. It can be seen that QCS-SGM outperforms the Neumann networks Gilton al. 2019) all cases. As in Gilton et al. 2019). the values reported are the median across
a test set of size 256. Reconstructed images of CCS-SGM are shown in Figure 16
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Figure 16: Results of QCS-SGM on Cifarl0 for CS2, i.c., M 1536, N 3072. Noiscless case.
I COMPARISON WITH NEUMANN NETWORKS GILTON ET AL. (2099
In the case of no quantization, we compare our method with one popular method called Neumann Networks (Gilton et al., 2019). The results for Cifar10 are shown in Table 2 It can be that our method outperforms Neumann Networks by a large margin. We do not show quantization comparisons for quantized CS since Neumann Networks does not support
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Figure 17: Results results of QCS- CCM-SGM Cifar10 for CS8, i.e., M 1536, N 3072. Noiseless case.
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Figure 18: Results QCS-SGM on CelebA for CS2, i.c. M 6144, 12288, Noiseless case.
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Figure 19: Results QCS-SGM on CelebA for CS8, i.c. M 1536, 12288. Noiseless case.
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Figure 20: Results of reconstructed images from 1-bit measurements on MNIST and CelebA images, following exactly the same setting Liu & Liu (2022). i.c., Aij (O0 1), and n, (O0 o?).
J COMPARISON IN THE EXACTLY SAME SETTING AS LIU & LIU (2022)
Note that there a slight difference in the modeling of measurement matrix and noise n between and that in Liu & Liu (2022). In Liu & Liu (2022). it is assumed that A is an i.i.d. Gaussian matrix where each element A.y follows A. ~ (N, 1) and that n is an i..d. Gaussian with variance o², ie., n4 ~ (O, )²). However, in practice, the measurement matrix is usually normalized so that the norm of each column equals 1. As result, in our setting, assume that each element of i.i.d. Gaussian matrix follows Aij ~ (O, 1/M). Mathematically, there is one-to-one correspondence between the two settings, but the simulation setting different due the different measurement size M. As a result, for an exact comparison with results in Liu & Liu (202)). we also conducted experiments assuming exactly the n, o²) The results are shown in Figures 20
setting Liu & Liu 2022
i.e., Aij NNO0, 1), and
21 22 23
An investigation of the effect of (pre-quantization) Gaussian noise is shown Figure 23 Interestingly, the results in Figure 23 empirically demonstrate that QCS-SGM is robust to pre-quantization noise, ie., similar performances have been achieved for a large range of noise, even when the noise is very large approaching zero. Note that different from the comparing methods, the results of QCS-SGM in Figure 23 are not normalized to range [0, 1]. which suggests that the "dithering" effect seems not apparent for QCS-SGM. A rigorous analysis of this is left as future work.
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Figure 21: Quantitative comparisons based cosine similarity for MNIST and CelebA images, following exactly same setting as Liu Liu (2022) i.c., Ai ~ No 1), and n, (0,00,
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Figure 22: Results of reconstructed images from 1-bit measurements on MNIST and CelebA images, following exactly the same setting Liu & Liu 2022). i.e., Aij NN 1), and n, <0 o2).
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NIUST: 1:44 Mounrreent Strd mw000
4.5
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(b) CelebA, M 4000
Figure 23: Results of reconstructed images from 1-bit measurements MNIST and CelebA images for different a,, following exactly the same setting as Liu & Liu (0022), i.c., Aij ~ NoO,, 1), and n4 ~ N/,00 o²).
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Reviews:
[{"review_id": "aUP6duwwZGz", "reviewer": "Reviewer_DMn5", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Motivated by the power of score-based generative models (SGM), this work studies quantized compressed sensing with score-based generative models and proposed an unsupervised data-driven approach that is denoted by QCS-SGM. One key to QCS-SGM is proposing an annealed likelihood score (noise-perturbed likelihood score), which incorporates information from the measurements while performing the posterior sampling. In particular, if $\\mathbf{A}\\mathbf{A}^T$ is a diagonal matrix, QCS-SGM reduces to a form similar to that of Jalal et al. (2021a), and corresponding guarantees are provided in Theorem 1 and Corollary 1.1. Numerical results are provided to show the superiority of QCS-SGM. ", "summary_of_the_review": "Although this work seems to be incremental compared to some closely relevant prior works, the main idea is interesting and the experimental results are promising. Therefore, I am inclined to acceptance.", "strength_and_weaknesses": "Strength:\n\nAs the authors mentioned, their work is the first one that utilizes SGM quantized CS. Such an idea is interesting, and the experimental results are promising. \n\nWeaknesses:\n\n(a). SGM has been used for linear CS in some recent works, including Jalal et al., 2021a;b and \n\nDaras, G., Dagan, Y., Dimakis, A. and Daskalakis, C., 2022, June. Score-Guided Intermediate Level Optimization: Fast Langevin Mixing for Inverse Problems. In International Conference on Machine Learning (pp. 4722-4753). PMLR.\n\nI hope the authors can provide more discussions on these closely relevant works and provide intuition on why the QCS-SGM can outperform the methods proposed in these works. Ideally, similar to Jalal et al., 2021a, also perform the experiments for some MRI datasets.\n\n(b). In Theorem 1, the assumption that $\\mathbf{A}\\mathbf{A}^T$ is diagonal is restrictive. Although this assumption holds approximately for some popular measurement matrices such as i.i.d. Gaussian and discrete cosine transform matrices, it is more desired to provide guarantees for these measurement matrices directly (i.e., for the case that $\\mathbf{A}\\mathbf{A}^T$ is approximately diagonal ).", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "TJehayqeBS", "reviewer": "Reviewer_CEkX", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors present a novel method for reconstruction from quantized compressive sensing measurements. The proposed method is a modification of score-based models to incorporate information about the forward measurement process and quantization noise. Overall, the method is interesting and generally well-explained. However, the experimental assessment seems somewhat incomplete. As a minor point, the quality of figures is poor and they are often hard to read.", "summary_of_the_review": "The authors present a generally novel and interesting idea to use score-based models for CS reconstruction in presence of quantization. However, I feel the idea is not thoroughly validated from the experimental standpoint, making it unclear how it is positioned in the wider literature on CS reconstruction.\n\n-----\n\nPost-rebuttal: the authors significantly improved the original submission. I raised my score accordingly.", "strength_and_weaknesses": "Strengths:\n- Novel and interesting method to adapt score-based generative models to compressive sensing reconstruction in presence of quantization\n- Good performance in the specific settings that are presented and on \"toy\" datasets like MNIST and CIFAR\n\nWeaknesses:\n- Experimental evaluation seems rather limited. Some of the alternative methods shown in the comparisons are not quite mainstream, while important and widely-used baselines for deep-learning CS image reconstruction are missing (for example, ISTA-net Zhang 2017, and Neumann networks Gilton 2019) \n- Authors only report comparisons with other techniques for the 1-bit case but it is unclear how the method is positioned with respect to the state-of-the-art for a wider range of quantization rates\n- For the purpose of quality comparisons of image reconstructions, I feel cosine similarity is not very useful. The most used metrics are just PSNR and SSIM. The space saved removing cosine similarity could be better used expanding the set of comparisons. \n- Training SGM models is known to be very computationally expensive. The need for a pretrained one limits the applicability of the proposed method to a wider range of data for which no pretrained SGM is available, or it would be too expensive to train or not enough data are available. The authors need to discuss this limitation of the approach with respect to existing techniques, especially the ones not relying on generative models.\n- Related to the previous point, results are presented on toy datasets for which pretrained SGMs are easily available, but it is unclear how the method generalizes to higher resolution images. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "D6FjJA6Ljci", "reviewer": "Reviewer_CmCq", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes an approach for signal recovery in quantized compressed sensing (CS) that leverages score-based generative models. This addresses a specific setting for the first time, as previous approaches to SGM-based CS do not assume quantization, and quantized CS approaches with generative models have used other options such as GANs and variational autoencoders. A new score is proposed that leverages knowledge of the measurements when assessing the likelihood of a particular candidate signal.\n\nSample candidates for the recovered signal are generated using a Angevin dynamics formulation where the probability term becomes conditioned on the measurements observed. A Bayes rule is leveraged to allow for the score training process to be independent of the measurement matrix.\n\nThe paper sets up a recovery method that leverages a score-based generative model and derives a closed form solution under an orthogonal-row measurement matrix assumption. This result is specialized to the one-bit and non-quantized cases. \n", "summary_of_the_review": "An interesting approach to quantized CS recovery with good analytical treatment and compelling experimental results. ", "strength_and_weaknesses": "The numerical results use real-world datasets of varying complexity and show significant performance improvements vs. other generative model-based approaches to quantized CS recovery. However, I would argue that the use of Daubechies-1 (i.e., Haar) wavelets for images does not provide a fair comparison, as other wavelet or patch DCT bases are widely known to be better suited for images. Other results show good scaling to multi-bit quantization and out-of-distribution samples. \n\nThe method opens some questions that the authors may want to briefly address, or consider in the future:\n\nIt would be interesting to comment on the robustness of these results when the rows of the measurement matrix are only approximately orthogonal, or perhaps validate it numerically.\n\nIt would be interesting to consider the effect of noise in the recovery, and how the different methods would perform on a \"fixed budget basis\" setting - e.g, M one-bit measurements vs. M/B B-bit measurements.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster