💻 computer science

Deep Probabilistic Unfolding for Quantized Compressive Sensing

This paper proposes a deep probabilistic unfolding model for quantized compressive sensing that improves reconstruction accuracy and efficiency by replacing traditional L2 projections with a closed-form likelihood gradient to respect quantization physics and incorporating a dual-domain Mamba module for effective multi-scale feature fusion.

Original authors: Gang Qu, Ping Wang, Siming Zheng, Xin Yuan

Published 2026-05-13

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CC BY 4.0

Original authors: Gang Qu, Ping Wang, Siming Zheng, Xin Yuan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant jigsaw puzzle, but someone has thrown away 90% of the pieces and then painted over the remaining ones with just a few shades of gray. This is essentially what Quantized Compressive Sensing (QCS) is: trying to reconstruct a clear, high-quality image from a tiny amount of very "blurry" and "low-color" data.

The paper introduces a new tool called DPUNet to solve this puzzle. Here is how it works, broken down into simple concepts:

1. The Problem: The "Pixelated" Mess

In the real world, cameras and sensors often need to save space and battery. So, instead of recording every detail of a signal (like a full-color photo), they take a few samples and round them off to the nearest simple number (like turning a smooth gradient into just "black" or "white").

The Challenge: When you try to rebuild the image from these rough, rounded-off numbers, it's like trying to guess the shape of a cloud based only on its shadow. Previous methods were either too slow (taking minutes to guess) or too rigid (forcing the image to fit a strict mathematical box that didn't match reality).

2. The Solution: A Smart, Step-by-Step Detective

The authors built DPUNet, which acts like a super-smart detective who solves the puzzle in a series of quick, logical steps rather than guessing randomly.

The "Soft" Guidance (Probabilistic Unfolding)

Old methods treated the rounding error as a hard wall: "If the number is 1, the pixel must be here." This often led to mistakes.

The New Trick: DPUNet treats the rounding as a "fuzzy" clue. Instead of a hard wall, it uses a soft probabilistic guide. Imagine being told, "The piece is likely in this area, but maybe slightly to the left or right."
The Result: The model calculates the most likely direction to move the image pieces toward the truth without getting stuck. It turns a rigid math problem into a flexible, guided search.

The "Dual-Eye" Brain (The Mamba Module)

To put the puzzle together, the model needs to see both the tiny details (like the texture of a shirt) and the big picture (like the shape of a face).

The Old Way: Previous models looked at the image like a snail crawling across a page, checking one pixel at a time. This is slow and misses the big picture.
The New Way (Dual-Domain Mamba): The authors gave the model two "eyes":
1. The Spatial Eye: Looks at the image normally, pixel by pixel, to catch local details.
2. The Spectral Eye: Looks at the image as a whole wave (like looking at the ripples in a pond). This eye sees the "global" patterns instantly.
The Magic: These two eyes work together. The "Spectral Eye" uses a special math trick (Fourier transforms) to understand the whole image at once, while the "Spatial Eye" fills in the fine details. They combine their views to reconstruct the image perfectly, even with very little data.

3. Why It's Better (The Race Results)

The paper tested this new detective against the current champions (other AI models).

Speed: The old champions were like marathon runners taking hours to finish. DPUNet is a sprinter, finishing the job in a fraction of a second.
Quality: The images reconstructed by DPUNet are much sharper and have fewer "artifacts" (weird glitches) than the others.
Efficiency: It uses less computer power (memory and processing) to do the same job, making it practical for real devices like phones or sensors.

Summary

Think of DPUNet as a master chef who can recreate a complex, delicious meal (the original image) even if the recipe was scribbled on a napkin with only a few ingredients listed (the quantized data). Instead of guessing blindly, it uses a smart, step-by-step process that understands the "flavor" of the data, looks at the dish from both a close-up and a distance, and serves up a perfect result almost instantly.

Key Takeaway: This method makes it possible to use very low-quality, low-power sensors to take high-quality pictures, which could help in things like single-pixel cameras, radar, and wireless networks, but the paper focuses specifically on the image reconstruction performance itself.

Technical Summary: Deep Probabilistic Unfolding for Quantized Compressive Sensing

Problem Statement
Quantized Compressive Sensing (QCS) addresses the challenge of reconstructing signals from compressed measurements that have undergone coarse quantization (e.g., 1-bit or low-bit). While coarse quantization significantly reduces bandwidth and power consumption—making it vital for applications like massive MIMO, radar, and wireless sensor networks—it introduces severe non-linearity and information loss regarding signal amplitude. Existing reconstruction methods face a dichotomy:

Traditional Model-Driven Algorithms: These rely on hand-crafted priors and iterative optimization. While robust, they often suffer from high computational costs and limited reconstruction quality.
Deep Learning-Based Generative Approaches: Recent methods (e.g., QCS-SGM, QCS-SGM+) utilize score-based generative models to perform posterior sampling via annealed Langevin dynamics. While these achieve high perceptual quality, they are computationally expensive, requiring iterative stochastic updates over multiple noise levels. This makes them unsuitable for real-time deployment due to high latency and throughput constraints. Furthermore, these methods often rely on restrictive assumptions (e.g., row-orthogonality of the sensing matrix) for tractable likelihood approximations, or require heavy pre-training on large-scale datasets that may not match specific domain settings.

Methodology: Deep Probabilistic Unfolding Network (DPUNet)
The authors propose DPUNet, a deterministic, end-to-end trainable deep unfolding framework that bridges the gap between principled Bayesian inference and efficient deep learning.

Closed-Form Likelihood Gradient Projection:
Unlike previous unfolding methods that apply $L_2$ projection or rely on Expectation Propagation (EP) approximations, DPUNet derives a numerically stable, closed-form likelihood gradient.
- Mechanism: The method treats quantization constraints as soft probabilistic guidance rather than hard constraints. It models the measurement domain as being perturbed by noise (either inherent or artificial).
- 1-bit Case: Utilizes the Mills ratio to compute the gradient of the log-likelihood for binary measurements ( $y = \text{Sign}(Mx + n)$ ).
- Multi-bit Case: Uses the difference of Gaussian Cumulative Distribution Functions (CDFs) over quantization intervals to derive the gradient.
- Advantage: This approach avoids the need for row-orthogonality assumptions required by EP-based methods and eliminates the iterative sampling steps of score-based models, enabling a single feed-forward pass.
Dual-Domain Mamba Block (DMB):
To serve as the refinement network within the unfolding stages, the authors design a novel Dual-Domain Mamba Block that captures both local and global features efficiently.
- Spatial Branch: Implements a standard State Space Model (SSM) on flattened spatial sequences to capture local dependencies.
- Spectral Branch (S3M): Introduces a complex-valued, frequency-conditioned spectral layer. Instead of sequential token scanning (which is order-dependent), S3M operates on Fourier modes. It parameterizes a stable complex state-space system in the frequency domain, yielding a closed-form solution for the spectral response.
- Cross-Frequency Interaction: To address the limitation of pure diagonal filtering (which assumes frequency independence), the S3M incorporates a structured low-rank projection–broadcast term. This enables efficient cross-frequency interactions with $O(LR)$ complexity, avoiding the quadratic cost of dense spectral mixing.
- Order Invariance: By operating in the Fourier domain, the global receptive field becomes order-invariant, overcoming the inherent order dependence of standard scan-based Mamba architectures.
Training Objective:
The network is optimized by minimizing a composite loss function consisting of the $L_2$ reconstruction error and a Negative Log-Likelihood (NLL) term. The NLL term ensures the output adheres to the quantized data-consistency constraints derived from the probabilistic model.

Key Contributions

DPUNet Framework: A new deep unfolding architecture that converts hard quantization constraints into soft probabilistic guidance via a closed-form likelihood gradient projection. This allows for deterministic, single-pass inference without the computational burden of iterative posterior sampling.
Dual-Domain Mamba Block: A novel refinement module that fuses spatial state-space modeling with a frequency-conditioned, order-invariant spectral operator. It achieves global feature interaction through structured spectral operators rather than repeated recurrences, significantly reducing computational cost.
Performance and Efficiency: The method demonstrates state-of-the-art (SOTA) performance in reconstruction accuracy (PSNR/SSIM) across 1-bit to 3-bit quantization levels while maintaining inference speeds orders of magnitude faster than score-based sampling baselines.

Experimental Results
The authors evaluated DPUNet on the CelebA (64×64) and FFHQ (256×256) datasets, comparing it against QCS-SGM, QCS-SGM+, and a vanilla deep unfolding baseline.

Accuracy: DPUNet achieved superior PSNR and SSIM scores across all quantization bits. For example, on FFHQ with 1-bit measurements, DPUNet achieved 27.18 dB PSNR compared to 13.26 dB for QCS-SGM+.
Efficiency: The inference time for DPUNet was approximately 0.25 seconds for 256×256 images, compared to 558 seconds for QCS-SGM and 1072 seconds for QCS-SGM+.
Ablation Studies: Removing the likelihood gradient projection caused the most severe performance degradation (approx. 10.57 dB drop), validating the core probabilistic formulation. The spectral branch and rank-R coupling also contributed significantly to performance.
Out-of-Distribution (OOD): The model demonstrated robust reconstruction capabilities on the CSet8 dataset (OOD) when trained on FFHQ, outperforming previous methods even in 1-bit scenarios.

Significance and Claims
The paper claims that DPUNet offers a principled alternative to existing QCS reconstruction methods by unifying the interpretability of model-based unfolding with the representational power of deep learning, without the computational overhead of generative sampling.

Practical Deployment: By eliminating iterative sampling and reducing reliance on restrictive sensing matrix assumptions, the method is positioned as a viable solution for real-world applications where latency and throughput are critical.
Future Potential: The authors suggest that this framework serves as a prototype for future task-aware or hardware-in-the-loop optimization, where the reconstruction pipeline can be jointly tuned for downstream objectives such as recognition or secure transmission. The method is presented as a step toward more efficient and reasonable end-to-end designs for quantized compressive sensing.