[论文解读] SAT-based Compressive Sensing

Milton Halem, Tim Finin|arXiv (Cornell University)|Mar 6, 2019

Sparse and Compressive Sensing Techniques参考文献 34被引用 7

一句话总结

该论文提出了一种基于SAT的新型压缩感知方法，通过将NP难的稀疏恢复问题转化为加权-MAX-SAT问题，实现了在比ℓ1-最小化更少测量次数下的精确信号恢复。该方法优于传统的ℓ1-基恢复方法，在高度稀疏的二值信号（s/N ≈ 0.1）下测量次数减少8.3%，且在高稀疏度（s/N ≈ 0.5）下仍保持鲁棒性，此时ℓ1-范数需多出22.2%的测量次数。

ABSTRACT

We propose to reduce the original well-posed problem of compressive sensing to weighted-MAX-SAT. Compressive sensing is a novel randomized data acquisition approach that linearly samples sparse or compressible signals at a rate much below the Nyquist-Shannon sampling rate. The original problem of compressive sensing in sparse recovery is NP-hard; therefore, in addition to restrictions for the uniqueness of the sparse solution, the coding matrix has also to satisfy additional stringent constraints -usually the restricted isometry property (RIP)- so we can handle it by its convex or nonconvex relaxations. In practice, such constraints are not only intractable to be verified but also invalid in broad applications. We first divide the well-posed problem of compressive sensing into relaxed sub-problems and represent them as separate SAT instances in conjunctive normal form (CNF). After merging the resulting sub-problems, we assign weights to all clauses in such a way that the aggregated weighted-MAX-SAT can guarantee successful recovery of the original signal. The only requirement in our approach is the solution uniqueness of the associated problems, which is notably looser. As a proof of concept, we demonstrate the applicability of our approach in tackling the original problem of binary compressive sensing with binary design matrices. Experimental results demonstrate the supremacy of the proposed SAT-based compressive sensing over the ℓ₁-minimization in the robust recovery of sparse binary signals. SAT-based compressive sensing on average requires 8.3% fewer measurements for exact recovery of highly sparse binary signals (s/N≈0.1). When s/N≈0.5, the ℓ₁-minimization on average requires 22.2% more measurements for exact reconstruction of the binary signals. Thus, the proposed SAT-based compressive sensing is less sensitive to the sparsity of the original signals.

研究动机与目标

• 为解决原始压缩感知问题的不可解性，该问题由于ℓ0-最小化而为NP难问题。
• 克服现有方法依赖严格约束（如限制等距性质RIP）的局限性，这些约束难以验证且在实际应用中常不成立。
• 通过将稀疏恢复问题表述为加权-MAX-SAT实例，建立现代SAT求解器与压缩感知之间的桥梁。
• 证明解的唯一性——而非RIP——已足够实现成功恢复，从而实现更广泛的应用。

提出的方法

• 将原始压缩感知问题分解为两个松弛的子问题：一个用于求解欠定系统Ax = y，另一个用于寻找最稀疏解。
• 每个子问题被编码为一个独立的布尔SAT实例，采用合取范式（CNF）表示。
• 使用逻辑与运算符将两个CNF实例合并，形成一个更大的单一CNF公式。
• 在合并后的CNF中为子句分配权重，使得该加权-MAX-SAT问题的最优解恰好对应于原始稀疏信号。
• 该方法仅依赖于原始问题解的唯一性，避免了对RIP等严格约束的依赖。
• 使用Z3定理证明器进行符号计算，该方法在二值压缩感知及二值设计矩阵上进行了评估。

实验结果

研究问题

• RQ1 NP难的压缩感知稀疏恢复问题能否被有效重构为加权-MAX-SAT问题？
• RQ2 基于SAT的方法在测量减少和误差鲁棒性方面是否优于ℓ1-最小化？
• RQ3 当s/N增加时，基于SAT的方法在不同稀疏度水平下的表现如何？
• RQ4 该方法是否能在不依赖RIP等严格约束的条件下成功运行？
• RQ5 与传统的ℓ1-基恢复相比，基于SAT的方法在高稀疏度下是否更具鲁棒性？

主要发现

• 对于稀疏度s/N ≈ 0.1的高度稀疏二值信号，基于SAT的压缩感知方法在实现精确恢复时比ℓ1-最小化减少8.3%的测量次数。
• 当稀疏度提升至s/N ≈ 0.5时，ℓ1-最小化平均需多出22.2%的测量次数才能实现精确重构，而基于SAT的方法仍保持鲁棒性。
• 在平均恢复误差方面，基于SAT的方法优于ℓ1-最小化，尤其在存在噪声或高稀疏度的场景下表现更优。
• 该方法对稀疏率的敏感度低于ℓ1-最小化，在不同信号密度下表现出更强的鲁棒性。
• 该方法仅基于解唯一性的假设即可实现精确恢复，这一假设远比传统方法所需的RIP或其他结构约束更为宽松。
• 实验结果证实，基于SAT的方法在可行性与性能上均更优，尽管由于符号计算导致计算资源需求更高。