[论文解读] Thresholded Basis Pursuit: Quantizing Linear Programming Solutions for Optimal Support Recovery and Approximation in Compressed Sensing

Venkatesh Saligrama, Manqi Zhao|arXiv (Cornell University)|Sep 29, 2008

Sparse and Compressive Sensing Techniques被引用 20

一句话总结

本文提出了一种阈值化基追踪方法，通过求解线性规划来从含噪测量中恢复稀疏信号，然后对解进行阈值化处理以实现精确支持恢复。理论证明表明，当信噪比 SNR = O(log n) 且测量数 m = O(k) 时，若稀疏度 k 随信号维度 n 线性增长，则可保证完美重建，该方法在信噪比和稀疏度要求方面优于先前的 LASSO 和 MAX-Correlation 方法。

ABSTRACT

We consider the Compressed Sensing problem. We have a large under-determined set of noisy measurements Y = GX + N, where X is a sparse signal and G is drawn from a random ensemble. In our previous work, we had shown that a signal-to-noise ratio, SNR = O(log n) is necessary and sufficient for support recovery from an information-theoretic perspective. In this paper we present a linear programming solution for support recovery. The solution of the problem amounts to solving min ‖Z‖1 s.t. Y = GZ, and quantizing/thresholding the resulting solution Z. We show that this scheme is guaranteed to perfectly reconstruct a discrete signal or control the element-wise reconstruction error for a continuous signal for specific values of sparsity. We show that in the linear regime when the sparsity, k, increases linearly with signal dimension, n, the sign pattern of X can be recovered with SNR = O(log n) and m = O(k) measurements. Our proof technique is based on perturbation of the noiseless ℓ1 problem. Consequently, the achievable sparsity level in the noisy problem is comparable to that of the noiseless problem. Our result offers a sharp characterization in that neither the SNR nor the sparsity ratio can be significantly improved. In contrast previous results based on LASSO and MAX-Correlation techniques assume significantly larger SNR or sub-linear sparsity. We also show that our final result can be obtained from Dvoretsky theorem rather than the restricted isometry property (RIP). The advantage of this line of reasoning is that Dvoretsky’s theorem continues to hold for non-singular transformations while RIP property may not be satisfied for the latter case. We also consider approximation in terms of ℓ2 and show that our bounds match existing bounds for LASSO in this case.

研究动机与目标

• 解决在噪声和欠定测量条件下压缩感知中可靠支持恢复的挑战。
• 开发一种在稀疏信号恢复中实现最优信噪比（SNR）和测量要求的方法。
• 对支持恢复中 SNR、稀疏度与测量数之间的权衡关系提供精确刻画。
• 证明该方法在稀疏度线性增长（k = Θ(n)）时，仅需最低信噪比即可工作，优于先前的 LASSO 和 MAX-Correlation 方法。
• 使用 Dvoretzky 定理替代受限等距性质（RIP）建立理论保证，从而扩展至非奇异变换的适用范围。

提出的方法

• 求解标准基追踪问题：min ‖Z‖₁ 受限于 Y = GZ，以获得初始稀疏估计。
• 对解 Z 应用阈值化操作，以提取原始信号 X 的支持。
• 利用无噪 ℓ₁ 问题的扰动分析，界定在噪声存在下的误差。
• 通过分析在稀疏度线性增长和 O(log n) 信噪比下的符号模式恢复，建立恢复保证。
• 以 Dvoretzky 定理替代对受限等距性质（RIP）的依赖，该定理在更广泛条件下成立，包括非奇异变换。
• 分析 ℓ₂ 近似误差，并证明其界与该情形下 LASSO 的最佳已知界一致。

实验结果

研究问题

• RQ1 基于线性规划并结合阈值化的算法，能否在最小信噪比和测量约束下实现完美支持恢复？
• RQ2 当稀疏度随信号维度线性增长时，可靠支持恢复所需的最小信噪比是多少？
• RQ3 该方法在信噪比和稀疏度要求方面与 LASSO 和 MAX-Correlation 技术相比表现如何？
• RQ4 能否以 Dvoretzky 定理为基础替代受限等距性质（RIP）来建立恢复保证？
• RQ5 该方法的 ℓ₂ 近似界在多大程度上与 LASSO 的界一致？

主要发现

• 当稀疏度 k 随信号维度 n 线性增长时，该方法在 SNR = O(log n) 且 m = O(k) 测量数下可实现完美支持恢复。
• 在相同信噪比和测量数条件下，该方案可保证精确符号模式恢复，达到信息论下限。
• 该方法优于 LASSO 和 MAX-Correlation 技术，后者需显著更高的信噪比或次线性稀疏度才能获得类似保证。
• 使用 Dvoretzky 定理替代 RIP 可确保在 RIP 可能失效的非奇异变换下仍具鲁棒性。
• 对于 ℓ₂ 近似，该方法的误差界与 LASSO 在相同设置下的最佳已知界一致。
• 理论分析表明，信噪比和稀疏度比均无法显著改进，表明对恢复极限的刻画是紧致的。