[论文解读] Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

Yonina C. Eldar, Holger Rauhut|ArXiv.org|Apr 3, 2009

Sparse and Compressive Sensing Techniques参考文献 38被引用 4

一句话总结

本文对使用混合 $\ell_{2,1}$ 范数的多通道稀疏恢复的平均情况分析表明，在稀疏性和字典一致性的温和条件下，恢复失败的概率随通道数呈指数衰减。这表明联合恢复通常优于单通道方法，即使信号数量较少时也是如此。

ABSTRACT

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed l_2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

研究动机与目标

• 分析凸松弛方法在多通道稀疏恢复中的平均情况性能，超越最坏情况的保证。
• 解释为何在实践中联合稀疏恢复优于单通道恢复，尽管在最坏情况设置下理论等价。
• 推导使用混合 $\ell_{2,1}$ 范数的多通道恢复失败概率的紧致、非渐近界。
• 扩展并收紧阈值法和同时正交匹配追踪（SOMP）的现有平均情况性能边界。

提出的方法

• 在信号向量上使用概率模型，分析期望情况下的恢复性能，而非对所有信号的最坏情况。
• 应用测度集中不等式，以界定 SOMP 中残差选择错误索引的概率。
• 采用混合 $\ell_{2,1}$ 范数最小化作为多通道稀疏恢复的凸松弛方法。
• 利用测量矩阵的一致性及子矩阵的谱性质，推导失败概率的界。
• 使用高斯向量的期望范数 $C_2(L) = \mathbb{E}\|Z\|_2$，量化残差选择步骤中的信噪比。
• 对所有可能的支持集应用并集界，以确保 SOMP 所有迭代中均能成功恢复。

实验结果

研究问题

• RQ1 在何种条件下，多通道稀疏恢复的混合 $\ell_{2,1}$ 松弛在平均情况下以高概率成功？
• RQ2 为何在实践中联合稀疏恢复优于单通道恢复，尽管在理论最坏情况设置下二者等价？
• RQ3 在随机信号模型下，SOMP 的失败概率如何随通道数衰减？
• RQ4 能否使用相同的概率框架，为阈值法和 SOMP 推导出更紧致的平均情况边界？
• RQ5 字典的一致性在决定多通道恢复成功率方面起什么作用？

主要发现

• 即使在稀疏性和一致性条件较温和的情况下，混合 $\ell_{2,1}$ 松弛的恢复失败概率也随通道数呈指数衰减。
• 对于固定数量的通道，失败概率被界为 $ (|S^c| + 1) \exp(-\epsilon^2 A_L^2) $，其中 $ A_L $ 依赖于每通道的测量数。
• 分析表明，联合恢复通常优于单通道恢复，因为在随机模型下最坏情况信号实例出现的可能性极低。
• 与先前工作相比，SOMP 和阈值法的推导边界更为紧致，提供了更精确的性能预测。
• 条件 $ (1+\epsilon)\frac{\mu_2(S)}{1-\delta(S)} \geq (1-\epsilon)\left(1 - \frac{\mu_2(S)^2}{1-\delta(S)}\right) $ 确保了 SOMP 以高概率成功恢复支持集。
• 即使通道数较少，结果依然有效且具有实际意义，使其在现实应用中具有实际相关性。