[论文解读] Exact Joint Sparse Frequency Recovery via Optimization Methods

Zai Yang, Lihua Xie|arXiv (Cornell University)|May 26, 2014

Sparse and Compressive Sensing Techniques参考文献 55被引用 38

一句话总结

该论文提出了一种基于原子范数的凸优化方法，用于从多个测量向量（MMVs）中精确恢复共享稀疏频率分量的连续频率，利用信号间共享的频率成分。该方法在理论条件下可实现精确恢复，减少所需测量数或放宽频率分离约束，在相关或噪声环境下优于传统方法（如MUSIC）。

ABSTRACT

Frequency recovery/estimation from discrete samples of superimposed sinusoidal signals is a classic yet important problem in statistical signal processing. Its research has recently been advanced by atomic norm techniques which exploit signal sparsity, work directly on continuous frequencies, and completely resolve the grid mismatch problem of previous compressed sensing methods. In this work we investigate the frequency recovery problem in the presence of multiple measurement vectors (MMVs) which share the same frequency components, termed as joint sparse frequency recovery and arising naturally from array processing applications. To study the advantage of MMVs, we first propose an $\ell_{2,0}$ norm like approach by exploiting joint sparsity and show that the number of recoverable frequencies can be increased except in a trivial case. While the resulting optimization problem is shown to be rank minimization that cannot be practically solved, we then propose an MMV atomic norm approach that is a convex relaxation and can be viewed as a continuous counterpart of the $\ell_{2,1}$ norm method. We show that this MMV atomic norm approach can be solved by semidefinite programming. We also provide theoretical results showing that the frequencies can be exactly recovered under appropriate conditions. The above results either extend the MMV compressed sensing results from the discrete to the continuous setting or extend the recent super-resolution and continuous compressed sensing framework from the single to the multiple measurement vectors case. Extensive simulation results are provided to validate our theoretical findings and they also imply that the proposed MMV atomic norm approach can improve the performance in terms of reduced number of required measurements and/or relaxed frequency separation condition.

研究动机与目标

• 解决多个测量向量（MMVs）共享相同稀疏频率分量时的精确频率恢复挑战。
• 通过直接在连续频率域工作，克服离散压缩感知中固有的网格失配问题。
• 利用MMVs之间的联合稀疏性提升恢复性能，减少所需测量数或放宽频率分离条件。
• 通过原子范数框架构建非凸联合稀疏问题的凸松弛，实现高效的半定规划求解。
• 在适当条件下提供精确频率恢复的理论保证，拓展了MMV压缩感知与连续超分辨框架。

提出的方法

• 提出一种类似$β_{2,0}$-范数的优化方法，以利用MMVs中的联合稀疏性，但该方法为非凸且求解为NP难。
• 引入一种基于原子范数的连续频率恢复凸松弛方法，称为MMV原子范数方法。
• 将恢复问题表述为在数据保真约束下最小化原子范数，可通过半定规划求解。
• 使用半定规划（SDP）松弛高效求解原子范数最小化问题。
• 在噪声场景下，将方法扩展为约束优化：在观测数据的Frobenius范数误差边界内最小化原子范数。
• 将所提方法与传统MUSIC及SMV原子范数方法进行比较，评估其在相干与噪声环境下的性能。

实验结果

研究问题

• RQ1 与单测量向量（SMV）方法相比，多个测量向量中的联合稀疏性是否能提升连续频率的精确恢复？
• RQ2 所提原子范数方法在MMV设置下，精确频率恢复的理论条件是什么？
• RQ3 在传统子空间方法（如MUSIC）失效的相干源或噪声环境中，所提方法表现如何？
• RQ4 通过MMV原子范数方法，可在多大程度上减少所需测量数或放宽频率分离条件？
• RQ5 在鲁棒性与虚假分量抑制方面，所提方法与MUSIC及SMV原子范数方法相比表现如何？

主要发现

• 所提MMV原子范数方法在适当条件下可实现精确频率恢复，将连续压缩感知框架拓展至多测量向量情形。
• 仿真结果表明，与SMV方法相比，该方法可减少所需测量数并放宽频率分离条件。
• 在存在相干源的噪声情况下，所提方法成功恢复了全部三个频率分量，而MUSIC无法分辨相干源。
• ANM方法中出现虚假频率分量，但其功率可忽略不计——SMV情况下约为总功率的0.4%，MMV情况下约为$10^{-6}$量级。
• 在噪声场景下，每道问题的计算时间约为1.5秒；在无噪声仿真中，每道问题约需13秒，大规模仿真总计算时间约为200小时。
• 理论分析表明，即使在理论上限$K = \frac{1}{2}(M+L)$以上，仍可实现成功恢复，表明其性能显著优于最坏情况边界。