[论文解读] Separation-Free Super-Resolution from Compressed Measurements is Possible: an Orthonormal Atomic Norm Minimization Approach

Weiyu Xu, Jirong Yi+4 more|arXiv (Cornell University)|Nov 4, 2017

Sparse and Compressive Sensing Techniques参考文献 25被引用 7

一句话总结

本文提出正交原子范数最小化（OANM），以实现从压缩的、非均匀时域采样中无需分离的R个复指数信号的超分辨重建。通过利用Hankel矩阵的核范数作为正交原子范数，该方法在任意频率间隔下均能实现精确恢复，克服了传统原子范数和总变差最小化方法在无噪声条件下仍需最小频率间隔的根本限制。

ABSTRACT

We consider the problem of recovering the superposition of $R$ distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the $R$ frequencies or the missing data. However, it is known that in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying $R$ frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the $R$ complex exponentials and their frequencies from compressed non-uniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values $ extbf{must}$ be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close. As a byproduct of this research, we provide one matrix-theoretic inequality of nuclear norm, and give its proof from the theory of compressed sensing.

研究动机与目标

• 解决现有超分辨方法在精确恢复时要求信号频率之间存在最小间隔的根本限制。
• 构建一种压缩感知框架，从非均匀、欠定的时域样本中恢复谱稀疏信号，而无需依赖频率间隔。
• 基于正交原子范数建立新的理论基础，实现对频率任意接近的复指数信号的精确超分辨。
• 证明Hankel矩阵核范数最小化之所以能实现无分离恢复，是因为其本质上是正交原子范数。

提出的方法

• 提出正交原子范数最小化（OANM）作为超分辨的新框架，其中原子范数通过基于信号结构的正交基向量定义。
• 证明由信号向量构造的Hankel矩阵的核范数等价于正交原子范数，从而实现从压缩测量中恢复信号。
• 利用压缩感知理论导出的核范数矩阵不等式，建立在低秩Hankel结构下的恢复保证。
• 证明精确恢复的对偶证书条件即使在频率任意接近时仍可满足，这与标准原子范数最小化形成对比。
• 采用提升技术将复值超分辨问题转化为实对称矩阵优化问题，从而可应用凸松弛技术。
• 证明对偶证书以满足特定零空间和谱范数约束的矩阵形式存在，确保精确恢复。

实验结果

研究问题

• RQ1 能否在无需频率间隔要求的前提下，从压缩的、非均匀测量中实现超分辨？
• RQ2 是否存在一类原子范数，使得即使原子之间距离任意接近，也能实现精确恢复？
• RQ3 为何Hankel矩阵核范数最小化能实现无分离超分辨，而标准原子范数最小化却失败？
• RQ4 Hankel矩阵的核范数能否被解释为正交原子范数？其对恢复保证意味着什么？

主要发现

• Hankel矩阵的核范数是正交原子范数，可实现从压缩测量中无需频率间隔的精确超分辨恢复。
• OANM即使在频率任意接近时仍能恢复R个复指数信号，而标准原子范数最小化要求最小间隔为2/(2N-1)。
• 在测量算子满足特定非相干性和秩恢复条件的前提下，方法可从M < 2N−1个非均匀时域样本中实现精确恢复。
• 推导出一个新的核范数矩阵不等式，该不等式通过压缩感知工具证明，是建立恢复条件的关键。
• 对偶证书存在且可显式构造，即使在无频率间隔时，通过满足对偶矩阵的零空间和谱范数约束，可确保精确恢复。
• 理论分析证实，通过核范数最小化实现的Hankel矩阵恢复等价于OANM，而这种等价性正是实现无分离恢复特性的根源。