Separation-free super-resolution from compressed measurements is possible: an orthonormal atomic norm minimization approach
Separation-free super-resolution from compressed measurements is possible: an orthonormal atomic norm minimization approach
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压缩测量的无分离超分辨率是可能的：正交原子范数最小化方法
DOI：
10.1093/imaiai/iaad033
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Xu, Weiyu
中科院分区：
文献类型：
--
作者：
Yi, Jirong;Dasgupta, Soura;Cai, Jian-Feng;Jacob, Mathews;Gao, Jingchao;Cho, Myung;Xu, Weiyu
本文研究了从压缩的非均匀时域样本中恢复不同复指数函数的叠加的问题。文献中提出了全变差（TV）最小化或原子范数最小化来恢复频率或缺失数据。然而，众所周知，为了使TV最小化和原子范数最小化来恢复丢失的数据或频率，即使当测量是无噪声的时，也需要很好地分离基础频率。本文证明了Hankel矩阵恢复方法可以从压缩的非均匀测量中超分辨出复杂的指数及其频率，而不管它们的频率彼此有多接近。提出了一种新的正交原子范数最小化（OANM）概念，并证明了在无分离超分辨中Hankel矩阵恢复的成功来自于Hankel矩阵的核范数是正交原子范数这一事实.更具体地说，我们表明，在传统的原子范数最小化，底层的参数valuesmustbe很好地分离，以实现成功的信号恢复，如果原子不断变化的连续值参数。相反，对于OANM，即使原始原子可以任意接近，OANM也可能是成功的。作为研究的副产品，我们给出了一个核范数的矩阵不等式，并利用压缩感知理论给出了证明。
We consider the problem of recovering the superposition ofdistinct 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 thefrequencies 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 underlyingfrequencies are required to be well separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve thecomplex 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 valuesmustbe 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 using the theory of compressed sensing.