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Incoherence is Sufficient for Statistical RIP of Unit Norm Tight Frames: Constructions and Properties
IEEE Transactions on Signal Processing ( IF 5.4 ) Pub Date : 2021-03-18 , DOI: 10.1109/tsp.2021.3066777
Pradip Sasmal , Chandra R. Murthy

In this work, we show that the incoherence alone is sufficient to establish the statistical restricted isometry property (StRIP) and statistical incoherence property (SInCoP) for unit norm tight frames (UNTFs). Further, we derive three simple properties that binary matrices need to satisfy, in order to produce incoherent UNTFs (IUNTFs) with high redundancy (ratio of the number of columns to the number of rows) via an existing embedding operation. We show that biadjacency matrices corresponding to biregular graphs satisfy the required properties. Thereby, we provide a connection between graph theory and the construction of IUNTFs. We also provide a bouquet of constructions of IUNTFs from finite fields and combinatorial designs. Another important aspect of our construction is that the sparse recovery guarantees for the embedded IUNTFs can in fact be translated to the constituent binary matrix. We show that if the constituent $m\times M$ binary matrix has constant row and column weight, it can support sparse recovery through $\ell_{1}$ -minimization for all but an $\epsilon$ -fraction of $t$ -sparse signals chosen from a random signal model, provided $m=O(t(\log(\frac{M}{\epsilon}))^{3}),$ which is a significant improvement over the existing $m=O(t^{2})$ bound, where $m$ denotes the number of measurements. Also, the StRIP and SInCoP based approach results in matrices whose column size is exponential in the fourth root of the row size. To the best of our knowledge, this is the first construction of deterministic matrices satisfying StRIP and SInCoP with such high redundancy.

中文翻译:

不一致性足以满足单位规范紧框架的统计RIP:结构和属性

在这项工作中,我们证明了单独的不相干性足以建立单位规范紧密框架(UNTF)的统计受限等距特性(StRIP)和统计不相干特性(SInCoP)。此外,我们得出了二进制矩阵需要满足的三个简单属性,以便通过现有的嵌入操作产生具有高冗余度(列数与行数之比)的非相干UNTF(IUNTF)。我们表明,对应于双正则图的双性矩阵满足所需的属性。因此,我们提供了图论与IUNTF构造之间的联系。我们还提供了来自有限域和组合设计的大量IUNTF构造。我们构建的另一个重要方面是,嵌入式IUNTF的稀疏恢复保证实际上可以转换为组成二进制矩阵。我们证明如果$ m \次M $ 二进制矩阵具有恒定的行和列权重,可以通过以下方式支持稀疏恢复 $ \ ell_ {1} $ -除了 $ \ epsilon $ 的分数 $ t $ -从随机信号模型中选择的稀疏信号 $ m = O(t(\ log(\ frac {M} {\ epsilon})^ {3}),$ 这是对现有技术的重大改进 $ m = O(t ^ {2})$ 绑定,在哪里 $ m $表示测量次数。同样,基于StRIP和SInCoP的方法会导致矩阵的列大小在行大小的第四根中呈指数形式。据我们所知,这是具有如此高冗余性的满足StRIP和SInCoP的确定性矩阵的第一个构造。
更新日期:2021-04-30
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