Exact recovery of $K$-sparse signals ${\boldsymbol{x}}\in \mathbb{R}^{n}$ from linear measurements ${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}$, where $\boldsymbol{A}\in \mathbb {R}^{m\times n}$ is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely used for reconstructing ${\boldsymbol{x}}$ based on ${\boldsymbol{y}}$ and $\boldsymbol{A}$ due to its excellent recovery performance and high efficiency. A fundamental question in the performance analysis of OMP is the characterizations of the probability of exact recovery of ${\boldsymbol{x}}$ for random matrix $\boldsymbol{A}$ and the minimal $m$ to guarantee a target recovery performance. In many practical applications, in addition to sparsity, ${\boldsymbol{x}}$ also has some additional properties (for example, the nonzero entries of ${\boldsymbol{x}}$ independently and identically follow a Gaussian distribution, or ${\boldsymbol{x}}$ has exponentially decaying property). This paper shows that these properties can be used to refine the answer to the above question. Toward this end, we first show that the prior information of the nonzero entries of ${\boldsymbol{x}}$ can be used to provide an upper bound on $\Vert {\boldsymbol{x}}\Vert _1^2/\Vert {\boldsymbol{x}}\Vert _2^2$. Then, we use this upper bound to develop a lower bound on the probability of exact recovery of ${\boldsymbol{x}}$ using OMP in $K$ iterations. Furthermore, we develop a lower bound on the number of measurements $m$ to guarantee that the exact recovery probability using $K$ iterations of OMP is no smaller than a given target probability. Finally, we show that when $K=O(\sqrt{\ln n})$, as both $n$ and $K$ go to infinity, for any $0< \zeta \leq 1/\sqrt{\pi }$, $m=2K\ln (n/\zeta)$ measurements are sufficient to ensure that the probability of exact recovering any $K$-sparse ${\boldsymbol{x}}$ is no lower than $1-\zeta$ with $K$ iterations of OMP. This improves the $m=4K\ln (2n/\zeta)$ result of Tropp et al. For $K$-sparse $\alpha$-strongly decaying signals and for $K$-sparse ${\boldsymbol{x}}$ whose nonzero entries independently and identically follow the Gaussian distribution, the number of measurements sufficient for exact recovery with probability no lower than $1-\zeta$ reduces further to $m=(\sqrt{K}+4\sqrt{\frac{\alpha +1}{\alpha -1}\ln (n/\zeta)})^2$ and to asymptotically $m\approx 1.9K\ln (n/\zeta)$, respectively.

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用于精确稀疏恢复的正交匹配追踪的信号相关性能分析

准确恢复 $K$-稀疏信号 ${\boldsymbol{x}}\in \mathbb{R}^{n}$ 从线性测量 ${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}$， 在哪里 $\boldsymbol{A}\in \mathbb {R}^{m\times n}$是一个传感矩阵，产生于许多应用。正交匹配追踪（OMP）算法被广泛用于重建${\boldsymbol{x}}$ 基于 ${\boldsymbol{y}}$ 和 $\boldsymbol{A}$由于其优良的回收性能和高效率。OMP 性能分析中的一个基本问题是精确恢复的概率的表征${\boldsymbol{x}}$ 对于随机矩阵 $\boldsymbol{A}$ 和最小的 百万美元以保证目标恢复性能。在许多实际应用中，除了稀疏性，${\boldsymbol{x}}$ 也有一些额外的属性（例如，非零条目 ${\boldsymbol{x}}$ 独立且相同地服从高斯分布，或 ${\boldsymbol{x}}$具有指数衰减特性）。这篇论文表明，这些属性可以用来完善上述问题的答案。为此，我们首先证明非零项的先验信息${\boldsymbol{x}}$ 可用于提供上限 $\Vert {\boldsymbol{x}}\Vert _1^2/\Vert {\boldsymbol{x}}\Vert _2^2$. 然后，我们使用这个上限来制定准确恢复概率的下限${\boldsymbol{x}}$ 使用 OMP $K$迭代。此外，我们制定了测量次数的下限百万美元 以保证使用的确切恢复概率 $K$OMP 的迭代次数不小于给定的目标概率。最后，我们证明当$K=O(\sqrt{\ln n})$， 既 $n$ 和 $K$ 去无穷大，对于任何 $0< \zeta \leq 1/\sqrt{\pi }$, $m=2K\ln (n/\zeta)$ 测量足以确保准确恢复任何 $K$-疏 ${\boldsymbol{x}}$ 不低于 $1-\zeta$ 和 $K$OMP 的迭代。这改善了$m=4K\ln (2n/\zeta)$ 特罗普的结果 等. 为了$K$-疏 $\alpha$- 强衰减信号和 $K$-疏 ${\boldsymbol{x}}$ 其非零项独立且完全遵循高斯分布，测量次数足以准确恢复，概率不低于 $1-\zeta$ 进一步减少到 $m=(\sqrt{K}+4\sqrt{\frac{\alpha +1}{\alpha -1}\ln (n/\zeta)})^2$ 并且渐近地 $m\approx 1.9K\ln (n/\zeta)$， 分别。