随机正交矩阵串接表示的稀疏向量的典型$l_1$ -恢复极限
Typical $l_1$-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices
摘要：
我们考虑从$M(< N)$维度的线性变换$\vm{y}=\vm{D} \vm{x}$恢复$N$维度稀疏向量$\vm{x}$的问题。在约束$\vm{y} = \vm{D} \vm{x}$下最小化$\vm{x}$的$l_{1}$ -范数是恢复问题的标准方法，早期的研究报告指出，通常成功的$l_1$ -恢复的临界条件在各种随机构造的矩阵$\vm{D}$上是通用的。为了检查的普遍性的程度，我们专注于$\vm{D}$的情况下，其中$\vm{D}$是通过连接$\nb=N/M$矩阵$\vm{O}_{1}, \vm{O}_{2},..., \vm{O}_\nb$均匀绘制根据Haar措施上的$M \times M$正交矩阵。通过使用副本的方法结合发展的一个积分公式处理的随机正交矩阵，我们表明，级联矩阵可以导致更好的恢复性能比普遍性预测时，非零信号的密度是不均匀的$\nb$矩阵模块。对于均匀非零信号密度的特殊情况，再现了普遍条件。大量的数值实验支持理论预测。
英文摘要：
We consider the problem of recovering an $N$-dimensional sparse vector $\vm{x}$ from its linear transformation $\vm{y}=\vm{D} \vm{x}$ of $M(< N)$ dimension. Minimizing the $l_{1}$-norm of $\vm{x}$ under the constraint $\vm{y} = \vm{D} \vm{x}$ is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful $l_1$-recovery is universal over a variety of randomly constructed matrices $\vm{D}$. For examining the extent of the universality, we focus on the case in which $\vm{D}$ is provided by concatenating $\nb=N/M$ matrices $\vm{O}_{1}, \vm{O}_{2},..., \vm{O}_\nb$ drawn uniformly according to the Haar measure on the $M \times M$ orthogonal matrices. By using the replica method in conjunction with the development of an integral formula for handling the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than what the universality predicts when the density of non-zero signals is not uniform among the $\nb$ matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.
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