The null space condition for ℓ1 minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via ℓ1 minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on these techniques to find upper bounds and further develop a new procedure-tree search algorithm-that is able to precisely and quickly verify the null space condition. Numerical experiments show that the execution speed and accuracy of the results obtained from our methods far exceed those of the previous methods which rely on linear programming (LP) relaxation and semidefinite programming (SDP).

中文翻译：

---

可压缩压缩矩阵的可计算性能保证。

压缩传感中将ℓ1最小化的零空间条件是在传感矩阵上的充要条件，在这种条件下，可以通过via1最小化从观测数据中唯一恢复稀疏信号。但是，验证空空间条件是已知的计算难题。大多数现有方法只能在表征空空间条件的比例参数上提供上限和下限。在本文中，我们提出了新的多项式时间算法来建立比例参数的上限。我们利用这些技术来找到上限，并进一步开发新的过程树搜索算法，该算法能够精确快速地验证空空间条件。