We study the ratio of ${\ell }_1$ and ${\ell }_2$ norms (${\ell }_1/{\ell }_2$) as a sparsity-promoting objective in compressed sensing. We first propose a novel criterion that guarantees that an s-sparse signal is the local minimizer of the ${\ell }_1/{\ell }_2$ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is satisfied for a class of random matrices. We also present analysis on the robustness of the procedure when noise pollutes data. Numerical experiments are provided that compare ${\ell }_1/{\ell }_2$ with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach support selection, and we demonstrate that it empirically improves the performance of existing ${\ell }_1/{\ell }_2$ algorithms.

我们研究的比率 ${\ell }_1$ 和 ${\ell }_2$ 规范（${\ell }_1/{\ell }_2$) 作为压缩感知中的稀疏促进目标。我们首先提出了一个新的标准，保证一个s稀疏信号是${\ell }_1/{\ell }_2$客观的; 我们的标准在实践中是可以解释和有用的。我们还使用测量矩阵的零空间的几何特征给出了第一个均匀恢复条件，并表明该条件对于一类随机矩阵是满足的。我们还对噪声污染数据时程序的稳健性进行了分析。提供了数值实验来比较${\ell }_1/{\ell }_2$与压缩感知中其他一些流行的非凸方法。最后，我们提出了一种新的初始化方法来加速数值优化过程。我们称这种初始化方法支持选择，并且我们证明它凭经验提高了现有的性能。${\ell }_1/{\ell }_2$ 算法。