压缩感知中$\ell_1$和$\ell_2$范数比值的分析
Analysis of The Ratio of $\ell_1$ and $\ell_2$ Norms in Compressed Sensing
摘要：
我们首先提出了一个新的标准，保证$s$稀疏信号是局部极小的$\ell_1/\ell_2$目标，我们的标准是可解释的，在实践中是有用的。利用测量矩阵的零空间的几何特征，给出了第一个一致恢复条件，并证明了这一条件对于一类随机矩阵是容易满足的.我们还分析了噪声污染数据时该过程的鲁棒性。数值实验提供比较$\ell_1/\ell_2$与其他一些流行的非凸方法在压缩感知。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。我们称之为这种初始化方法\emph{support selection}，我们证明，它经验性地提高了现有的$\ell_1/\ell_2$算法的性能。
英文摘要：
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 easily 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 \emph{support selection}, and we demonstrate that it empirically improves the performance of existing $\ell_1/\ell_2$ algorithms.
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