论文标题
分析$ \ ell_1 $和$ \ ell_2 $ norms in Compressed Sensing的比率分析
Analysis of The Ratio of $\ell_1$ and $\ell_2$ Norms in Compressed Sensing
论文作者
论文摘要
我们首先提出了一个新颖的标准，可以保证$ s -sparse信号是$ \ ell_1/\ ell_2 $ objective的本地最小化器；我们的标准在实践中是可解释的，并且有用。我们还使用测量矩阵的空空间的几何表征给出了第一个均匀的恢复条件，并证明对于一类随机矩阵很容易满足此条件。当噪声污染数据时，我们还对程序的鲁棒性进行了分析。提供了数值实验，可以将$ \ ell_1/\ ell_2 $与压缩传感中的其他一些流行的非凸方法进行比较。最后，我们提出了一种新的初始化方法，以加速数值优化程序。我们称这种初始化方法\ emph {支持选择}，并证明它从经验上改善了现有$ \ ell_1/\ ell_2 $算法的性能。
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.
