In many applications we seek to recover signals from linear measurements far fewer than the ambient dimension, given the signals have exploitable structures such as sparse vectors or low rank matrices. In this paper we work in a general setting where signals are approximately sparse in an so called atomic set. We provide general recovery results stating that a convex programming can stably and robustly recover signals if the null space of the sensing map satisfies certain properties. Moreover, we argue that such null space property can be satisfied with high probability if each measurement is subgaussian even when the number of measurements are very few. Some new results for recovering signals sparse in a frame, and recovering low rank matrices are also derived as a result.

翻译：在许多应用中,我们寻求从线性测量中恢复信号,远远少于环境层面,因为信号具有可利用的结构,如稀有的矢量或低级矩阵。在本文件中,我们在一个总的环境中工作,在所谓的原子集中,信号几乎很少。我们提供了一般的恢复结果,指出如果遥感地图的空格满足某些特性,曲线编程可以刺死和有力地恢复信号。此外,我们争辩说,如果每次测量都是低水平的,即使测量数量非常少,这种空域财产可以满足很高的概率。恢复一个框架所缺少的信号的一些新结果,因此也得出了恢复低级矩阵的结果。