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，我们旨在从远低于环境维度的线性测量中恢复信号，前提是这些信号具有可开发的结构，例如稀疏向量或低秩矩阵。本文在广义框架下工作，其中信号在所谓的原子集中近似稀疏。我们提供了一般性的恢复结果，指出若感知映射的零空间满足特定性质，则凸规划能够稳定且鲁棒地恢复信号。此外，我们论证了当每个测量值呈次高斯分布时，即使测量次数极少，该零空间性质也能以高概率成立。由此，我们还推导出关于帧稀疏信号恢复及低秩矩阵恢复的一些新结果。