Sparse signal recovery deals with finding the sparsest solution of an under-determined linear system $\vx = \mQ\vs$. In this paper, we propose a novel greedy approach to addressing the challenges from such a problem. Such an approach is based on a characterization of solutions to the system, which allows us to work on the sparse recovery in the $\vs$-space directly with a given measure. With $l_2$-based measure, an orthogonal matching pursuit (OMP)-type algorithm is proposed, which significantly outperforms the classical OMP algorithm in terms of recovery accuracy while maintaining comparable computational complexity. An $l_1$-based algorithm, denoted as $\text{Alg}_{GL1}$, is derived. Such an algorithm significantly outperforms the classical basis pursuit (BP) algorithm. Combining with the CoSaMP-strategy for selecting atoms, a class of high performance greedy algorithms is also derived. Extensive numerical simulations on both synthetic and image data are carried out, with which the superior performance of our proposed algorithms is demonstrated in terms of sparse recovery accuracy and robustness against numerical instability of the system matrix $\mQ$ and disturbance in the measurement $\vx$.

翻译：稀疏信号恢复旨在寻找欠定线性系统 $\vx = \mQ\vs$ 的最稀疏解。本文提出了一种新颖的贪婪方法来解决该问题带来的挑战。该方法基于对系统解的表征，使我们能够直接在 $\vs$ 空间中利用给定的度量进行稀疏恢复。基于 $l_2$ 度量，我们提出了一种正交匹配追踪（OMP）型算法，该算法在恢复精度上显著优于经典 OMP 算法，同时保持了相当的计算复杂度。还推导了一种基于 $l_1$ 的算法，记为 $\text{Alg}_{GL1}$。该算法显著优于经典基追踪（BP）算法。结合用于选择原子的 CoSaMP 策略，我们进一步推导了一类高性能贪婪算法。通过在合成数据和图像数据上进行的大量数值模拟，结果表明我们提出的算法在稀疏恢复精度以及针对系统矩阵 $\mQ$ 数值不稳定性和测量值 $\vx$ 扰动的鲁棒性方面均具有优越性能。