Consider the compressed sensing setup where the support $s^*$ of an $m$-sparse $d$-dimensional signal $x$ is to be recovered from $n$ linear measurements with a given algorithm. Suppose that the measurements are such that the algorithm does not guarantee perfect support recovery and that true features may be missed. Can they efficiently be retrieved? This paper addresses this question through a simple error-correction module referred to as LiRE. LiRE takes as input an estimate $s_{in}$ of the true support $s^*$, and outputs a refined support estimate $s_{out}$. In the noiseless measurement setup, sufficient conditions are established under which LiRE is guaranteed to recover the entire support, that is $s_{out}$ contains $s^*$. These conditions imply, for instance, that in the high-dimensional regime LiRE can correct a sublinear in $m$ number of errors made by Orthogonal Matching Pursuit (OMP). The computational complexity of LiRE is $O(mnd)$. Experimental results with random Gaussian design matrices show that LiRE substantially reduces the number of measurements needed for perfect support recovery via Compressive Sampling Matching Pursuit, Basis Pursuit (BP), and OMP. Interestingly, adding LiRE to OMP yields a support recovery procedure that is more accurate and significantly faster than BP. This observation carries over in the noisy measurement setup. Finally, as a standalone support recovery algorithm with a random initialization, experiments show that LiRE's reconstruction performance lies between OMP and BP. These results suggest that LiRE may be used generically, on top of any suboptimal baseline support recovery algorithm, to improve support recovery or to operate with a smaller number of measurements, at the cost of a relatively small computational overhead. Alternatively, LiRE may be used as a standalone support recovery algorithm that is competitive with respect to OMP.
翻译:考虑压缩感应设置 压缩感应设置 : 美元 美元 美元, 美元, 美元, 美元, 美元 。 在无噪音的测量设置中, 建立了足够的条件, 保证利 ree 恢复全部支持, 即 $, 美元 包含 $ 美元 。 这些条件意味着, 例如, 在高度的 BRE 系统中, 利 re 可以纠正以美元计算的子线性错误数 。 利RE 将真实支持 $, 美元 和 产出 精确的支持 $, 美元 美元 。 在无噪音的测量设置中, 建立了足够的条件, 利RE 保证恢复全部支持, 也就是 $ 美元 。 利RE, 利 real deal disalation, 在 Orational recocococouptection 中, 使用 IM AS, 以 IMBRILI 的恢复, 以 IMRI 的精确性支持 。