Greed is good: Algorithmic results for sparse approximation
Greed is good: Algorithmic results for sparse approximation
复制标题
DOI：
10.1109/tit.2004.834793
复制
发表时间：
2004-10-01
影响因子：
2.5
通讯作者：
Tropp, JA
中科院分区：
文献类型：
--
作者：
Tropp, JA
本文介绍了使用贪婪算法、正交匹配追踪 (OMP) 来解决冗余字典上的稀疏逼近问题的新结果。它提供了 OMP 和 Donoho 的基追踪 (BP) 范式能够恢复精确稀疏信号的最佳表示的充分条件。它利用这一理论来证明 OMP 和 BP 对于来自各种字典的每个稀疏输入信号都是成功的。这些准不连贯词典提供了不连贯词典的自然概括，并且引入了累积连贯函数来量化不连贯程度。该分析统一了 BP 上的所有最新结果，并将其扩展到 OMP。此外，本文还提出了 OMP 可以从非稀疏信号的最佳近似中识别原子的充分条件。从这里开始，它认为 OMP 是一种针对准不相干字典上的稀疏问题的近似算法。也就是说，对于每个输入信号，OMP 计算一个稀疏近似值，其误差仅比使用相同项数可获得的最小误差差一小部分。
This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP.Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.