This paper proposes an exact recovery analysis of greedy algorithms for non-negative sparse representations. Orthogonal greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS) consist of gradually increasing the solution support and updating the nonzero coefficients in the least squares sense. From a theoretical viewpoint, greedy algorithms have been extensively studied in terms of exact support recovery. In contrast, the exact recovery analysis of their non-negative extensions (NNOMP, NNOLS) remains an open problem. We show that when the mutual coherence $\mu$ is lower than $\frac{1}{2K-1}$, the iterates of NNOMP / NNOLS coincide with those of OMP / OLS, respectively, the latter being known to reach $K$-step exact recovery. Our analysis heavily relies on a sign preservation property satisfied by OMP and OLS. This property is of stand-alone interest and constitutes our second important contribution. Finally, we provide an extended discussion of the main challenges of deriving improved analyses for correlated dictionaries.

本文提出了一种针对非负稀疏表示的贪婪算法的精确恢复分析。正交贪婪算法，例如正交匹配追踪 (OMP) 和正交最小二乘法 (OLS)，包括逐渐增加解支持和更新最小二乘意义上的非零系数。从理论的角度来看，贪婪算法在精确支持恢复方面得到了广泛的研究。相比之下，其非负扩展（NNOMP、NNOLS）的准确恢复分析仍然是一个悬而未决的问题。我们证明，当相互一致性$\mu$ 低于 $\frac{1}{2钾-1}$，NNOMP / NNOLS 的迭代分别与 OMP / OLS 的迭代一致，后者已知达到 $钾$-步精确恢复。我们的分析在很大程度上依赖于 OMP 和 OLS 满足的符号保留属性。该财产具有独立的利益，是我们的第二个重要贡献。最后，我们对相关字典的改进分析的主要挑战进行了扩展讨论。