On the Fundamental Limit of Multipath Matching Pursuit

Multipath matching pursuit (MMP) is a recent extension of the orthogonal matching pursuit algorithm that recovers sparse signals with a tree-searching strategy. In this paper, we present a new analysis for the MMP algorithm using the restricted isometry property. Our result shows that if the sampling matrix $\mathbf {A} \in \mathbb {R}^{m \times n}$ satisfies the RIP of order $K + L$ with isometry constant $ \delta _{K+L}<\sqrt{\frac{L}{{K} + {L}}} $ then the MMP accurately recovers any $K$-sparse signal $\mathbf {x} \in \mathbf {R}^n$ from the samples $\mathbf {y}=\mathbf {A}\mathbf {x}$, where $L$ is the number of child paths for each candidate of the algorithm. Moreover, through a counterexample, we show that the proposed bound is optimal in that the MMP algorithm may fail under $\delta _{K+L} = \sqrt{\frac{L}{{K} + {L}}}.$