The performance of orthogonal multi-matching pursuit under RIP
This work addresses signal recovery in compressed sensing, offering incremental improvements in computational efficiency for sparse signal reconstruction algorithms.
The paper tackles the problem of recovering sparse signals using orthogonal multi-matching pursuit (OMMP) under restricted isometry property (RIP) conditions, showing that OMMP can recover s-sparse signals within s iterations for certain RIP parameters and within O(s/M) iterations for slowly-decaying signals, with computational complexity reduced by factors like O(s^{1-a}) for M = s^a.
The orthogonal multi-matching pursuit (OMMP) is a natural extension of orthogonal matching pursuit (OMP). We denote the OMMP with the parameter $M$ as OMMP(M) where $M\geq 1$ is an integer. The main difference between OMP and OMMP(M) is that OMMP(M) selects $M$ atoms per iteration, while OMP only adds one atom to the optimal atom set. In this paper, we study the performance of orthogonal multi-matching pursuit (OMMP) under RIP. In particular, we show that, when the measurement matrix A satisfies $(9s, 1/10)$-RIP, there exists an absolutely constant $M_0\leq 8$ so that OMMP(M_0) can recover $s$-sparse signal within $s$ iterations. We furthermore prove that, for slowly-decaying $s$-sparse signal, OMMP(M) can recover s-sparse signal within $O(\frac{s}{M})$ iterations for a large class of $M$. In particular, for $M=s^a$ with $a\in [0,1/2]$, OMMP(M) can recover slowly-decaying $s$-sparse signal within $O(s^{1-a})$ iterations. The result implies that OMMP can reduce the computational complexity heavily.