Confined Orthogonal Matching Pursuit for Sparse Random Combinatorial Matrices

Orthogonal matching pursuit~(OMP) is a commonly used greedy algorithm for recovering sparse signals from compressed measurements. In this paper, we introduce a variant of the OMP algorithm to reduce the complexity of reconstructing a class of $K$-sparse signals $\boldsymbol{x} \in \mathbb{R}^{n}$ from measurements $\boldsymbol{y} = \boldsymbol{A}\boldsymbol{x}$. In particular, $\boldsymbol{A} \in \{0,1\}^{m \times n}$ is a sparse random combinatorial matrix with independent columns, where each column is chosen uniformly among the vectors with exactly $d~(d \leq m/2)$ ones. The proposed algorithm, referred to as the confined OMP algorithm, leverages the properties of the sparse signal $\boldsymbol{x}$ and the measurement matrix $\boldsymbol{A}$ to reduce redundancy in $\boldsymbol{A}$, thereby requiring fewer column indices to be identified. To this end, we first define a confined set $\Gamma$ with $|\Gamma| \leq n$ and then prove that the support of $\boldsymbol{x}$ is a subset of $\Gamma$ with probability 1 if the distributions of nonzero components of $\boldsymbol{x}$ satisfy a certain condition. During the process of the confined OMP algorithm, the possibly chosen column indices are strictly confined to the confined set $\Gamma$. We further develop the lower bound on the probability of exact recovery of $\boldsymbol{x}$ using the confined OMP algorithm. Furthermore, the obtained theoretical results can be used to optimize the column degree $d$ of $\boldsymbol{A}$. Finally, experimental results show that the confined OMP algorithm is more efficient in reconstructing a class of sparse signals compared to the OMP algorithm.