Confined Orthogonal Matching Pursuit for Sparse Random Combinatorial Matrices

TL;DR

This paper introduces a confined OMP algorithm tailored for sparse signals and sparse random combinatorial matrices, reducing complexity and improving efficiency in signal recovery compared to standard OMP.

Contribution

The paper proposes a novel confined OMP algorithm that leverages matrix and signal properties to reduce redundancy and enhance recovery efficiency for sparse signals.

Findings

Confined OMP reduces the number of column indices needed for reconstruction.

Theoretical bounds on exact recovery probability are established.

Experimental results demonstrate improved efficiency over standard OMP.

Abstract

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.