Confined Orthogonal Matching Pursuit for Sparse Random Combinatorial Matrices
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915633675894784 |
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| author | Zhao, Xinwei Wen, Jinming Yang, Hongqi Ma, Xiao |
| author_facet | Zhao, Xinwei Wen, Jinming Yang, Hongqi Ma, Xiao |
| contents | 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 $Γ$ with $|Γ| \leq n$ and then prove that the support of $\boldsymbol{x}$ is a subset of $Γ$ 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 $Γ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Confined Orthogonal Matching Pursuit for Sparse Random Combinatorial Matrices Zhao, Xinwei Wen, Jinming Yang, Hongqi Ma, Xiao Signal Processing 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 $Γ$ with $|Γ| \leq n$ and then prove that the support of $\boldsymbol{x}$ is a subset of $Γ$ 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 $Γ$. 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. |
| title | Confined Orthogonal Matching Pursuit for Sparse Random Combinatorial Matrices |
| topic | Signal Processing |
| url | https://arxiv.org/abs/2501.01008 |