Conditions for Reconstructing Sparse-Flat Matrix Signals
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摘要:
建立基于凸优化方法重构矩阵信号X=[x1，…，xn]的一组充分条件，X具有列稀疏性和平坦性的结构特征，即每个列向量xj至多具有s个非零分量、同时所有列向量的l1范数具有相同数值。所采用的矩阵范数是‖|X|‖1≡maxj|xj|1。工作分两部分，第一部分分别对无观测误差和有观测误差的情况，针对求解min-‖|·|‖1型凸优化问题重构以上类型矩阵信号的方法，建立保障稳定性和鲁棒性的充分条件；第二部分工作建立随机观测情况下观测空间维数的下界，用以保证信号以高概率被正确重构。所得结果优于将重构向量信号的min-l1方法直接推广到针对矩阵信号的min-l1方法所得到的结果，并给出数值仿真验证。所针对的信号模型出现在具有定常或缓变包络波形的多输入/多输出雷达及合成孔径雷达等新应用领域，本文工作针对这类应用提供一组实用的信号重构条件。
Abstract:In this study, a group of sufficient conditions is established for reconstructing the structured matrix signal X=[x1, …, xn] with the column-wise sparsity and flatness features via convex programming, where each column xj is a vector of s-sparsity and all columns have the same value of l1 norms. The regularizer in use is a matrix norm ‖|X|‖1≡maxj|xj|1. The contributions have two parts: First, sufficient conditions are established for stability and robustness in signal reconstruction by solving the min-‖|·|‖1 convex program from noise-free or noisy measurements. Second, fundamental lower bounds on the dimensions of the measurement space are established for correct reconstruction with high probability. The results are superior to those obtained by directly extending the vector signal's min-l1 approach to the matrix signal's min-l1 approach. Numerical simulations are presented. The investigated signal model appears in some emerging application fields, e.g., multiple-input and multiple-output and synthetic aperture radars with constant or slowly varying envelope waveforms. The work in this paper provides practical conditions to reconstruct such signals.
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Keywords:
- compressive sensing /
- matrix signal /
- convex programming /
- column-wise sparsity /
- l1-column-flatness
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图 1 min-‖|·‖|1成功重构信号的概率与观测维数m的关系
Figure 1. Relationship between measurement dimensions m and success probability of reconstruction via min-‖|· ‖|1
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图 2 min-l1成功重构信号的概率与观测维数m的关系
Figure 2. Relationship between measurement dimensions m and success probability of reconstruction via min-l1
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