基于分段Toeplitz矩阵的秩最小化检测
Piecewise Toeplitz Matrices-based Sensing for Rank Minimization
摘要：
本文提出了一组分段Toeplitz矩阵作为线性映射/感知算子$\mathcal{A}: \mathbf{R}^{n_1 \times n_2} \rightarrow \mathbf{R}^M$，用于从少量测量中恢复低秩矩阵。我们证明了这样的运营商有效地编码的信息，所以存在一个唯一的重建矩阵在温和的假设。这项工作提供了一个显着的扩展的压缩感知和秩最小化理论，它实现了一个折衷之间减少存储所需的存储器从$\mathcal{O}(n_1n_2M)$到$\mathcal{O}(\max(n_1,n_2)M)$的采样算子，但在增加$r$的测量的数量为代价。仿真结果表明，该算子能有效地恢复低秩矩阵，重建性能接近于使用随机非结构化算子的情况。
英文摘要：
This paper proposes a set of piecewise Toeplitz matrices as the linear mapping/sensing operator $\mathcal{A}: \mathbf{R}^{n_1 \times n_2} \rightarrow \mathbf{R}^M$ for recovering low rank matrices from few measurements. We prove that such operators efficiently encode the information so there exists a unique reconstruction matrix under mild assumptions. This work provides a significant extension of the compressed sensing and rank minimization theory, and it achieves a tradeoff between reducing the memory required for storing the sampling operator from $\mathcal{O}(n_1n_2M)$ to $\mathcal{O}(\max(n_1,n_2)M)$ but at the expense of increasing the number of measurements by $r$. Simulation results show that the proposed operator can recover low rank matrices efficiently with a reconstruction performance close to the cases of using random unstructured operators.
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