Piecewise Toeplitz Matrices-based Sensing for Rank Minimization

Abstract

This paper proposes a set of piecewise Toeplitz matrices as the linear mapping/sensing operator $\mathcal{A}:{\mathbf{R}}^{n_1\times n_2}\to {\mathbf{R}}^M$ 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 $\mathcal{O}(n_1{n}_{2}M)$ 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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• Structured Compressed Sensing: From Theory to Applications
• Quantum state tomography via compressed sensing
• On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing