Piecewise Toeplitz matrices-based sensing for rank minimization

This paper proposes a set of piecewise Toeplitz matrices as the linear mapping/sensing operator A : Rⁿ₁×ⁿ₂ → 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 O(n₁n₂M) to O(max(n₁, n₂)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.