Piecewise Toeplitz Matrices-based Sensing for Rank Minimization

TL;DR

This paper introduces piecewise Toeplitz matrices as a new sensing method for low rank matrix recovery, reducing memory usage while maintaining high reconstruction accuracy.

Contribution

It extends compressed sensing and rank minimization theory by proposing structured sensing operators that balance memory efficiency and measurement requirements.

Findings

Efficient recovery of low rank matrices demonstrated in simulations.

Proposed sensing operator achieves near-random operator performance.

Reduces memory from O(n1n2M) to O(max(n1,n2)M) for storing the sampling operator.

Abstract

This paper proposes a set of piecewise Toeplitz matrices as the linear mapping/sensing operator $\mathcal{A}: \mathbf{R}^{n_1 \times n_2} \rightarrow \mathbf{R}^M$ 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.