Symmetric Toeplitz-Structured Compressed Sensing Matrices

TL;DR

This paper demonstrates that symmetric Toeplitz-structured matrices can serve as effective measurement matrices in compressed sensing, requiring fewer random entries and enabling more efficient signal recovery compared to traditional random matrices.

Contribution

It proves that symmetric Toeplitz matrices and their transforms are suitable as measurement matrices with high probability, reducing randomness needs and improving recovery efficiency.

Findings

Symmetric Toeplitz matrices can be used as measurement matrices in compressed sensing.

Fewer independent entries are needed compared to Gaussian and Bernoulli matrices.

Signal recovery is more efficient with these structured matrices.

Abstract

How to construct a suitable measurement matrix is still an open question in compressed sensing. A significant part of the recent work is that the measurement matrices are not completely random on the entries but exhibit considerable structure. In this paper, we proved that the symmetric Toeplitz matrix and its transforms can be used as measurement matrix and recovery signal with high probability. Compared with random matrices (e.g. Gaussian and Bernullio matrices) and some structured matrices (e.g. Toeplitz and circulant matrices), we need to generate fewer independent entries to obtain the measurement matrix while the effectiveness of recovery does not get worse. Furthermore, the signal can be recovered more efficiently by the algorithm.