Circulant and Toeplitz matrices in compressed sensing

TL;DR

This paper demonstrates that partial random circulant and Toeplitz matrices can reliably recover sparse signals via -minimization, with measurement requirements scaling linearly with sparsity and only logarithmically with ambient dimension.

Contribution

It extends compressed sensing theory to arbitrary subsets of rows of circulant and Toeplitz matrices, showing improved measurement bounds for sparse recovery.

Findings

Measurement bounds scale linearly with sparsity.

Recovery guarantees hold for arbitrary row subsets.

Uses a new non-commutative Khintchine inequality.

Abstract

Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by $\ell_1$-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by $\ell_1$-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a $\log$-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality.