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 ℓ 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 ℓ 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.
In REF H. Rauhut mainly focused on sensing and recovering a time domain sparse signal by using arbitrary subset of rows of a random circulant or Toeplitz matrix.
Circulant and Toeplitz matrices in compressed sensing
ArXiv
ArXiv
Mathematics Computer Science