## Abstract

Consider a Bernoulli-Gaussian complex n-vector whose components are V _{i} = X_{i} B_{i}, with X_{i} ∼ CN (0, P_{x}) and binary B_{i} mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix bf U, and a randomly chosen subset, of average size np, p ε[0,1], of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically a matrix with iid components, to allow U satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdú, as well as a number of information-theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of n → ∞. We also extend the scope of the large deviation approach of Rangan and characterize the performance of a class of estimators encompassing thresholded linear MMSE and l_{1} relaxation.

Original language | English (US) |
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Article number | 6472315 |

Pages (from-to) | 4243-4271 |

Number of pages | 29 |

Journal | IEEE Transactions on Information Theory |

Volume | 59 |

Issue number | 7 |

DOIs | |

State | Published - Jul 15 2013 |

## All Science Journal Classification (ASJC) codes

- Information Systems
- Computer Science Applications
- Library and Information Sciences

## Keywords

- Compressed sensing
- free probability
- random matrices
- rate-distortion theory
- sparse models
- support recovery