### Abstract

Minimum mean square error (MMSE) estimation of block sparse signals from noisy linear measurements is considered. Unlike in the standard compressive sensing setup where the non-zero entries of the signal are independently and uniformly distributed across the vector of interest, the information bearing components appear here in large mutually dependent clusters. Using the replica method from statistical physics, we derive a simple closed-form solution for the MMSE obtained by the optimum estimator. We show that the MMSE is a version of the Tse-Hanly formula with system load and MSE scaled by a parameter that depends on the sparsity pattern of the source. It turns out that this is equal to the MSE obtained by a genie-aided MMSE estimator which is informed in advance about the exact locations of the non-zero blocks. The asymptotic results obtained by the non-rigorous replica method are found to have an excellent agreement with finite sized numerical simulations.

Original language | English |
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Title of host publication | 2011 IEEE Information Theory Workshop |

Pages | 553-557 |

ISBN (Electronic) | 978-1-4577-0437-6, 978-1-4577-0436-9 |

DOIs | |

Publication status | Published - 21 Dec 2011 |

MoE publication type | A4 Article in a conference publication |

Event | 2011 IEEE Information Theory Workshop, ITW 2011 - Paraty, Brazil Duration: 16 Oct 2011 → 20 Oct 2011 |

### Conference

Conference | 2011 IEEE Information Theory Workshop, ITW 2011 |
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Country | Brazil |

City | Paraty |

Period | 16/10/11 → 20/10/11 |

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## Cite this

Vehkaperä, M., Chatterjee, S., & Skoglund, M. (2011). Analysis of MMSE estimation for compressive sensing of block sparse signals. In

*2011 IEEE Information Theory Workshop*(pp. 553-557). [6089563] https://doi.org/10.1109/ITW.2011.6089563