Analysis of MMSE Estimation for Compressive Sensing of Block Sparse Signals

TL;DR

This paper derives a closed-form MMSE estimation formula for block sparse signals in compressive sensing, accounting for dependent clusters, and validates it with numerical simulations.

Contribution

It introduces a replica method-based closed-form solution for MMSE in block sparse compressive sensing, considering dependence among non-zero entries.

Findings

MMSE formula resembles Tse-Hanly with scaled parameters

MMSE matches that of a genie-aided estimator with known block locations

Asymptotic results agree well with finite-size simulations

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 parameters that depend 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.