Performance Limits for Noisy Multi-Measurement Vector Problems

Here is some analysis (and attendant phase diagrams) for BP and AMP solvers in the MMV and SMV (CS) cases. As opposed to previous work featuring phase diagrams, this analysis goes into the noisy setting. It's been added to the Advanced Matrix Factorization Jungle page.  

Performance Limits for Noisy Multi-Measurement Vector Problems by Junan Zhu, Dror Baron, Florent Krzakala

Compressed sensing (CS) demonstrates that sparse signals can be estimated from under-determined linear systems. Distributed CS (DCS) is an area of CS that further reduces the number of measurements by considering joint sparsity within signal ensembles. DCS with jointly sparse signals has applications in multi-sensor acoustic sensing, magnetic resonance imaging with multiple coils, remote sensing, and array signal processing. Multi-measurement vector (MMV) problems consider the estimation of jointly sparse signals under the DCS framework. Two related MMV settings are studied. In the first setting, each signal vector is measured by a different independent and identically distributed (i.i.d.) measurement matrix, while in the second setting, all signal vectors are measured by the same i.i.d. matrix. Replica analysis is performed for these two MMV settings, and the minimum mean squared error (MMSE), which turns out to be identical for both settings, is obtained as a function of the noise variance and number of measurements. Multiple performance regions for MMV are identified where the MMSE behaves differently as a function of the noise variance and the number of measurements. Belief propagation (BP) is a CS signal estimation framework that often achieves the MMSE asymptotically. A phase transition for BP is identified. This phase transition, verified by numerical results, separates the regions where BP achieves the MMSE and where it is suboptimal. Numerical results also illustrate that more signal vectors in the jointly sparse signal ensemble lead to a better phase transition. To showcase the application of MMV models, the MMSE of complex CS problems with both real and complex measurement matrices is analyzed.

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