Blind Calibration for Sparse Regression: A State Evolution Analysis

Sparse regression, such as the one used in compressed sensing, allows to acquire compressible signals with a small number of measurements. As such, a correct calibration of a potential hardware problem is a central issue. Blind calibration, that is performing at the same time calibration and compressed sensing when the training signals are sparse but unknown, is thus particularly appealing. A potential approach was suggested by Schülke et al, using an approximate message passing (AMP) for blind calibration (cal-AMP). Here, we show that the asymptotic performances of this algorithm can be analysed by an exact state evolution equation. It allows to confirm that cal-AMP requires a smaller number of measurements and/or signals in order to perform with respect to standard convex approaches, and opens the way to more complex message passing techniques.