Blind calibration for compressed sensing: state evolution and an online algorithm

Abstract

Compressed sensing, allows to acquire compressible signals with a small number of measurements. In applications, a hardware implementation often requires a calibration as the sensing process is not perfectly known. Blind calibration, that is performing at the same time calibration and compressed sensing is thus particularly appealing. A potential approach was suggested by Schülke and collaborators in [1, 2], using approximate message passing (AMP) for blind calibration (cal-AMP). Here, the algorithm is extended from the already proposed offline case to the online case, where the calibration is refined step by step as new measured samples are received. Furthermore, we show that the performance of both the offline and the online algorithms can be theoretically studied via the State Evolution (SE) formalism. Through numerical simulations, the efficiency of cal-AMP and the consistency of the theoretical predictions are confirmed.

Mean-field inference methods for neural networks

“…We here present the derivation of the multi-value AMP and its SE motivated in Section 5.3, focusing on the multi-value GLM. These derivations also appear in [GBKZ19].…”

“…In [AMB + 18], the teacherstudent matched setting of the committee machine is examined through the replica approach and the Bayes optimal State Evolution equations are obtained as the saddle point equations of the replica free energy. In Appendix D we present the alternative derivation of the State Evolution equations from the message passing and without assuming a priori matched teacher and student, as done in [GBKZ19].…”