Blind Calibration in Compressed Sensing using Message Passing Algorithms

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Blind Calibration in Compressed Sensing using Message Passing Algorithms by Christophe Schülke, Francesco Caltagirone, Florent Krzakala, Lenka Zdeborová

Compressed sensing (CS) is a concept that allows to acquire compressible signals with a small number of measurements. As such it is very attractive for hardware implementations. Therefore, correct calibration of the hardware is a central is- sue. In this paper we study the so-called blind calibration, i.e. when the training signals that are available to perform the calibration are sparse but unknown. We extend the approximate message passing (AMP) algorithm used in CS to the case of blind calibration. In the calibration-AMP, both the gains on the sensors and the elements of the signals are treated as unknowns. Our algorithm is also applicable to settings in which the sensors distort the measurements in other ways than multiplication by a gain, unlike previously suggested blind calibration algorithms based on convex relaxations. We study numerically the phase diagram of the blind calibration problem, and show that even in cases where convex relaxation is possible, our algorithm requires a smaller number of measurements and/or signals in order to perform well.

One notes the potential use of this calibration framework to deal with a much larger set of cases (other than multiplicative noise). In particular, if you recall the recent point of view given in the Sunday Morning Insight on a Quick Panorama of Sensing from Direct Imaging to Machine Learning, there does not seem to be a problem to extend the current function h to those found in nonlinear compressive sensing, in particular quantization or even the successful  autoencoders as found in deep learning. 

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