  The ubiquity of approximately sparse data has led a variety of com- munities
to great interest in compressed sensing algorithms. Although these are very
successful and well understood for linear measurements with additive noise,
applying them on real data can be problematic if imperfect sensing devices
introduce deviations from this ideal signal ac- quisition process, caused by
sensor decalibration or failure. We propose a message passing algorithm called
calibration approximate message passing (Cal-AMP) that can treat a variety of
such sensor-induced imperfections. In addition to deriving the general form of
the algorithm, we numerically investigate two particular settings. In the
first, a fraction of the sensors is faulty, giving readings unrelated to the
signal. In the second, sensors are decalibrated and each one introduces a
different multiplicative gain to the measures. Cal-AMP shares the scalability
of approximate message passing, allowing to treat big sized instances of these
problems, and ex- perimentally exhibits a phase transition between domains of
success and failure.
