Cal-AMP : Blind Sensor Calibration using Approximate Message Passing

From the paper: 

One issue that can arise in CS is a lack of knowledge or an uncertainty on the exact measurement process. A known example is dictionary learning, where the measurement matrix F is not known. The dictionary learning problem can also be solved with an AMP-based algorithm if the number P of available signal samples grows as N [13].
A different kind of uncertainty is when the linear transformation F, corresponding to the mixing process, is known, but the sensing process is only known up to a set of parameters. In that case, it is necessary to estimate these parameters accurately in order for exact signal reconstruction to be possible. In some cases, it might be possible to estimate these parameters prior to the measurements in a supervised sensor calibration process, during which one measures the outputs produced by known training signals, and in this way estimate the parameters for each of the sensors. In other cases, this might not be possible or practical - think in particular of a setting in which these parameters can change over time, which would make a calibration step necessary prior to each measure. This is known as the blind sensor calibration problem, as the input signals are not known, and is it schematically shown on Fig. 2.

What you see is unsupervised learning of the transfer function between signals with particular properties and recorded data. If you would not know better, you'd think we were talking about a machine learning task. Let us note the use of sharp phase transitions to evaluate the capability of the algorithm to do the work. This is a line of investigation that ought to be looked into from the Machine Learning side if you ask me ( see Sunday Morning Insight: Sharp Phase Transitions in Machine Learning ? , Sunday Morning Insight: Randomization is not a dirty word) but then nobody asks me :-)

Blind Sensor Calibration using Approximate Message Passing by Christophe Schülke, Francesco Caltagirone, Lenka Zdeborová

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.

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