Blind Calibration in Compressed Sensing using Message Passing Algorithms
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 issue. 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.
The "blind calibration" problem involves a 
      compressed sensing problem where, when observing a M-dimensional 
      measurement of N-dimensional sparse signals x_1,x_2,..., each of the M 
      measurement sensors is distorted in some parametrized way, where the 
      parameter can vary over the sensors. Therefore the goal is to be able to 
      recover these distortion parameters d_1,...,d_M at the same time as 
      estimating the sparse signals. This paper proposes an effective 
      message-passing algorithm "C-AMP" to solve the two problems 
      simultaneously. The authors use a simple dimensionality argument to derive 
      a lower bound alpha_min on the sampling ratio alpha=(measurement dimension 
      M)/(signal dimension N), which is needed for recovery to be possible. In 
      the simulation results, C-AMP performs very well, with successful signal 
      recovery nearly everywhere above the threshold max{alpha_CS,alpha_min} 
      where alpha_CS is the phase transition for ordinary compressed sensing 
      (i.e. the setting where there are no unknown calibration parameters). In 
      contrast, L1 minimization (which does not take the calibration issue into 
      account) does not perform nearly as well. I did find the initial 
      presentation of the problem to be somewhat confusing. Because blind 
      calibration is presented after background is given on supervised 
      calibration, it gives the impression that the goal is again the estimation 
      of the distortion parameters d_mu, rather than signal recovery (the sparse 
      x_l's). It would perhaps be clearer to say something like, "Given a sparse 
      signal recovery problem, if we were not able to previously estimate the 
      distortion parameters via supervised calibration, we will need to estimate 
      the unknown signals and the unknown distortion parameters simultaneously - 
      this is known as blind calibration." 
      Other comments... -- 062: 
      z_{\mu l} should be z_{\mu}? -- 063-064: not clear if w is the 
      noise or w is noise+z? I assume that w is the noise Delta; might be 
      clearer to say "one usually considers an iid Gaussian noise with variance 
      Delta, which will then be added to z". Also there is an extra "one" in 
      this sentence. -- 101-102: this is hard to follow. Perhaps specify 
      what is rho. -- 108: "the use if AMP" - > "the use of AMP" 
      -- 132: very nice intuitive derivation of the lower bound 
      -- equation in line 146 is very clear and easy to read; in-line 
      equations in lines 178-180 are harder to read & understand -- 
      149: "Z" is not the best notation for the normalizing constant since "z" 
      is used for something unrelated -- 158-159: "NPM messages" and "PM 
      messages" should maybe be "pairs of messages" since ther are m's and 
      \tilde{m}'s? -- 263-264: This line is a bit awkward to read, 
      perhaps rewrite -- 268-269: not sure what it means to have a 
      uniform distribution with a mean & variance parameter - I assume it 
      means that sigma^2 determines the width of the interval, i.e. the width is 
      chosen such that variance is equal to sigma^2? This is an unusual 
      parametrization and it might be easier to just give the width itself. 
      -- 300: what is the green line in the central figure? -- 
      380: these figures & results are very clear. It might be nice to 
      discuss alpha_min a bit more - give some intuition for how alpha_min is 
      not a phase transition, since alpha_CS sometimes lies above alpha_min in 
      these figures; perhaps the maximum of alpha_min and alpha_CS functions as 
      a phase transition? Or is there some single bound that might unite the 
      two? -- I think it's more standard, in phase transition diagrams, 
      to use light regions to indicate successful signal recovery and dark 
      regions when the signal cannot be recovered.